a-simple-universe-math.tex

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\title{A Simple Universe Argument}
\author{Virgil Șerbănuță\thanks{\href{mailto:design-and-chance@poarta.org}{design-and-chance@poarta.org}}}
%\date{June 2015}
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\def\infordinala{\omega}
\def\infordinalb{\omega_1}
\def\reale{\mathbb{R}}
\def\intregi{\mathbb{Z}}
\def\complexe{\mathbb{C}}
\def\naturale{\mathbb{N}}
\def\rationale{\mathbb{Q}}
\def\descriptions{D_L}
\def\designer{\mathbb{D}}
\def\our_description{OURD}
\newcommand{\paper}[1]{paper}
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\newcommand{\definitie}[1]{\textbf{#1}}
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\begin{document}

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\maketitle
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 \begin{abstract}
  This \paper{} argues that if our universe is not designed, it should
  have a high level of a certain kind of complexity that should be visible
  everywhere. Given that we don't observe this, our universe is most probably designed.
  Moreover, it's probably designed for intelligent beings.
  \svn{This is valid regardless
  of how many universes exist, whether it is one, or all logically
  possbile ones, or anything in between.}
\end{abstract}

\section{Introduction}

There is a common belief
that there are certain laws that explain our universe's behavior fairly well,
that we can compute its age,
and that we can make predictions about the distant future, and so on.
In a way, all of these are really surprising.
In the words of \textcite{Feynman2009}:
\begin{quote}
Incidentally, the fact that there are rules at all to be checked
is a kind of a miracle; that it is possible to find a rule,
like the inverse square law of gravitation, is some sort of miracle.
It is not understood at all, but it leads to the possibility of
prediction --- that means it tells you what you would expect in
an experiment you have not yet done.
\end{quote}

This \paper{} attempts to figure out statistically
why we observe such a surprising universe and
what we can reasonably believe about it.

The argument can be described as a statistical approach to Aquinas' fifth way.
It tries to avoid some of the issues encountered by other statistical approaches
attempting to show that our universe is designed
(e.g. the fine-tuning argument).

To be specific, it argues that, in the context of all the possible universes
that could contain human-like beings, our universe is an extremely unlikely
case unless
a Designer intended it, and furthermore, it's probable that the
Designer wanted to design for human-like beings.

In order to show this, we first need to specify what would count as a universe.
This is a difficult question in general but, fortunately, this paper's argument
also works if we take into account only a subset of all possible universes.
In particular, we only need to take into account universes that are able
to sustain intelligent beings
and, assuming that they contain intelligent beings, there should be mathematical
theories (called \ghilimele{descriptions}) that these beings could use to
predict (either fully, or approximately) how their universes behave.
We allow for both deterministic and non-deterministic universes.

% Diff with the edit.
%The section entitled Possible Universes and Their Descriptions introduces these ideas in a more
%rigorous way, together with some of the reasoning behind them.
Section \ref{sec:possible-universes} introduces these ideas in a more
rigorous way, together with some of the reasoning behind them.

% Diff with the edit.
% The section Options for Our Universe discusses the ideas central to this paper.
Section \ref{sec:options} discusses the ideas central to this paper.

% Diff with the edit.
%Merely defining the ideas presented in the section Possible Universes and Their Descriptions
%is not enough;
%we also need to make some assumptions about our universe and about
%the link between mathematical theories and reality.
Merely defining the notions presented in section \ref{sec:possible-universes}
is not enough;
we also need to make some assumptions about our universe and about
the link between mathematical theories and reality.

First, we specify what it means to observe events that have a probability of $0$,
then we specify what happens when there is no way to assign probabilities to
the events that we observe.
Next, we assume that our universe can be modeled by a finite-dimensional
space based on real numbers or by something close to that.
We also assume that it is possible to predict, approximately, the behavior of the space
around us, assuming it to be satisfactorily isotropic,
i.e., the same physical laws apply everywhere.
We also need to assume that the set of possible universe descriptions
is large enough, i.e., it has the same size as the set of real numbers.

% Diff with the edit.
% The section entitled Axioms contains axioms for all the assumptions mentioned
% above, each containing an argument for how they make sense.
Section \ref{sec:axioms} contains axioms for all the assumptions mentioned
above, each containing an argument for how they make sense.

With these arguments made, we can start the main part of the discussion
which can be stated as follows: that, in the context of all the possible universes that could
have human-like beings, our universe is extremely unlikely to exist
unless a Designer
intended it to, and that the most obvious reason for its existence is
that the Designer created it for human beings.

The main issue is that our universe seems to behave according to relatively
simple laws that do not change with time and space.
It's completely implausible for this to happen by chance so there should be
an explanation for it.
If our universe is not designed, then the only other possible explanation is that
there is something outside of our universe that made it likely for our universe
to exist, i.e., a meta-universe.
However, this paper will argue that a meta-universe that
would explain our universe is also unlikely, so, in the same way,
it would require a meta-meta-universe explaining the meta-universe,
which in turn would need an explanation and leave us with an infinite chain
of meta-universes.
Since this chain would also need an explanation, perhaps we could make a
claim for a meta-infinity universe that explains it all.
But wouldn't this meta-infinity universe also need an explanation?
We could, of course,
continue this process for quite a while,
which makes these meta-universes also somewhat implausible.

This means that if our universe is not designed, then it will not behave
according to simple laws, and, in principle, this behavior should be
noticeable to us.
Yet it isn't evident that our universe behaves outside of simple laws.
Of course, our current means of space-travel allows us to observe only a limited
part of our entire universe, so it's theoretically possible that we live
in a bubble covering
a large amount of time and space in which simple laws apply.

Even if it were the case that ours was a bubble universe,
it's still very unlikely among all possible bubble universes
because our bubble is much larger than it needs to be.
A conservative estimate produces a probability so low for the existence of our universe,
that almost any other explanation is preferable.
As an example, if that bubble includes the planet Mars for at least one second,
then our universe has a probability lower than $1$ to $10$ to the power $3·10^18$.
That power is proportional to the volume of space included in the bubble,
so one can figure out what happens if, say,
our entire galaxy is included in the bubble.

It's harder to estimate the same probabilities when there is a Designer, but
it is likely that the probability for the existence of our universe is much higher than in the
non-design case.

% Diff with the edit.
% The section Valid Options for Our Universe describes this argument in detail.
Section \ref{sec:valid-options} describes this argument in detail.

% Diff with the edit.
% Objections and Clarifications presents some exposition and possible
% answers to various objections to this paper,
% and the final section, Conclusion, draws this paper's argument to a close.
Section \ref{sec:objections} presents some exposition and possible
answers to various objections to this paper,
and the final section, \ref{sec:conclusion}, draws this \paper{}'s argument to a close.

This \paper{} uses some mathematical concepts related to set cardinalities,
probabilities and ordinals which can be found in, for example,
\textcite{sep-set-theory}, \textcite{Cohen1966} and \textcite{Billingsley1995}.
These are summarized in section \ref{sec:background}, together with some
mathematical statements related to them.

\section{The Ordered Universe Argument}
\label{sec:ordered-universe}

The great Catholic theologian Thomas Aquinas,
in his \ghilimele{fifth way}, attempts to show God's existence from
the order of the universe, arguing that almost all bodies, almost always
behave according to simple natural laws.
One can find a good exposition of this argument in
\citetitle{swinburne1968}
\parencite{swinburne1968}, but let us look at a few ideas which are
interesting in the context of this \paper{}.

There are two types of order one may consider for showing God's existence,
the spatial order and the temporal order.
The former is the order that can be seen in, say, a watch:
even if one would not know what a watch is or does,
the way it is built has a certain order among its pieces
and within each piece that makes it highly unlikely to be a natural product.
One could argue that some part of our universe,
either something as large as our galaxy, or as small as a bacteria,
has some similar kind of order that is unexpected and which suggests
the work of a Designer.
For example, an animal, or a plant usually has a lot of components that
fit and work together in a rather complex way,
and people have argued that this makes it look like the work of a Designer.
The latter–temporal order–can be seen in the behavior of things, including the laws of
nature, and it's the one that will be used in this \paper{}.

While it addresses many possible objections to Aquinas' fifth way,
% Diff with the edit
% Swinburne’s argument from design still leaves it vulnerable to criticism
Swinburne’s Argument from Design still leaves it vulnerable to criticism
by virtue of its basis on an analogy between the order of the world and the order
produced by people, which limits its strength.
I think that this \paper{}'s argument, although it builds on the same
foundation, does not need a high probability for our world,
so it’s much less dependent on analogies between people
and a potential Designer
% Diff with the edit
% (see the Design Probability section).
(see Section \ref{sec:design-probability}).
Depending on how one reads this section, it may even work without an analogy.

The following quote from \textcite{swinburne1968}, made when addressing
Hume's objection that the order which can be observed in this universe
is just an accident, makes a nice introduction for the argument
described in this \paper{}:
\begin{quote}
But if we say that it is chance that in 1960 matter is behaving in a
regular way, our claim becomes less and less plausible as we find that in
1961 and 1962 and so on it continues to behave in a regular way. An appeal
to chance to account for order becomes less and less plausible
the greater the order.\footnote{
  % TODO: Cite using the Latex tools.
  Swinburne, R. G. (1968). The Argument from Design. Philosophy, 43(165), page 210.}
\end{quote}

% TODO: Maybe list some of Hume's issues with this argument.

\section{Possible Universes and Their Descriptions}
\label{sec:possible-universes}

If our universe is designed, then it's likely to be the way it is because
its Designer wanted it to have certain properties.
In order to understand why our universe works the way it does,
one would need to understand the intent of its Designer.
While that's interesting in itself, I will not try to pursue it in depth here,
except for outlining a few brief ideas.

For most of the remainder of this \paper{} let us consider the other case
and assume the hypothesis that our universe is not designed,
and try to make a prediction based on it.
How would a non-designed universe look?
Would it be similar to our universe?
Maybe an infinity of universes exist and ours is just one of many,
or maybe our universe is the only one.
Even if ours is the only one, we could easily imagine that it worked
in a different way, for example, maybe some constant like the speed of light would be
different, or maybe gravity would work differently.

There are people who claim that all logically possible universes exist,
either because they think that it simply makes sense or because they want to
give a good account of modality, or for other reasons.
Were it indeed the case that all logically possible worlds exist, making predictions about
non-designed universes would seem, at first sight, rather challenging.
This \paper{} argues, however, that there are certain things that can be said
regardless of how many universes exist (just one, every logically
possible universe, or any number in between).
%The argument in this \paper{} does not assume that more than
%one universe exists, but it should work either way.

