# The necessity of mathematical infinity: the [M]-model of Set Theory from Laurent Schwartz and some philosophical consequences

Daniel Parrochia

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# The necessity of mathematical infinity: the [M]-model of Set Theory from Laurent Schwartz and some philosophical consequences

Daniel Parrochia

University of Lyon (France)

To Pierre Anglès who introduced me to Laurent Schwartz's [M]-model.

**Abstract.**

For a long time now, philosophy has abandoned the idea of medieval philosophers (Anselm) or classical philosophers (Descartes, Malebranche), to provide proof of the existence of an infinity in the world. This question of infinity, when it is a question of an *actual* infinity, was however raised again in the time of Cantor, whom the Church forced to distinguish the purely mathematical "transfinite" from a philosophical-theological "absolute infinity". Even if these debates no longer occupy the forefront today, it remains that there are links between philosophy and mathematics, as proven by the existence of this curious model of set theory that is the [M]-model of Laurent Schwartz. This, as we are going to discover, risks in fact revealing the unavoidable and absolutely necessary character of the axiom of infinity in set theory. From then on, should the old question not be asked anew? Should we not conclude that if we are obliged to introduce such an axiom in mathematics, it is because infinity necessarily exists? If this were the case, Laurent Schwartz's [M]-model would constitute a sort of indirect proof of this axiom, which would be confirmed, beyond the quarrel between finitism and infinitism, by the idea of a "classifying topos".

**Key words.** Infinity, Laurent Schwartz, Models of set Theory, God's existence, theology.
# 1 Introduction

The mathematician Laurent Schwartz, the first Frenchman to have obtained the Fields medal in 1950 for his work on the theory of distributions (see [Schwartz 78]), was also interested, as we know, in topology and set theory (see [Schwartz 97]). Concerning the latter, he is at the origin of a particular model, little known because undoubtedly mathematically a little strange<fnref n="1" />, but interesting from an extra-mathematical point of view. This model was first presented during a conference at the Henri Poincaré Institute in 1967 (see [Schwartz 81]). Then it gave rise to an article from which we draw inspiration here (see [Schwartz 68]), reported and commented on by P. Anglès in a recent book (see [Anglès 25], 56-60 and 73-74).

We will describe this model, comment on it and draw all its philosophical consequences.

Let us only recall here, before all, that this article from Schwartz is part of a reflection on the necessity of axiomatics in mathematics, a necessity which mathematicians still doubted at the beginning of the century<fnref n="2" /> and whose relevance Schwartz recalls in the first pages, since, in a theory without axioms (resp. poor in axioms), all (resp. almost all) the formulas become undecidable. Even in a theory like set theory, undecidable formulas are in the overwhelming majority (see [Schwartz 68], 89).

Schwartz continues his article by defining the notion of an infinite set and explaining what conditions the set theory [M]-model he wants to construct will obey (see [Schwartz 68], 89-90):

The best definition of an infinite set is probably the following:

**Definition 1.1** (Infinite set). *E* is an infinite set if and only if there exists a bijection of *E* onto one of its parts distinct from itself. In formula, we have:

$$ (\exists F)(\exists f) : (F \cup E) \land (F \neq E) \land (f \text{ is a bijection of } E \text{ on } F). $$

Now let us define the conditions to be satisfied by the [M]-model.

<footnote n="1">¹It is also possible that the date of publication of the article from Schwartz, in March-April 1968, just before the events of May, when the French, including mathematicians, were preoccupied with entirely other problems, contributed to its passing unnoticed.</footnote>

<footnote n="2">²Schwartz recalls (see [Schwartz 68], 87) that the fiery letters exchanged by Borel and Hadamard on the subject of the paradoxes of set theory could not lead to much, for lack of having clearly specified what the initial axiomatics was.</footnote>
## Conditions

1. The objects and symbols of [M] will be taken from usual mathematics and will therefore be subject to the processing rules familiar to the mathematician;

2. Inside the [M]-model, the objects and symbols will have special meanings well specified *a priori* and such that all the first axioms of set theory are verified in [M];

3. On the existence of infinite sets, nothing will be posited axiomatically in [M]. This existence will be undecidable in [M], whereas, within the framework of usual mathematics, that is to say from outside the model, it will be possible to prove that, in [M], there is no infinite set.