A possible universe
could have exactly the same fundamental laws as ours but with matter
organized differently.
It could have similar laws but with different universal constants.
It could have different fundamental particles (or fields, or whatever the basic
building blocks of our universe are, assuming that there are any).
Or it could be completely different, i.e., different in all possible ways.

It could be that our logic and reasoning are universal instruments,
but it could also be that some of these possible universes are
beyond what our reasoning can grasp and others have properties
for which our logic is flawed.
Even if that's the case, let us see if we can say anything about
the possible universes that we could understand and model in some way.

In the following, the \definitie{possible universes} term will denote
only the logically possible universes that we could model (with a few
more constraints that will be added below). However, the word
\ghilimele{possible} is ambiguous so when the distinction between logical
possibility and actual possibility is important, the
term \definitie{conceivable universes} may be used instead.

This concept of a model is not precise enough.
Let us further restrict the possible universes term
to possible universes that could be modeled mathematically–even
if that leaves out some of them.
This may seem too restrictive,
especially since this paper only needs universes that can be approximated
by mathematical models.
We are going to relax this when discussing approximations, but, for now,
let's consider only universes
which are modelable with sets of axioms that are at most countable.\footnote{
  A set of axioms is countable if it is infinite,
  but the axioms can be counted, i.e.,
  one can assign distinct natural numbers to them.
  A set is at most countable if it is finite or countable.
}

Let us restrict the universes we are considering even further to universes that
have something remotely resembling time and space and for which
the statement, \ghilimele{the state of the universe at a given moment in time}, or something
close, makes sense; and which can plausibly contain intelligent beings.
% And let us restrict again, to universes where there is no action at a
% distance.
Any such universe is, for the purpose of this paper, a conceivable universe.

To keep the exposition simple, I will use
\ghilimele{the state of the universe at a given moment in time},
but one can replace it with one's preferred alternative concept, e.g.,
\ghilimele{the state of a hyperplane whose points only have spacelike
intervals between them}.

Let us define a \definitie{universe description} as a
consistent mathematical theory that has
a set of axioms which is at most countable and which allows making
predictions about the future state of the universe–given its state
at a certain moment in time. A \definitie{universe region description}
is something similar, but limited to a given space-time region of a universe,
with extra axioms to take into account interactions with the rest of the
universe (e.g., by using the state of the region boundary when predicting).
In the best case, for a deterministic universe, there might exist
a description which allows one to correctly predict the entire future
given the state of the universe at some moment in time,
but a universe description
as defined here does not have to predict everything and,
even when it predicts something, it does not always have to be correct.

\ghilimele{Predicting}, as used above, would normally mean that one
starts from the theory and makes some formal inferences and computations, producing
the prediction as the result.
% Diff with the edit
%However, as Calude et al.(2013) shows, there are many
However, as \cite{Calude2013} shows, there are many
things that can't be proven this way and
we don't know if the state of the universe at a given moment in time is one of
those things, although we could restrict our descriptions to cases where this
is possible, at least up to a reasonable level.
Regardless, let us use a different meaning: a theory predicts something if
that something is true in all mathematical models of that theory (i.e., all
sets in which the theory's axioms are satisfied).

Note that usually the data available for making predictions is dependent
on who is making the prediction. For example, if we assume that
all predictions are about things that can be perceived, directly or indirectly,
then
each kind of intelligent being (e.g., humans) will make predictions
about the universe as perceived through their senses. If a universe contains
multiple kinds of intelligent beings, with different kinds of
sense organs, then that universe may have descriptions that are
very different.
Of course, things that are not observable directly can sometimes be mapped
to things that are observable, but this may not always be true.

In order to handle this dependence on who observes a universe
in a reasonable way, in the remainder of this paper we will work with universes
that contain intelligent beings,
and all predictions will be relative to what these intelligent beings
could observe.
If there are multiple kinds of intelligent beings in the
universe, whenever we are talking about its description
we will assume that we picked one such kind,
i.e., although we
will talk about universes and their descriptions, we'll actually mean
universes as perceived by their intelligent beings, and descriptions corresponding to those perceptions.

Next, let us try to specify how good a universe description
should be. First, predictions speak about the future, but expecting to
predict everything until the end of the universe (if any) may not be
reasonable. We may want to fix an amount of time $\Delta t$,
focusing on predictions about things that are at most
$\Delta t$ in the future. Second, we shouldn't expect to
be able to describe everything with full precision, so we may want to
have a precision $\eta>0$ for all the values that are predicted.
Third, we shouldn't expect predictions to always be correct, so
we should require that they are true with probability $p>0$.
The exact meaning of \ghilimele{true with probability} here is left open,
except that we require $p$ to be the same for each location where
predictions are being made.
Of course, we may add other similar constraints if needed.

As an example, we could demand that, of all predictions we can make
at a given space-time location, a fraction of $p$ must turn out to be true.
If we are making
statistical predictions, then the observed outcomes would be consistent
with a fraction $p$ of all sentences representing statistical predictions
being correct.

Then let us say that an \definitie{approximate universe description} with a
\definitie{level of approximation} $L=(\eta>0,$ $p>0$ and $\Delta t>0)$
is a universe description which allows approximating the future
state of the universe with a precision $\eta$, with a probability
$p>0$ for a prediction to be correct
and for a limited amount of time $\Delta t$.

There is a distinction that we should make.
When predicting weather, for example, we can't make long-term precise predictions
because weather is chaotic and, more specifically, a small difference
in the start state can create large differences over time.
This would happen–even if the universe was deterministic
and we knew the laws of the universe perfectly–as long as we didn't know
the full current state of the universe.
However, high precision predictions may be possible for a deterministic
universe if the full state
is taken into account and, as mentioned, we assume that we know that full
state when making predictions.
% TODO: Distinction with what?

For a given universe or region of a universe,
at an agreed level of approximation, we will pick a canonical description
in the following way: Let $S$ be the set of descriptions that account for
the universe with the given level of approximation. If $S$ contains
at least one finite description, then, for this paper,
we could pick any finite description from $S$ as
\ghilimele{the canonical description}.
However, in order to make this more precise,
let's just pick the description with the minimum length
that comes first in the dictionary.
Otherwise, we simply say that the
universe (region) has an infinite description, and we will abuse the
terminology a bit by picking the entire set
$S$ as the canonical description (we could
also pick a random description from the set).
If the level of approximation is obvious from the context, we will call
this canonical description \definitie{the universe's description}
or \definitie{the universe region's description}.

One could also use a well-ordering on the set of real numbers to choose the
lowest description as the universe's description, but that would
complicate things without any benefit.

\section{Options for Our Universe}
\label{sec:options}

The remainder of this paper will analyze what we can reasonably believe about
the following issues:
\begin{itemize}
  \item Our universe is designed or not.
  \item Our universe has a finite or infinite description.
  \item There is an uncountable chain of meta-universes, in which
        our universe is the first, and in which any meta-universe
        includes, directly or not, all the previous ones.
  \item The set of possible descriptions for a finite chunk of space-time
        that is also compatible with life has at least the cardinal
        of the set of real numbers, $\reale$, or a smaller one.
\end{itemize}

\section{Axioms}
\label{sec:axioms}

\subsection{Observing Events}
In the main argument we will try to use the fact that the set of laws describing
the behavior of our universe is a member of a large set of universe
descriptions, a set large enough to have the same cardinality as the set of
real numbers.

For sets with the same cardinality as real numbers,
the most natural probability distributions are the continuous
ones, i.e., probabilities for which any element of the set has probability $0$.
Practically, this means that we would lose any bet that we would make on a
single element of the set.

In order to make some sense of this, we will use two terms,
\ghilimele{generic} and \ghilimele{peculiar},
% Diff with the edit
% which are defined precisely in the section Probabilities.
which are defined precisely in section \ref{sec:probabilities}.
Informally,
an object is peculiar if it satisfies a peculiar predicate, and a peculiar
predicate is one that has a zero probability for any continuous probability
distribution.
For example, \ghilimele{has a finite number of digits} is a peculiar
predicate over real numbers, and any real number with a finite number
of digits, like $12.5$, is peculiar. An object or predicate is generic if
it's not peculiar.

Axiom \ref{ax:zeroisgeneric} below specifies this in a more formal way.

However, it might happen that for some of the sets used in this \paper{}
no reasonable probability distribution can be defined.
Axiom \ref{ax:noprobability} specifies how choice works in those cases.

Throughout this paper we will implicitly use only separated probability
measures, i.e., they can measure singletons (single-element sets).

\begin{axiom}
  \label{ax:zeroisgeneric}
  If $P$ is a probability over the set of real numbers\footnote{Note
    that here, and in all the axioms in this paper, it's not required that
    $P$ is a probability over the
    Borel algebra of $\reale$, although people often implicitly
    assume it when talking about probabilities over $\reale$.
  }
  (or a set with the same cardinal)\footnote{
    Readers should
    keep in mind that, in most cases throughout this paper, what is being said
    about the set of real numbers is similarly valid for any set with the same
    cardinality.},
  we observe a real number $x$, and $P(x)=0$, then $x$ is generic.
\end{axiom}

The set of events for which $P(y)$ is
% Diff with the edit
% greater than $0$ is at most countable (see the Probabilities section),
greater than $0$ is at most countable (see section \ref{sec:probabilities}),
and, if we remove them from $\reale$, what remains will still have
the same cardinality.

Let us consider only this later set containing all elements with probability $0$.
If its probability is $0$, then we will not observe any of its elements,
which means that we will not be able to apply the axiom above.
Otherwise, let us note that the probability
of all peculiar events taken together is $0$, so there is no chance of us
observing one.
In other words, within this set, the probability of all generic events is $1$,
so we can be sure that we observed a generic event.

Of course, the (logical) possibility of observing a peculiar event still exists,
but, practically, we will not observe it as long as the set of our
observations is at most countable.

\begin{axiom}\label{ax:noprobability}
  If we observe a specific real number $x$, when we could have
  observed any real number, and there is no probability distribution that could
  describe how $x$ was chosen, then $x$ is generic.
\end{axiom}

Note that this axiom doesn't say that we do not know this probability
distribution; it says that there is no such probability distribution.
Anyone believing that this cannot happen should treat the cases where
this axiom applies as invalid.

Also note that this cannot happen when using subjective probabilities.

If there is nothing that could favor
peculiar numbers over generic ones, it's absurd to think that we could have
observed an element of such a tiny set among something infinitely larger.
Also, the similar axiom for probabilities above suggests that this is the only
reasonable assumption in this case.

\subsection{$\reale^4$ Universe}

The universes we are interested in are universes that can be
modeled on top of a finite-dimensional space based on real numbers,
or something close enough to that.

In order to not define what \ghilimele{close enough} means, we will use
the following axiom, which is true for any finite dimensional space
based on the set of real numbers, but it also works on spaces based on,
say, the set of rational numbers, and on many similar finite dimensional spaces.

Let us define a \definitie{generalized rational number} as being either a
rational number, or one of $-\infty$ and $+\infty$. Let us have
\ghilimele{cuboid} stand for a corner-based shape in the $n$-dimensional space
(we could take it to mean \ghilimele{hyperrectangle}, \ghilimele{hypercube},
or any similar shape).

\begin{axiom}\label{ax:rationalcovering}
  The set of (generalized) cuboids using the same dimensions as our space-time
  and whose corners' coordinates are generalized rational numbers (i.e., they
  belong to $\rationale\cup\multime{-\infty, +\infty}$),
  is countable and covers our universe.
\end{axiom}

Note that any space-time based on real numbers, i.e., included
in $\reale^\alpha$, will be included in the generalized cuboid having
its corners at plus or minus infinity.

The axiom above does not require the space-time to include the cuboids or
their corners.
If, let's say, our universe is based on the set of integers, i.e.,
it would be included in $\intregi^\alpha$,
we could consider it as being included in $\rationale^\alpha$,
where we could check if it's included in one of the cuboids mentioned
above.

The following definition could also be written as an axiom.

\begin{definition}\label{finitecuboid}
  A part of our universe is \definitie{finite} if
  it can be covered with a finite cuboid, i.e., one
  for which all corner coordinates are rational numbers.
\end{definition}

There are a lot of possible definitions for \ghilimele{finite} which are not
covered by this definition.
While this paper could probably be extended to also handle many of these,
most likely there is no point in doing so.
For example, in $\reale^3$ one can define it as
\ghilimele{having a finite volume}, which would
mean that there are finite things that do not fit in finite cuboids.
In order to handle this we could change the way we define a level of approximation
by allowing ourselves to ignore what happens in a small part of the region we model.

\subsection{Logically Possible Universes}

The following axiom states that, for a given level of approximation,
there is a large set of conceivable universes, which in some narrow respects
are similar to ours, but which are, in general, wildly different.
Also, our universe belongs to this set.

\begin{axiom}\label{ax:uncountable}
  For any level of approximation $L$ above a certain minimum level (see below)
  there is a set $\descriptions$
  of universe descriptions such that the following are true:
  \begin{enumerate}
    \item $\descriptions$ has the same cardinality as
          the set of real numbers $\reale$.
    \item For all descriptions $d$ in $\descriptions$
          there is at least one conceivable
          universe $U_d$ which
      \begin{enumerate}
        \item is based on space-time or something similar enough;
        \item can plausibly contain intelligent beings that use mathematics;
        \item for the intelligent beings mentioned above and
              for the level of approximation $L$,
              $d$ is $U_d$'s description.
      \end{enumerate}
    \item A description can be used only for its universe, i.e.,
          if $d$ and $d'$ are descriptions from $\descriptions$,
          then $d$ does not work
          for $U_{d'}$, the universe corresponding to $d'$.
    \item $\descriptions$ contains a description for our universe.
  \end{enumerate}
  The same is true for universe regions, except that
  $\descriptions$ may have a lower
  cardinality.
\end{axiom}

The minimum level for which this is true is left unspecified, but we should
include some common-sense restrictions, e.g., the minimum length we would
need to measure is not below Planck's length.
All levels of approximation used below will be above this
minimum level, even if this is not mentioned explicitly.

To see why this axiom is reasonable, we will first show that our universe's
description could be part of such a set, then we will identify a large set
of descriptions that fulfills all the requirements except for containing
our universe's description. We will then show that, by removing some of the
descriptions in that set and adding our own, we get a set that fulfills all
requirements.

Let us note that
we most likely have an approximate description for the observable part
of our universe
given by classical mechanics, maybe with some additions.
Alternatively, one could use a description based on, say, quantum field theory.

We can then assume that there is a description
(possibly different from the one above) that works for our entire universe,
as we are theoretically able to observe it. In the worst case, the description
would be just a recording of what we would observe (i.e., at a given time, in
a given place, a certain event takes place).

For almost any countable axiom system that still
has an $n$-dimensional real space $\reale^n$ as a base,
one could imagine an alternate universe
which, in the present, is exactly like ours inside (say) the orbit of Mars,
but what is outside of this orbit is described by that axiom set.

Some of these axiom sets would describe laws of nature which are similar enough
to ours to allow us to observe what happens outside of the orbit of Mars,
but different enough that
we would notice (e.g., gravity could work differently, depending on the region
of space in which one travels). Some of them would also allow life
to exist inside the orbit of Mars (this is not a given since, for example, the zone
outside of this orbit may produce large quantities of heat that
would make sure that inside the orbit only hot gas or plasma exists).

So we could then try to take all hypothetical universes with infinite space
or time, and we could split them into an infinite number of regions. If we take
all ways of having sets of laws (descriptions) which are different enough
for these regions,
we get a set of universes whose set of descriptions has the same cardinality
as the set of real numbers and which fulfills all the conditions of the axiom
above, except the last one.

To be more precise, for any approximation level $L$, and any region that
is not trivial for $L$
(the meaning of \ghilimele{trivial} is left open, but, for example,
if we use an approximation level that doesn't distinguish things smaller
than a size $l$, then the region must be significantly larger than $l$),
there are multiple possible descriptions that
are different enough for $L$, i.e., there is no possible region where both
descriptions
would be valid within the level of approximation $L$
% Diff with the edit
% (see the section Not Enough Descriptions).
(see \ref{sec:not-enough-descriptions}).
We can find two descriptions that are not valid for the region of space around
us, which means that they are different enough from any approximate
description that we would use.

By splitting an infinite space-time
into disjointed regions defined by a set of finite rational coordinates
(for example, we could split it into cubes whose edges have length $1$ and whose
corners are integers)
and taking all possible ways of assigning descriptions to these regions,
we get a set of universe descriptions with infinitely countable axioms
which we will denote by $\descriptions$.
Since the set of regions is countable, and we can assign at least two
descriptions to each region, we get a set of assignments with the same
% Diff with the edit
%cardinality as $\reale$ (also see the section Cardinals).
cardinality as $\reale$ (also see section \ref{sec:cardinals}).

We will still get a set with the same cardinality as $\reale$ if we identify a region $A$ that can sustain intelligent life,
whose description is fixed,
and we require that the other regions around it have descriptions that are compatible with life
in region $A$.

Also, let us note that, from the way we constructed it, we can add our
universe's description to this set without breaking any of the requirements.

This means that $\descriptions$, the set of descriptions mentioned in the axiom,
has the same cardinality as $\reale$. See
% Diff with the edit
% Few Universes Exist for another take on this issue.
section \ref{sec:fewuniverses} for another take on this issue.

% Diff with the edit
% Also see Manson (2003) who suggests that something similar
Also see \parencite{Manson2003}, who suggests that something similar
might be happening in our universe.

\subsection{Neighborhood Modeling}

Since we currently cannot observe our entire universe, saying that it has a
high level of complexity is of limited use.
The second part of our argument will try to draw some conclusions from
what we see in the observable part of our universe.
In order to do that we need an axiom saying that, around us, perhaps in the
entire observable part of our universe, there is a relatively simple
set of laws that describe it.

\begin{axiom}\label{ax:finiteneighbourhood}
  There is a large compact space-time region of our universe that
  includes our solar
  system, and there is a level of approximation $L$ such that:
  \begin{enumerate}
    \item Any cuboid included in that region has a finite approximate
          description for the level $L$ (i.e., we can make non-trivial
          approximate predictions in all such cuboids).
    \item A description for one of the cuboids also works for all other
          cuboids with the same size in the given space-time region
          (i.e., the space region is isotropic enough).
  \end{enumerate}
\end{axiom}

Usually we assume, implicitly or explicitly,
that this statement is true, and, moreover, we assume that a cuboid's description works
for the entire universe.
This is especially the case when claiming, for example,
that the universe is around $14$ billion years old, that the sun will,
in some distant future, become a white dwarf; or that standard-candle supernovae
are not illusions.
In order to believe this, one must assume that our universe has a
finite approximate description or, at least, that our
solar system/galaxy/observable part of the universe has such a description.

\section{Valid Options for Our Universe}
\label{sec:valid-options}

This section will try to develop the axioms above so as to establish what
is reasonable to believe about the issues presented in
% Diff with the edit.
% section Options for Our Universe.
section \ref{sec:options}.

We will focus mostly on what happens when our universe is not designed
since in this case it's easier to make predictions about our universe
and to falsify them, but we will also take a look at created universes.

% Perhaps a diff with the edit
\textcite[][The section entitled \ghilimele{Why a world with human bodies is unlikely
if there is no God}]{Swinburne2003} comes quite close to the argument
presented here, but while Swinburne
argues that human bodies are unlikely, I am arguing that, in the context
of all possible universes that could have human-like beings,
our universe is extremely unlikely unless a Designer intended it,
and, what's more, the Designer wanted to design for human-like
beings.
Since the existence of human bodies is not directly related to the subject
of this \paper{}, I won't discuss that section more
than is strictly needed.

% Perhaphs a diff with the edit
% Note that (Swinburne, 2005) says that individual sets of laws
Note that \textcite{Swinburne2003} says that individual sets of laws
have non-zero probability whereas I'm claiming that their probability is $0$.
It seems to me that Swinburne’s paper implicitly assumes that
such a set has a finite number of laws, while I am explicitly removing
that constraint. I contend therefore that both Swinburne and I can be right within our own contexts.

\subsection{Peculiar Descriptions and Meta-universes}
\label{fdaumu}

\subsubsection{Informal argument}

From what we observe around us, it seems that our universe is fairly homogeneous,
having a relatively simple approximate description.
In this section we will try to see how plausible it is that this description
(or any finite description that still approximates what we see around us)
applies to the entire universe, not only the space and time around us.
To do this, we will first note that a finite description is peculiar,
which means that there is no chance of
us observing it both in the case where we use a continuous probability
distribution over universes (axiom \ref{ax:zeroisgeneric}),
and in the case when we can't use any probability
distribution (axiom \ref{ax:noprobability}).