# 2 Schwartz [M]-model of set theory

Schwartz constructs this [M]-model of set theory such that the objects of [M] are the natural numbers 0, 1, 2, ..., *n*... of usual mathematics.

In this model, the connectives are the usual connectives for the Predicate Calculus with equality, namely: ∨, ∧, ¬ and "=" which means, of course, "is equal to".

To avoid confusion between the model formulas and the formulas of usual mathematics, Schwartz agrees to write the model formulas in square brackets. Thus, [3] denotes an object of [M], while 3 is simply the natural number element of N. Likewise, [*m* = *n*] will express the equality of two objects of [M], while *m* = *n* will be only, as usual, the equality of two natural integers.

A logico-mathematical theory is, formally speaking, a symbolism that can be interpreted in different ways. It all depends on the meaning that we give to the symbols. In the context of set theory, we can therefore ask ourselves what would happen if we interpreted the membership sign ($\in$) in a different way than usual. Schwartz, for example, observes in particular this:

**Proposition 2.1.** If we choose, for $[p \in q]$ the meaning $p \le q$, then one of the first axioms of set theory would not be verified. Indeed, in [M], there would be no empty set.

*Proof.* Let us recall that the axiom of the empty set is one of the possible axioms of
the axiomatic set theory from Zermelo Fraenkel (called ZF)<fnref n="3" />. As its name indicates, it allows us to posit the existence of an empty set. In the formal language of the theory, it is written:

$$ \exists A \forall B (B \notin A), $$

or in words:

There exists a set A such that, for any set B, B is not an element of A, that is, there exists a set of which no set is an element<fnref n="4" />. It is usually written : ∅.

If we transpose this axiom to [M], we see that it tells us that there should exist a number [p], such that, for any number [q], [q] is not less than or equal to [p]. But there is no number of this kind, since, on the contrary, it always exists $p \in \mathbb{N}$ such that, for all q, q is less than or equal to p. In symbols.

$$ \exists p \in \mathbb{N} | \forall q (q \le p). $$

So the axiom is not satisfied.

□

**Proposition 2.2.** If we now choose for $[p \in q]$ the meaning $[p < q]$, then there does exist an empty set which is zero, but another important axiom is not verified: that of the existence of a set with only one element.

*Proof.* Let us take some example: the number [6] indeed checks $[6 \in 7, 7 \in 8]$, but there exists no object [a] in [M] such that [6] is the only object to check $[6 \in a]$, since there exist other numbers which can satisfy this expression as well. □

Schwartz will choose an interpretation of the membership predicate based on the following theorem:

**Theorem 2.3.** Every non-zero natural number admits a unique dyadic development.

This theorem means: to every natural number $m \neq 0$ corresponds a unique finite sequence of distinct naturals $p, q, r, s, ..., w$ such that we can write:

$$ m = 2^p + 2^q + 2^r + ... + 2^w. $$

With these notations, we are led, in [M] to the following definition:

<footnote n="3">³In modern presentations, it is true that it is no longer mentioned among the axioms of Z or ZF, because it is a consequence in first-order logic of the axiom schema of comprehension.</footnote>

<footnote n="4">⁴The axiom of extensionality can be used to show that this set is unique.</footnote>
**Definition 2.1.** For all $m = 2^p + 2^q + 2^r + ... + 2^w$, the definition of membership is:

$$[p \in m, q \in m, r \in m, ..., w \in m].$$

Example: As $5 = 2^0 + 2^2$, we can write: $[2 \in 5, 0 \in 5]$, or again, since $64 = 2^6$, we can write: $[6 \in 64]$ and even $[64 = \{6\}]$.

# 3 Properties of [M]

Schwartz shows that the first axioms of ZF theory are verified in [M].

**Empty set** Zero is the only natural number of [M] which does not admit non-empty dyadic expansion.

**One-element set** $[\{p\}]$ exists in [M] since there exist natural numbers which are powers of 2. Here, $2^p$.

**Set with two elements** Let $p \neq q$ be two distinct natural numbers. Then, $[\{p, q\}]$ exists in [M] since the natural number $2^p + 2^q$ exists.