This result should probably be good enough, and if you agree with
% this, please skip to the section Peculiar Descriptions for Universe Regions.
this, please skip to section \ref{sec:peculiarregions}.
Otherwise, the only option left is that the correct probability
distribution used over universe descriptions is discontinuous.
That, in itself, is not enough because if our universe's description is not
one of these discontinuities, then its probability is still $0$, which,
as stated in axiom \ref{ax:zeroisgeneric}, means that it's still generic.

If our universe is not designed, then we don't normally have any
reason for using a discontinuous probability distribution.
For many people this may be enough to
show that any approximate description for our universe's description is
infinite, and it should settle at least the case when this probability
distribution is a subjective one.

However, we will look deeper into this issue.
Let us assume that there is a probability distribution over universe
descriptions, and that this probability distribution has discontinuities,
and that our universe's description is such a discontinuity.

First, we can ask ourselves why there is a probability distribution over
universe descriptions.
A probability distribution means that which universes exist and which do not,
and how many of each type exist, is not completely chaotic,
i.e., there is a minimal amount of order to them.
We will call this order a \ghilimele{meta-universe}, and we will say that
this meta-universe includes our universe, but keep in mind that the usual
intuition is that a meta-universe is something more complex than just
a minimal amount of order.
On the other hand, the most natural explanation for this order
that does not involve design is an actual
meta-universe, which may contain many other universes besides ours.
To condense this entire paragraph into fewer words: the probability distribution over
universe descriptions is determined by something we will call
a \ghilimele{meta-universe}, which includes our universe.

Now we have something that explains this probability distribution, which,
as mentioned above, is not continuous.
We can ask ourselves if this distribution's discontinuities are what
we would normally expect or not.
It's natural to ask whether we have a probability distribution telling us
what we should expect from these discontinuities, and what kind of probability
distribution it would be.

The set of discontinuities for a probability distribution is at most countable
% Diff with the edit
% (see Probabilities) so, if they are described by a
(see section \ref{sec:probabilities}) so, if they are described by a
continuous probability
distribution, or if there is no such probability distribution, we would
expect them to be generic.
Since our universe's description is one of them, then it should be generic.

Since it's not generic, we can try to explain it (or its discontinuity
in the meta-universe)
using a discontinuous probability distribution over discontinuities
and assume that our universe's description is also one of the
discontinuities for this new probability distribution.
Having this probability distribution would mean that there is
a meta-meta-universe that includes the meta-universe and, indirectly,
our universe.

However, by using the same reasoning, having peculiar discontinuities for the
meta-meta-universe probability
distribution is plausible only if there is a meta-meta-meta-universe
with a discontinuous probability distribution.
By repeating this reasoning, we get an infinite chain of meta-universes,
each with a discontinuity for our universe's
description.
Applying the same reasoning as above to this chain, this discontinuity
must be generic,
unless there is something explaining it, i.e., a meta-infinity universe
whose probability distribution has a discontinuity for our universe's
description.

In order to make this chain more precise, we will assign a number corresponding
to the meta-level of each meta-universe.
We will assign $0$ to the meta-universe including ours, $1$ to the
meta-meta-universe including (directly or indirectly) the meta-universe and our
universe, $2$ to the next level, and so on.
Then we can denote by $\omega$ the meta-infinity universe that explains why all
the meta-universes corresponding to finite numbers have a certain discontinuity.

But, if the $\omega$ meta-universe probability distribution has a peculiar
discontinuity, then it also needs an explanation, so, again,
either our universe's
description is generic, or there is a meta-universe with a discontinuous
probability distribution that includes the
$\omega$ meta-universe.
We will call this the $\omega + 1$ meta-universe.

It follows then that we need meta-universes corresponding to $\omega + 2$,
$\omega + 3$, and so on.
So we get another infinite chain of meta-universes that also needs an
explanation which, in turn, should be provided by another meta-universe, including this
second chain. We will call it the $\omega + \omega$ meta-universe
or the $2\omega$ meta-universe.
But this new meta-universe still needs an explanation,
which means that we also need
meta-universes corresponding to $2\omega + 1$, $2\omega + 2$, ..., and,
in the end, corresponding to $3\omega$.
Next, we will need meta-universes
corresponding to $4\omega$, $5\omega$, and so on.
But to explain this (doubly) infinite chain,
we need another meta-universe which we
will call $\omega\omega$, or $\omega^2$.

Continuing, we will need meta-universes up to $\omega^3$, then up to
$\omega^4$, and so on.
In the end, we will need one for the entire
infinite chain, denoted by $\omega^\omega$.
However, we can't stop here, and we will not be able to stop for quite a while.

Each of these meta-universes corresponds to a statistical observation and,
as long as we make a countable number of observations, we either expect to
observe generic discontinuities, or we expect to find yet another meta-universe
above everything that we have heretofore observed.

% Possible diff with the edit.
% This construction corresponds to ordinals, see Cohen (1966), Section 3,
This construction corresponds to ordinals, see \textcite{Cohen1966}, Section 3,
to find out more.
The process will break down at the first uncountable ordinal (let us call it
$\omega_1$), since we would make
uncountable many observations, one for each ordinal between $0$ and $\omega_1$.

So, in order for our universe description to have a chance to be finite,
we have to postulate a fairly improbable and complicated configuration
of meta-universes and probability distributions, which,
in itself, doesn't seem reasonable.
However, even if we postulate it, the probability of our universe description
is, most likely, still $0$.

To support this contention, let's observe that in this entire construction,
the probability of our universe description can be obtained by multiplying the
probabilities given by each meta-universe. However, it is unreasonable
to ask that any of these probabilities is $1$, since $1$ is a peculiar value.

It may be difficult to multiply an uncountable set of numbers, but,
since these numbers are between $0$ and $1$, then whatever value we should
assign to their product
should be less than or equal to the product of any countable subset.

If there would exist a countable subset of probabilities such that all of them were below a certain threshold,
$0.9$, let's say, then multiplying them produces $0$, which, again, would mean
that our universe's description is generic. To have a chance of having
a peculiar description, for any threshold less than $1$, there should be at
most a finite number of probabilities less than it.

Let us take a countable sequence of thresholds, e.g.
$0.9$, $0.99$, $0.999$, \dots, each of which has a finite number of
probabilities less than it.
If we take all of the probabilities which are below at least
one such threshold together, we will get a set that is at most countable.
But note that each number less than $1$ is below at least one such threshold,
which means that the set of
probabilities which are not $1$ is at most countable.
Also, since the entire set of
probabilities is uncountable, we must have an uncountable set of distributions
for which our universe description has probability $1$.

Each meta-universe probability distribution only gives the
probability that a certain description is a discontinuity, i.e., that
the probability of that description should be greater than $0$, but does not say
anything about the probability itself,
so we should expect these probabilities to
be distributed in a random way, perhaps uniformly distributed between $0$ and
$1$.
This means that it's completely implausible to have almost only
probabilities that are $1$.
It's also completely implausible that, when looking at the
probabilities which are not $1$, for any threshold below $1$, only a finite
number are below that threshold.

Postulating that such an improbable and complicated construction exists does
not seem in any way reasonable,
so any description for our universe must be peculiar.

Next section will contain most of this argument, presented in a more formal way.

\subsubsection{Formal argument}

Let's have our universe's description be denoted by
$\our_description$, and the set of descriptions from
axiom \ref{ax:uncountable} by $\descriptions$,
with $\our_description$ belonging to $\descriptions$.
As mentioned above, let us also consider that our universe might be
contained in a meta-universe, which is itself, perhaps, contained in
a meta-meta-universe, and so on, and let us label these as meta-universes
with ordinal numbers, starting with $0$ for the first meta-universe.

If one of these meta-universes does not exist, then, obviously, no meta-universe
that would include it exists. A less obvious conclusion is that if one
meta-universe does not exist, then there is a \ghilimele{least meta-universe}
that does not exist, i.e., a meta-universe that does not exist,
but all meta-universes
included in it (directly or not) do exist\footnote{In any set of ordinals
  one can find the smallest ordinal as ordered by set inclusion.
  Let us start with an ordinal corresponding to one of the
  meta-universes that do not exist.
  This hypothetical meta-universe includes, directly or not,
  all meta-universes (existing or hypothetical)
  corresponding to smaller ordinals.
  Let us consider the set containing the starting ordinal mentioned above,
  together with the ordinals for all the meta-universes
  that should be included in it.
  This set contains the ordinal for at least
  one meta-universe that does not exist,
  but it may contain more than one,
  so let us select all elements of that set corresponding
  to meta-universes which do not exist, producing a new set.
  In this latter set we can find the smallest ordinal, which corresponds
  to the smallest meta-universe that does not exist.
  Since it's the smallest, all universes that would be
  included in it (if any) must exist.
}. This meta-universe is denoted by $\alpha$. Note
that if no meta-universe exists, then $\alpha=0$. If all
meta-universes exist, then let $\alpha$ denote a meta-universe labeled
with a large enough ordinal, e.g., the first uncountable ordinal
(an ordinal or a meta-universe is countable if the set of ordinals or
meta-universes included in it, directly or not, is countable).

Next, let us take a possible meta-universe $MU$ whose ordinal is at most
countable, and is not $0$, and let us
ask ourselves how likely it is to observe that all meta-universes
included therein, directly or not, have a discontinuity for
the same peculiar universe description $U$.

Any such common discontinuity would be generic if $MU$ does not exist
(axiom \ref{ax:noprobability}), or if $MU$'s probability distribution
is continuous (axiom \ref{ax:zeroisgeneric}),
or if the probability distribution is discontinuous
but doesn't have the same peculiar discontinuity $U$
(also axiom \ref{ax:zeroisgeneric}).
The only case when all the universes included in $MU$ have $U$ as a
(peculiar) discontinuity is when $MU$ exists and its probability distribution
also has $U$ as a discontinuity.

We know that the $0$ meta-universe has a discontinuity for
our universe's description, and our universe's description is peculiar, so,
% Possible diff with the edit
% from transfinite induction (Cohen, 1966, section 3),
from transfinite induction (\textcite{Cohen1966}, section 3),
if the first meta-universe,
corresponding to ordinal $0$, has a peculiar discontinuity, then all countable
meta-universes exist, and have the same discontinuity.

\subsection{Partial conclusion}

As shown in the argument, in order to claim that our universe's description
($\our_description$) is peculiar,
we would need to postulate the
existence of an uncountable chain of meta-universes, all of them favoring a
peculiar $\our_description$, which is, by default,
prohibitively unlikely for any of them. Furthermore, almost all of them
should be extremely favorable to $\our_description$, i.e.,
they should assign a probability of $1$ to $\our_description$.

Although we cannot expand the chain of universes when it becomes uncountable,
intuitively the peculiarity problem
still remains: why would it suddenly become reasonable to make
only peculiar observations if we make enough of them? A few, maybe, but all
of them? Normally, when we make
observations with a continuous probability distribution, which is the default;
we expect to observe only generic elements as long as we make a
countable number of observations, and only when the observation set
becomes uncountable do we expect to, perhaps, also observe some peculiar elements.

From now on, I will assume that the possible objection in the preceding
paragraph is unreasonable, which means that, practically speaking,
either $\our_description$ is generic or our universe is designed.

Since we don't know the limits of our universe, and how similar
the entire universe is to the part we can observe, let us consider next
what happens for universe regions.

\subsection{Peculiar Descriptions for Universe Regions}
\label{sec:peculiarregions}

In this section we will try to show that it is implausible to have a large
region of space-time around us whose behavior is described by the same laws
of physics that apply here on Earth.
How implausible it is depends on how many
distinct descriptions a finite region of space can have in a universe
that contains intelligent beings.

We have then the following two cases:
\begin{enumerate}
  \item The set of descriptions for a finite region such that human-like life
      can exist in the universe has the same cardinality as $\reale$.
  \item The same set has a lower cardinality.
\end{enumerate}
Only one of these can be true, but we do not know which one, so we should
normally consider only the worst case. However, the first case is plausible,
and there is a more efficient way of settling it, so it will be treated
separately.

In the first case we can apply the same argument
% Diff with the edit.
% as in the Peculiar Descriptions and Meta-universes section. The main difference would be the way we assign
as in section \ref{fdaumu}. The main difference would be the way we assign
ordinals: the region would correspond to the universe in the previous argument,
our universe would correspond to $0$,
the first meta-universe would correspond
to $1$, and so on. In effect, we would treat the region as a universe
and our universe as the region's meta-universe.

We can then apply the previous argument to the observable universe, for example, or
to the solar system.
In the same way that we showed the entire universe to have
an infinite description, we will find that both the observable universe
and our solar system should have an infinite description.
However, this contradicts axiom \ref{ax:finiteneighbourhood}, so either the
set of descriptions for a finite region must have a lower cardinality, or our
universe is designed.

Let us focus now on what occurs when the set of descriptions mentioned above
has a cardinality lower than $\reale$.

First, let us note that our universe has a generic approximate description,
while the region around us has a finite (which means peculiar) approximate
description (axiom \ref{ax:finiteneighbourhood}).
This means that we can't
extend our region's description to the entire universe, which means that
there is at least one finite region in our universe with a different
description.
Furthermore, if we split our universe into a countable set of finite regions
(cuboids of the same size, for example),
it's not plausible that all of them except a finite set have the same
description.
If that would be the case, then we would be able to
have a finite description for our universe
(i.e., \ghilimele{region $X$ has the second description,
region $Y$ has the third description, ..., and all the other cuboids have the
first description}, where we use the actual description instead of the words
\ghilimele{nth description}), which would imply that our universe's description
is not actually generic, which is a contradiction.

Let $A$ be the set of possible approximate descriptions for
a cuboid with a size of $1\;second \times meter^3$, for example.
Unless our standard of approximation is extremely rough,
$A$ will have multiple elements.
We will assume that we are working with a reasonable approximation level,
for which $A$ has more than one element.
For any possible description $d$ in the set $A$,
let $P(d)$ be the probability of encountering
a cuboid with $d$ as its description in our universe.

If $A$ is finite, we should, by default, pick the uniform probability
distribution on $A$, assigning equal probabilities to all elements of $A$.
However, $A$ can have an infinite cardinal,
so we have to consider more general probabilities.

In any case, since $A$ has multiple elements, and, as mentioned above,
if we split our universe into finite regions
(i.e., cuboids with a size of $1\;second \times meter^3$) then there will be
at least two descriptions that each belong to an infinite number of regions,
and we can't reasonably expect to have one element with probability $1$.
Then let $p_1<1$ be the probability of the description that we use for the
cuboids around us.
The probability of observing $n$
non-overlapping cuboids
with this description without observing any other description is\footnote{
  This assumes independence between the cuboids, but, given that our
  % Diff with the edit
  % universe's description is a generic one–chosen randomly from all possible
  % descriptions, and not by a Creator–this is a reasonable assumption.
  universe's description is a generic one---chosen randomly from all possible
  descriptions, and not by a Creator---this is a reasonable assumption.
}
$p_1^n$.

Even if $p_1$ is very close to $1$, $p_1^n$ converges quickly to $0$.
For example, if $p_1=0.9$, then observing $n=70$ consecutive cuboids with
the same description is enough to make $p_1^n$
(the probability of those $70$ cuboids having the same description)
drop below one in one thousand;
$n=140$ is enough to make it drop below one in one million,
$n=210$ takes it below one in one billion. However, let's consider that
Mars has a volume of more than $10^{20}$ cubic meters, for example, so, for each second
in which it behaves according to the same laws of physics that work on
Earth, it decreases the probability that our universe is not designed by a
very large number\footnote{Using $p_1=0.9$ as before, for each second,
it decreases the
probability that our universe is not designed by more than $1$ to
$10$ to the power $3\times 10^{18}$. That is a $1$ followed by $3\times 10^{18}$ zeros.}.

Then, if we compare the non-design hypothesis
with another non-zero probability one,
almost always the consistency of a relatively small space-time region around us
is enough to make the non-design hypothesis unlikely enough to
disregard\footnote{
  Let $ND$ be the hypothesis that our universe is not designed,
  $D$ be the hypothesis that it is designed, $\our_description$ be our
  universe's description,
  $OUR$ be the description for the region of space around us.
  Note that $P(OUR)\ge P(\our_description)$ since $OUR$ occurs in any
  universe having $\our_description$ as its description, but may also
  occur in other universes.
  % Diff with the edit.
  % We will try to show in Design Probability
  We will try to show in section \ref{sec:design-probability}
  that $\our_description$ has a non-zero probability
  if our universe is designed, which would mean that $\our_description$ has a
  non-zero probability in general, so
  $P(OUR) \ge P(\our_description) > 0$.
  From Bayes' rule, $P(D|OUR) = \frac{P(OUR|D)\cdot P(D)}{P(OUR)} > 0$.
  Similarly,
  $P(ND|OUR) = \frac{P(OUR|ND)\cdot P(ND)}{P(OUR)}
    = \frac{P(ND)\cdot p_1^n}{P(OUR)}$.

  To compare the two we have to compare $P(OUR|D)\cdot P(D)$ with
  $P(ND)\cdot p_1^n$.
}.

Next, let us see what happens if we assume the design hypothesis.

\subsection{Design Probability}
\label{sec:design-probability}

Let us examine the hypothesis that our universe is designed.
In this case, the way our world works would be based
on the designer's intent,
so it can no longer remain obvious that continuous probability distributions
should be the default option for universe descriptions.

On the other hand, it's also not obvious that we have a better option.
Objections to previous similar arguments include that,
from a natural theology point of view, one can't
know what the Designer wanted
(e.g. one can't know that a universe designer would want to create
% Diff with the edit
% a universe having life [Sober, 2009; Narveson, 2005]), so –by default–
a universe having life \parencites{Sober2009}{Narveson2003}), so ---by default---
any argument showing that the probability of our universe is small
if it's not designed would also show that the probability is small
even if it is designed.
To fix this, one would need an independent way to show that the Designer
% Possible diff with the edit.
% wanted the universe to have life (Sober, 2008).
wanted the universe to have life \parencite{Sober2003}.

In order for this paper's argument to work, one would actually need a weaker
requirement, i.e., it would be enough to have a non-zero probability
to the hypothesis that the Designer wanted the universe to have intelligent
life. If we can't provide that weaker requirement, then this argument can't explain
that the universe around us seems to be consistent.
Instead, it would just point out how extremely odd it is to observe it.

It would be possible to argue for an analogy between us and
an intelligent Designer of worlds, saying that we would find it interesting
to create a world with intelligent beings in it, and that there is a chance,
maybe not that high but still greater than zero, that this would also be
the case for the Designer.
Given how unlikely it is that we live in a non-designed universe, this may be
a fairly attractive option.

However, let us try an approach based on the same principles used throughout
this \paper{}.
This argument will have the following structure:
\begin{itemize}
  \item The probability that a universe designed for intelligent beings
        has a finite description is non-zero.
  \item There is a good chance that a Designer would not design two things
        that are too similar.
  \item The set of similarity thresholds relevant to this paper is
        countable.
  \item Given the above, there is a good chance that the set of things that
        are designed (called purposes) is at most countable.
  \item There is a good (non-zero) chance that, for a similarity threshold
        which is high enough,
        any purpose which is \ghilimele{too similar} to the purpose wherein the Designer
        wants intelligent beings for their own sake, also involves intelligent
        beings.
  \item Combining all the above, there is a non-zero chance that a designer
        would want to design a world at least in part for the intelligent
        beings that would live in it.
\end{itemize}

Let us start with what we want to estimate:
how large is the probability of having
a large consistent space-time region when a Designer is involved?

We can separate this probability in two: the probability that a Designer
would want rational beings\footnote{A Designer could want a consistent universe
region with a finite
description without wanting intelligent beings, but this possibility is not
analyzed in this paper.}, denoted by $p_r$, and
the probability that the universe region containing
those intelligent beings is consistent
given that it was designed for them.

I think it's safe to assume that the latter probability is positive,
and, probably, fairly high. See as an example the following quote from
% Possible diff with the edit
% Swinburne (2005), in which he argues that, if God exists,
\textcite{Swinburne2003}, in which he argues that, if God exists,
there is a fairly good chance that humans can understand their universe.
The same argument can be applied to show that if a Designer wanted
rational beings, a large region of the universe around them is understandable
(perhaps the entire universe), i.e., that region probably has a finite
description.

\begin{quote}
  So, in order to have significant freedom and responsibility, humans need
  at any time to be situated in a \ghilimele{space} in which there is a
  region of basic control and perception, and a wider region into which
  we can extend our perception and control by learning which of our
  basic actions and perceptions have which more distant effects and causes
  when we are stationary, and by learning which of our basic actions cause
  movement into which part of the wider region.
  If we are to learn which of our basic actions done where have which
  more distant effects (including which ones move us into which parts
  of the wider region), and which distant events will have which basically
  perceptible effects, the spatial world must be governed by laws of nature.
  For only if there are such regularities will there be recipes for changing
  things and recipes for extending knowledge that creatures can learn and
  utilize.
  So humans need a spatial location in a law governed universe in which to
  exercise their capacities, and so there is an argument from our being thus
  situated to God.\footnote{
    Swinburne, R. G. (2005). The Argument to God From Fine-Tuning Reassessed.
    God and Design: The Teleological Argument and Modern Science,
    Routledge, page 111.
  }
\end{quote}

Next, let us note that the design and non-design cases
have some differences.

One such difference is that, for an intelligent Designer/Creator, what is being
created corresponds, one way or another, to a purpose.
To be consistent with
the approach used in this \paper{}, let's assume that the set of possible
purposes has at most the cardinality of real numbers, $\reale$. Certainly,
the set of purposes that could be described in words, even if we use an infinite
number of words, has this cardinality.

There could be a number of universes with
distinct descriptions for each purpose, but, since intelligence and design
imply choice, we should not expect that each
possible purpose is an actual purpose.
It's reasonable to assume that, for each actual purpose,
there were many similar purposes that were discarded when the actual one
was chosen.

If that is the case, we could ask ourselves two things:
how large is the set of actual
purposes, i.e., is it at most countable,
and does one of the actual purposes involve creating intelligent beings?

I think it's hard to estimate the likelihood of any of these, but, for the
purpose of this \paper{}, it's probably good enough to show that, by default,
we should assign a non-zero probability to both.

First, let us note that, for real numbers, the similarity of two numbers
is most often linked to the distance between them
(usually the similarity is the inverse of the distance).
Also, if we want to pick a subset of real numbers such that no two of them are
closer than a given distance (i.e., no two numbers are too similar),
then that subset is at most countable.

It's not obvious that
similarity between purposes works in the same way as similarity between
real numbers, but I think there is a non-zero chance
that if the Designer does not choose purposes which are too similar,
then the set of chosen purposes is at most countable.
Arguably, that chance is fairly high.
In the following paragraphs we will implicitly focus only on the case in which
the set of non-similar purposes is at most countable.

Given a countable set of purposes, we can (and should) assign by default
a non-zero probability to all of them.

We don't know what purposes would be included in this set.
However, we can cover all the interesting cases by taking all maximal sets of
purposes that are not too similar
(maximal meaning that we cannot add any other purpose to it without breaking
the similarity constraint), and we can ask ourselves whether one of these
purposes would involve creating rational beings.

To do that, let us take a closer look at the similarity threshold.
For any such threshold $t$,
there will be a natural number $n$ such that the similarity threshold is between
$1/n$ (exclusive) and $1/(n+1)$ (inclusive).
Since the set of purposes that were chosen by the Designer using $t$ as a
threshold can also be chosen
using $1/(n+1)$ as a threshold, in the following paragraphs we will use only
thresholds derived from natural numbers.

Since the new set of possible similarity thresholds is countable,
we should assign a non-zero probability to each of them.

There is a possible purpose where the designer wants intelligent beings
for their own sake which I will call
\textit{the main intelligent beings purpose}.
With a similarity threshold
high enough, there is a good chance that any purpose that is \ghilimele{too similar} to \textit{the
main intelligent beings purpose} (i.e., the similarity is above that threshold) will also involve
intelligent beings.

Given any maximal set of purposes that are not too similar, at least one
of them (called $P$) will be \ghilimele{too similar} to the main intelligent
beings purpose.
As mentioned above, if the similarity threshold is high enough, then there is
a good chance that everything in $P$ involves intelligent beings.

So far, we have a non-zero probability that the set of chosen purposes is countable,
a non-zero chance of choosing a purpose that is \ghilimele{too similar} to the main
intelligent beings purpose, a non-zero chance that below a threshold $1/n$
all purposes \ghilimele{too similar} to the intelligent beings one also involve intelligent beings, and a
non-zero chance that the actual threshold that we should use is below $1/n$.
Then we have a non-zero overall chance of having a universe designed for
intelligent beings.

\svn{5. TODO: Use Swinburne's argument about why would God create human-like beings.}

\svn{Second, a good question is whether, in a designed world, intelligent beings
would be able to identify this design by looking at their world.
TODO: citation to whoever talked about this previously.
If the intelligent beings would see many things that they design, they
may be able to identify something as being designed by them.
However, when they observe their world, they only see one thing that may or
may not be designed, so they may not have enough experience to identify designed
worlds.
However, if the Designer's purpose was to create intelligent beings,
i.e. they are not just an accident, then there's some chance that the
things they design are analogous enough with some of the things the Designer
designs that they would be able to figure that out. TODO: rephrase.
}

\svn{
  Did I say this somewhere else?

  If the Designer intends to create intelligent beings, then the Designer will
create a universe for them.
As argued in \cite{Swinburne2004},
it's very likely that this universe is, indeed, consistent on a large scale.
But if there is a non-zero probability of observing
consistency around us, then there is a non-zero probability of observing
a universe like ours.
}

\svn{
  TODO: Why does non-zero consistency imply non-zero probability for our universe?
}

\section{Objections and Clarifications}
\label{sec:objections}

This section includes various possible objections to this argument. Since the
% Diff with the edit.
% fine-tuning argument (summarized by Friederich, 2017) addresses the same problem, and also uses a
fine-tuning argument (summarized by \parencite{sep-fine-tuning}) addresses the same problem, and also uses a
probabilistic argument (though in a completely different way), some of the
objections below are similar to those against fine-tuning, and it may be helpful
to compare the two approaches.

% Diff with the edit
% For the fine-tuning argument see Friederich (2017).
For the fine-tuning argument see \parencite{sep-fine-tuning}.
For objections to the fine-tuning argument that are relevant here, see
% Diff with the edit
% Manson (2003 and 2009), McGrew et al (2001), Narveson (2005), Sober (2009).
\parencites{Manson2003}{Manson2009}{McGrew2001}{Narveson2003}{Sober2009}.
Examples of responses to such objections are to be found in
% Diff with the edit
% Leslie (2005), Swinburne (2005), Monton (2006), Kotzen (2012).
\textcites{Leslie2003}{Swinburne2003}{Monton2006}{Kotzen2012}.

%Manson2009 McGrew2001 - probabilities not applicable to cosmic parameter
%values.
%Manson2009 Sober2009 - observation selection effect
%Manson2009 - multiple universes
%Manson2003 - nonhomogenous spacetime, different fundamental constants
%Sober2009 Narveson2003 - unknown designer's intent, no difference to design

% TODO: Cite \ghilimele{Should we care about fine-tuning} by Jeffery Koperski.

\subsection{Observation Selection Effect and Multiple Universes}

In this \paper{}, we only look at universes that contain intelligent life,
and that restricts the set of possible universe descriptions.
Furthermore, we don't see a universe as it is, instead we see it
through the eyes of the intelligent beings inhabiting it.
It can be argued that, in a non-created universe, beings might be intelligent
only if their intelligence is useful to them.
But this likely means that those beings live in a space-time region which seems
consistent from their point of view, so maybe it's not that unlikely to see
consistency around us.

Let's look more carefully at how much consistency we would expect.
There are possible universes with consistent regions in which
intelligent life can exist and whose consistency ends abruptly at some
random time. There are possible universes whose consistent regions are
strictly the size needed for allowing intelligent life, and there are possible
universes with large consistent regions.

Let us assume that our existence means that some consistency is required.
Is there any non-required consistency around us; consistency which is due to
chance? To be more precise,
how large is the space-time region whose consistency is required? Well,
perhaps at a very distant time in the past, the consistent region included
the entire observable universe but, right now, there's no reason to require
full consistency outside of, say, the Earth's orbit. Moreover,
it's likely that we don't even need full consistency inside the Earth's orbit.

As far as we know, however, our solar system is consistent, our galaxy is
quite consistent, and distant galaxies are also fairly consistent.

This \paper{} argues that, since we observe much more consistency
than we would expect, design is the right explanation.

However, there is a possible objection to this: if multiple universes
exist, perhaps all possible ones, there will be some beings living in the
implausibly consistent ones.

While this is correct, the probability of an intelligent being
living in a fully consistent universe is still $0$.
In almost all universes the nonhomogeneity of the universe would be
easily observable, meaning that, for the relatively few intelligent beings
living in the other universes, as long as the design hypothesis
has some plausibility, it would be unreasonable to think that
their universe is not designed (assuming that the argument presented in
this \paper{} is correct).

\subsection{Few Universes Exist}
\label{sec:fewuniverses}

Another possible objection is that we used a meta-universe definition that's too
restrictive. What would happen if, for example, we had a meta-universe that, instead
of providing a probability distribution over universes, simply restricts the
possible universe descriptions to a given set?

We have these options: the set is countable; the set has the same cardinality
as $\reale$; and, if we reject the continuum hypothesis\footnote{
  The continuum hypothesis states that there is no set whose cardinality is
  strictly between that of the natural numbers and the real numbers.
}, the set can be somewhere between the two.

If the set is countable, then let us note that, by observing this meta-universe,
we are actually observing a countable set of elements chosen from the full
set of universe descriptions, so, by default, all of them should be generic.
If not, we can repeat the same argument above to show that either we have
an uncountable chain of improbable meta-universes or these descriptions should be
generic.

If the set has the same cardinality as $\reale$,
then we observe an element of this set, namely our universe's description, and there
is no probability distribution describing how to pick a description, so,
according to axiom \ref{ax:noprobability}, it should be generic.

If the set is not countable, but its cardinality is less than $\reale$, then
let us note that this paper's axioms should still be valid when using this
cardinality instead of $\reale$.
For example, the set of discontinuities would still be at most countable,
and, by default, we should use continuous probability distributions, so
if we observe any element with probability $0$, it should be generic.

To be specific, if we replace the set of real numbers with a set whose
cardinality is above countable but no greater than the real numbers
in axioms \ref{ax:zeroisgeneric}, \ref{ax:noprobability} and
\ref{ax:uncountable}, the argument should still work.

% However, there is anothe possible answer:
% if there is no Designer, we should actually
% expect the set of
% possible descriptions for the meta-universe to have the same cardinal as
% $\reale$, and that's true even
% if we ask these descriptions to be \ghilimele{different enough},
% i.e. different even though we work with approximations.
% To see why, let us consider the following argument, although it's a bit more
% technical than the others:

% The set of all conceivable descriptions has the same cardinality as the
% set of real numbers, $\reale$. This means that the
% set of all sets of conceivable descriptions
% has the same cardinality as the power set of $\reale$, i.e. $2^\reale$.

% A claim about what is possible specifies a particular set of descriptions,
% and any set of descriptions can be such a claim,
% which means that the set of possible claims has the same cardinal as
% the power set of $\reale$, $2^\reale$.
% The cardinal numbers remain the same even if we take into account only claims
% according to which our universe is possible.

% Unlike in the main topic discussed in this
% \paper{}, where any set of universes can, in principle, exist,
% this time it's clear that only one of the claims above can be true.

% Let us take all the claims in which the set of possible descriptions
% is less than $\reale$.
% The set of these claims (let us denote it by $S$)
% has a cardinality which is less than $2^\reale$.
% The set of claims not in $S$ (let us denote it by $T$) has the same cardinality
% as $2^\reale$.
% But $T$ is infinitely larger\footnote{
%   $S$ is infinite, which means that, if we take a distinct copy of $S$ for each
%   element of $S$, and we put them together, we get a set that still has the same
%   cardinal as $S$ (i.e. $S\times S$ has the same cardinal as $S$). In order to
%   get the same cardinal as $T$ we need to take even more distinct copies of $S$,
%   which means that $T$ is infinitely larger than $S$.

%   How many copies? We actually need one for each element in $T$, and no lower
%   cardinal would do.
% } than $S$, which means that it's reasonable to assume that $S$ has probability $0$.
% If so, then, with probability $1$, the set of possible descriptions has the same
% cardinality as $\reale$.

We should be able to get the same results when working with approximations
if, instead of looking at the set
of all distinct descriptions, we take a level of approximation $L$
and we group descriptions that, given $L$, work for the same universes.
% Diff with the edit
% As argued in the section Not Enough Descriptions,
As argued in section \ref{sec:not-enough-descriptions},
there are $\reale$ conceivable descriptions which are different
enough so that each of these would be in a different group --- which means that
there are $\reale$ groups of conceivable descriptions.
Using the same reasoning as above, with probability $1$, the set of groups of
conceivable descriptions that are possible according to the meta-universe
rules has the same cardinality as $\reale$.
From these groups we can extract a similar cardinal of
possible descriptions, which should be different enough since we grouped them
based on similarity.

\subsection{Not Enough Descriptions}
\label{sec:not-enough-descriptions}

When working without approximations, there are $\reale$ possible descriptions
that are essentially different. However, this is less obvious when working with
approximations, especially since it means we can probably have incompatible
descriptions for the same universe.

Let's assume (as an axiom) that, for a given level of approximation $L$, we can
find a finite cuboid size for which there is a finite set of
(full-precision)
descriptions, let us call it $D_f$, which cannot all be approximated with a
single finite approximate description.
In other words, if we have a set of cuboids whose descriptions include
the $D_f$ set, we can't
find a single finite approximate description that works for all of them
within the level of approximation $L$.

In the following example, the difference between
cuboid descriptions and universe descriptions is not always explicit,
but can be inferred from the context.

Here is an incomplete example that shows why the axiom outlined above is reasonable.
First, let us assume that we have a set of
\ghilimele{primary measurements} that we
can apply to our model, such that any other measurement that we could use can be
computed from those.
Then let us take two descriptions, the first stating that
\ghilimele{the value of each primary measurement is $0$},
and the second saying,
\ghilimele{the value of each primary measurement oscillates quickly
  between $0$ and something large enough to be easily detected within
  the approximation level}.

Then, let us consider a hypothetical infinite universe with an $n$-dimensional
space-time based on real numbers.
Let's divide it into cuboids of the size outlined above
and assign the incompatible
descriptions in the set above to these cuboids in a roughly even
way.
The set of cuboids is countable, so there are $\reale$ ways of
assigning these descriptions.

Let us pick an assignment and choose a space-time point very close to the
time end of a space-time cuboid boundary.
Let's take that entire point's past and try to make a prediction based
on it.
First, based on just that past, we can't predict the type of the next cuboid
since there will be many possible assignments that have the same past but in
which the next cuboid will be different from the current assignment.

Still, a universe description must predict the future from the past, and
the future depends only on the next cuboid type. Then the universe
description must allow predicting the next cuboid type, so it must include,
implicitly or explicitly,
a function predicting the next cuboid from a cuboid's past
and its context (i.e., position). Now we must find out how many such functions
we need in order to describe all possible assignments.

To make things simpler, let's assume that the number of dimensions, $n$,
is $1$ (i.e., we have only events ordered by time --- a group of such events taking
place between two moments in time would be a space-time cuboid.
We could also consider such a
case where the space contains only one space-cuboid, and the time dimension is
infinite --- a space-time cuboid would consist of the space cuboid taken for a
given time interval) and that we have only two cuboid types.
It should be obvious that
the remainder of this example can be generalized to any number of dimensions
and to any number of cuboid types greater than $2$.

Having $n=1$ means that we have a one-to-one correspondence between
integers and cuboids, so let's identify a cuboid with its number through this
correspondence.
Allow also the two cuboid types be denoted by $0$ and $1$ and
let us represent a universe as a function from the set of integers
(representing the set of cuboids) $\intregi$,
to the set $\multime{0, 1}$, identifying, for each cuboid, its type.

We can represent the information available for predicting the type of a future
cuboid by a pair between the cuboid and its past, denoted
by $(x, p_x)$, where $x$ is the cuboid's number, $0$ or $1$;
and $p_x$, which represents $x$'s past, is a function from the set of natural
numbers to $\multime{0,1}$.

However, we are not trying to predict just one cuboid's future;
we are trying to predict the future of all cuboids.
Any way of making these predictions can be put in a $1$:$1$ correspondence
with a function from the past
of cuboids to their future, i.e., from $(x, p_x)$ pairs to $\multime{0, 1}$.

Let's find out how many distinct functions we need.
To make things simple, allow us to consider only universes which, for negative
integers, have only cuboids of type $0$.

The past of cuboid $0$ is perfectly identical in all these universes, so,
in order to predict the type of the cuboid with index $0$,
we need two distinct prediction functions: one which predicts that the type is
$0$, and another that predicts that the type is $1$.
In order to predict the types of the first two cuboids, $0$ and $1$, we need
four distinct prediction functions.
In general, in order to predict the types of all cuboids between $0$ and $n$
we need $2^{n+1}$ distinct prediction functions.
And, in order to predict the type of all cuboids which are greater or equal
to $0$, we need $2^\naturale$, i.e., the power set of the natural numbers,
i.e., $\reale$ prediction functions.

As mentioned previously, intelligent beings may need some consistent space-time
around them. Even if that's the case, fixing the cuboid type for a finite
chunk of an infinite universe doesn't change the cardinal for the set of
functions mentioned above.

\subsection {Multiple Universes Based on the Same Laws}

As mentioned above, our universe could be included in a meta-universe in which,
colloquially speaking, all universes have the same laws
but different fundamental constants, and no other universes are allowed.
Of course, using this paper's definitions,
the fundamental constants would be part of a universe's description.

When using approximate universe descriptions---since our universe
seems to be based on laws that are continuous in these fundamental constants---the
set constants that are "different enough" should be at most countable.

Since no other universe is allowed, and the set of allowed approximate
descriptions is countable, the meta-universe's probability
has to be non-zero for some of these descriptions. In other words, the
meta-universe probability has some peculiar discontinuities; something which is, as
mentioned above, implausible.

%\subsection{Fractions of Space}

%Note that above I used a specific definition of what I mean by
%predictions being \ghilimele{true with probability $p>0$}.
%However, it's valid to ask, say,
%what happens if we want our predictions to be true
%in only a fraction $p$ of spacetime, with probability $p$.

%Since what we mean by \ghilimele{a fraction $p$} is not fully defined,
%there may be definitions for which the argument below does not work,
%but, most likely, it works for all reasonable definitions.
%Also, the argument below presents only the idea, without working out all
%details, since there is no point of being fully precise without a precise
%hypothesis.

%Let us take, again, $\reale^n$ as a hypothetical universe.
%Let us denote by $n$ the number of chunks we get if we split the entire
%spacetime
%in chunks covering a fraction $p$, one of which may be smaller than $p$.
%As above, we can probably find a cuboid size for which we
%have $2n$ different cuboid descriptions when working with a given level
%of approximation $L$.
%Let us consider again that
%we split a universe into cuboids of this size, and that we assign
%cuboid descriptions randomly.

%We are interested in cuboid type assignments that assign the cuboid types
%such that each of them covers a fraction less than $p/2$ of the universe.

%Let us take a universe chunk covering a fraction $p$.
%We can choose
% If that that includes at most one
%partial cuboid, then, with a single cuboid description we can cover at
%most half of it.

%We'll further assume that when we divide the spacetime in chunks of the same
%size (cuboids are a good canditate),
%we can get at least two assignments (i.e. two assignment types)
%for the cuboids inside those
%chunks such that the two assignments do not assign the same value to the
%same cuboid in both.

%Now, any description which works in a fraction $p$ of space must make
%predictions that, roughly, work in at least two cuboid types.
%So, in order to make predictions, a definition must identify
%the cuboid type, as in section \ref{sec:fewuniverses}.
%This means that we must have at least
%$\reale$ definitions.

%\svn{Is \ghilimele{true} appropriate here? and in the entire paper?
%Truth is a word that's hard to use.
%Would it be better to use \ghilimele{verified} or something similar?}

\subsection{Complicated universe}

We have argued above that we can divide the universe region around us into small
identical pieces, all of which have finite approximate descriptions, and that
any description that works for one of them, works for all of them.

Of course, in practice, we can only work with finite descriptions when
modeling our universe.
Fortunately, quantum mechanics, together with
relativity (although they may be incompatible), form a finite description which
seems to fit our universe almost perfectly.
Even Newtonian physics seems to work pretty well.

However, we can wonder if our space-time
region is actually much more complex, perhaps infinitely so;
it's just that we haven't become aware of that complexity yet.

This is possible, but the main part of this paper's argument can be applied here thus:
We notice a lot of consistency in the space-time around us
because maybe we do not (or cannot) pay attention to enough aspects of the universe.
It's conceivable too that the space-time we
witness is not consistent in the aspects that we do observe.
This means that if we divide our space-time into regions and we choose one to be our point of reference,
it's unreasonable to have a probability of $1$ that any other
region is consistent with our region of reference in those aspects.

% Diff with the edit
% But, much as it was described in the Peculiar Descriptions for Universe Regions section,
But, much as it was described in section \ref{sec:peculiarregions},
this means that we are observing
something having a really low probability, and it's more reasonable to
think that our universe is designed.

\section{Conclusion}
\label{sec:conclusion}

We have the following reasonable possibilities:

\begin{enumerate}
\item The probability that a Designer would want intelligent beings
      % Diff with the edit.
      % is non-zero (see the section Design Probability),
      is non-zero (see section \ref{sec:design-probability}),
      our universe is designed (at least in part) for us,
      and the universe around us is, roughly speaking, as homogeneous as we
      think it is.
% Diff with the edit.
% \item The argument in Design Probability doesn't make sense;
\item The argument in section \ref{sec:design-probability} doesn't make sense;
      we don't know whether our universe is designed or not,
      our universe is not homogeneous, but we observe some
      extremely unlikely homogeneity around us.
\end{enumerate}

The most reasonable option seems to be that our universe is designed.

\section{Mathematical Background}
\label{sec:background}

This section recalls some well-established mathematical definitions, axioms and theorems that are used in this paper.
Most of them should be covered in introductory math courses
on the topic of each subsection, but each subsection also
has references to mathematics books.

This section also contains precise definitions for some of the concepts that
are specific to this paper
(for example, \ghilimele{generic} and \ghilimele{peculiar})
and a generalization of an existing concept (continuous probability).

\subsection{Probabilities}
\label{sec:probabilities}

For the following definitions, see
% Diff with the edit
% Billingsley (1995, “Probability and Measure”, Chapter 1, section on “Probability Measures”).
\textcite{Billingsley1995} (\ghilimele{Probability and Measure}, Chapter 1, section on \ghilimele{Probability Measures}).

A probability measure $P$ is a function defined on some of the subsets of a
space $\Omega$. $P$ should satisfy three conditions:
\begin{itemize}
  \item $0 \le P(A) \le P(\Omega)$ for all $A$ for which $P$ is defined;
  \item $P(\emptyset) = 0$ and $P(\Omega) = 0$;
  \item If $A_1, A_2, \dots$ is a disjoint sequence of sets for which $P$ is
        defined, then $P$ is defined for their union, and
        $P(\bigcup\limits_{i=1}^{\infty} A_i)=\sum_{i=1}^{\infty}P(A_i)$
\end{itemize}

The subsets for which $P$ is defined are called \ghilimele{events}.

%TODO: Cite Theorem 31.7 from the same book. Perhaps also page 422. Example 38.1.

A probability $P$ is continuous if it assigns $0$ to each single element set
(this is a generalization of the \ghilimele{absolutely continuous} property
% Diff with the edit
% used in Billingsley [1995]).
used in \textcite{Billingsley1995}).
For example, both the uniform probability for a real number interval and the
normal probability over the set of real numbers are continuous.

Any single element set for which the probability assigns a non-zero value is
called a discontinuity.

The terms \ghilimele{probability measure} and
\ghilimele{probability distribution} will be used interchangeably.

\subsubsection{Peculiar and generic events}

Let $A$ be a set with the same cardinality as the set of real numbers $\reale$,
with a probability distribution $P$.
Let $F$ be the set of
all mathematical predicates of one variable over $A$ that can be written as a
finite formula.
If $f$ is such a predicate then let $A_f$ be the subset of
$A$ where $f$ is true, i.e., $A_f=\multime{a\in A\mid f(a) \mbox{ is true}}$.
Whenever $P(A_f)$ is defined, we will also denote it by $P(f)$, and we will
call it \ghilimele{the probability of $f$}.

Let $F_0$ be the set of predicates in $F$ whose probability is
$0$ for all continuous probability distributions over $A$, i.e.,
$$F_0=\multime{
  f\in F
  \mid P(A_f)=0
    \mbox{ for all continuous probability distributions over } A}.
$$
For example, any predicate $f$ which is true for a finite subset of elements
(i.e., $A_f$ is finite) would belong to $F_0$.

Let us identify by $A_0$ the
set of elements of $A$ for which at least one predicate of $F_0$ is true, i.e.,
$$A_0=\multime{a \in A\mid \exists f\in F_0 \mbox{ with } f(a)\mbox{ true} }.$$
Let us also denote $A_1$ as the complement of $A_0$.

In the following, let us assume that $P$ is a continuous probability distribution.

Since $F$, the set of all predicates, is countable, $F_0$ must be at most
countable.
This means that $A_0$ is an at-most-countable union of sets, one set for each
element of $F_0$, i.e., $A_0=\cup_{f\in F_0} A_f$.
This means that $P(A_0)$ is less or equal to the sum of the probabilities of
these sets, $P(A_0)\le\sum_{f\in F_0} P(A_f)$.
Since this sum is $0$, then $P(A_0)$ is $0$.
This means that $P(A_1)=1$.

Let us say that an element $a\in A_1$ is \definitie{generic}
and an element $a\in A_0$ is \definitie{peculiar}. Then we could rewrite
the equalities above as $P(x\mbox{ is generic}) = 1$ and
$P(x\mbox{ is peculiar}) = 0$.

When the probability for a subset of $A$ is $0$, $P(E)=0$, where $E\subset A$,
it does not mean that observing an element of $E$ is logically impossible;
it means that,
if we make a set of (independent) observations of elements of $A$
that is at most countable, we have no chance at all of observing an element of
$E$.

\subsubsection{Countable discontinuities}

Let us now consider a different issue.
Let $P$ be a probability over the set of real numbers $\reale$ (or
any infinite uncountable set).
Let $A$ be the set of real numbers with non-zero probability.
% Diff with the edit
% Then $A$ is at most countable, see (Billingsley, 1995, page 162, Theorem 10.2).
Then $A$ is at most countable, see \textcite{Billingsley1995} (page 162, Theorem 10.2).

\subsection{Ordinals}
\label{sec:ordinals}

For the definitions and properties presented in this section,
% Diff with the edit.
% see Bagaria (2020) and Cohen (1966).
see \textcite{sep-set-theory} and \textcite{Cohen1966}.

Ordinals are generalizations of natural numbers. Natural numbers can be
defined by identifying each natural number with the set of natural numbers less
than it. Then $0$ is identified with the empty set $\emptyset$,
$1$ with $\multime{0}=\multime{\emptyset}$,
$2$ with $\multime{0, 1}=\multime{\emptyset, \multime{\emptyset}}$,
and so on.

Each of the natural numbers defined as above is an ordinal.
However, we can define ordinals that do not correspond to natural numbers.
The smallest such ordinal, denoted by $\omega$, is defined with the same rule:
let $\omega$ be the set of ordinals smaller than it, i.e
the set of natural numbers, $\omega=\multime{0, 1, 2, \dots}$.

Of course, the next ordinal, called $\omega + 1$, will be the set
$\multime{0, 1, 2, \dots \omega}$ and the next one,
$\omega+2$, will be $\multime{0, 1, 2, \dots \omega, \omega+1}$. We can
continue and, in the same way, define $\omega\cdot 2=\omega+\omega$ to be
the ordinal that comes after all ordinals of the form $\omega+n$
where $n$ is a finite ordinal (i.e., a natural number).

Then we can define $\omega\cdot 3$, $\omega\cdot 4$ and so on, and we can
take $\omega\cdot \omega$ to be the ordinal that comes after all those
defined by using the rules above.

Let us note that $\omega$ is countable, and that all the ordinals mentioned
above that come after it are also countable.
By using the same kind of
reasoning we can produce other countable ordinals like
$\omega^\omega$ (containing ordinals like $\omega\cdot\omega\cdot\dots\cdot\omega$)
and $\epsilon_0$ (containing $\omega^{\omega^{\cdots^\omega}}$).

After going through many similar processes, at some point we obtain the
smallest uncountable ordinal $\omega_1$, which is the set of all
countable ordinals.

Some ordinals, like all the finite ones except $0$,
and like $\omega+1$, can be obtained from the previous one by using
a successor relation, i.e., $successor(\alpha) = \alpha\cup\multime{\alpha}$.
All ordinals have a successor, but not all are successors. Some, like
$\omega$ and $\omega\cdot 2$, can be defined only as the set of all
smaller ordinals. The former are called \ghilimele{successor ordinals};
the later are called \ghilimele{limit ordinals}. Note that $0$ is a limit
ordinal.

\subsection{Cardinals}
\label{sec:cardinals}

The set of natural numbers $\naturale$, the set of integers $\intregi$, and
the set of rational numbers $\rationale$ are countable.

The set of real numbers $\reale$ has a larger cardinality, the same cardinality
as the power set of a countable set. To put this in a different way, let's say
that we have a countable set, like $\naturale$,
and for each element we can label it with one of two
labels (let's call them $a$ and $b$).
For example, we can label $0$ with $a$, $1$ with $b$, $2$ with $a$, and so on.
Let's call \ghilimele{a set labeling} what we get if we assign a
label to each element in the set.
Then, if we take all possible labelings for the countable set,
we get another set with the same cardinal as the set of real numbers\footnote{
    This is also valid for more than two options, as long as the cardinality
    of their set is at most countable.
  }.
That is, the set of all functions
$$
f : \naturale \longrightarrow \multime{a, b}
$$
has the same cardinality as $\reale$.

If we have a set with the same cardinality as the set of real numbers, and
we remove a countable set, the resulting set still has the same cardinality
as the set of real numbers. For example, $\reale$, $\reale\setminus\naturale$,
$\reale\setminus\intregi$ and $\reale\setminus\rationale$ all have the same
cardinality. In general, if $A$ and $B$ are infinite sets and
$B$'s cardinality is lower than $A$, then removing $B$ from $A$ does not change
its cardinal, i.e., $A\setminus B$ has the same cardinal as $A$.

If $A$ has an infinite cardinal, then the product set $A\times A$ has the same
cardinality as $A$.

\printbibliography

\end{document}