*Note*: we absolutely must specify $p \neq q$ because $2^p + 2^q = 2^{p+1}$ and, in [M], $[\{p, p\} = \{p\} = 2^p]$ (we saw above that $[\{p\}]$ exists in [M] and corresponds to $2^p$).

**Axiom of extensionality** In [M], the statement "two objects having the same number of elements are equal" is written:

$$[\forall m, \forall n, (m = n \iff (\forall x, (x \in m \iff x \in n)))].$$

From outside [M], this simply means: "the dyadic decomposition of a natural number is unique", which is a well-known theorem.

**Inclusion and power set** By definition:

$$[m \subset n \iff (\forall x, (x \in m \Rightarrow x \in n))].$$

Example: $[5 = \{0, 2\}]$. The parts of [5] are therefore: $0, 2^0, 2^2, 2^0 + 2$.

So the power set of [5] is written: $[P5 = \{0, 1, 4, 5\}]$, that is to say $P5 : 2^0 + 2^1 + 24 + 2^5 = 51$.

It is easy to verify that the power set still exist.
**Couple** We choose the usual definition: $(a, b) = \{\{a\}, \{a, b\}\}$, which results in the fundamental property of the couple, with two cases:

1. If $a = b$, $(a, a) = \{\{a\}, \{a, a\}\} = \{\{a\}, \{a\}\} = \{\{a\}\}$;

And, therefore: $(a, a) = (a', a') \iff a = a'$.

2. If $a \neq b$, $(a, b) = (a', b') \iff \{\{a\}, \{a, b\}\} = \{\{a'\}, \{a', b'\}\}$.

This last equality implies that:

- Let $\{a\} = \{a'\}$, in other words, $a = a'$ and, consequently $\{a, b\} = \{a', b'\}$, therefore $b = b'$ (since we already know that $a = a'$);

- Let $\{a\} = \{a', b'\}$, which implies that: $a' = b' = a$ and leads to a contradiction with $a \neq b$ since we found that $b = b'$.

Example of calculating a couple in [M]:

Let the couple be: $[(3, 5)]$. Now $\{\{3\} = 8\}$ and $[(3, 5) = 8 + 32 = 40]$.

It follows that:

$$[(3, 5) = \{8, 40\} = 2^8 + 2^{40} = 1050935628032.$$

**Cartesian product** We will show on an example that we can define such a product.

$[2 \times 3 = \{(x, y)/x\} \in 2 \text{ and } y \in 3]$. We know that $[2 = \{1\}, 3 = \{0, 1\}]$.

The pairs of $[2 \times 3]$ will therefore be:

$$[(1, 0) = \{\{1\}, \{1, 0\}\} = \{2, 3\} = 12]; [\{\{1\}, \{1, 1\}\} = \{\{1\}\} = 4].$$

Therefore:

$$[2 \times 3 = \{4, 12\} = 4112].$$

And we can verify on this example that: $[2 \times 3 \neq 3 \times 2]$.

Indeed, as above, we find that:

$$[3 \times 2 = \{(0, 1), (1, 1)\} = \{10, 4\} = 1040].$$
Just like Laurent Schwartz, we will not continue the explicit interpretation of the
other axioms of ZF in [M]. We will admit, according to him, that [M] has exactly
the same axioms as ZF.

4 The Schwartz [M]-model and the question of in-
finity

1) We know that the dyadic development of a natural number $m$ necessarily includes a finite number of powers of 2, whose exponents are natural numbers strictly less than $m$. We deduce that, in the Schwartz [M]-model, an object $[m]$ has as elements only a finite number of objects $[p, q, r, s, ...w]$.

From the outside, therefore, there exists no infinite object in [M].

Within the model itself, however, nothing of the sort is demonstrated. Otherwise,
such an assertion would follow from the axioms given above. Now these axioms are
the same as those of classical set theory, in which the non-existence of infinite sets
is not demonstrated.

We deduce that we cannot, from the usual axioms of set theory without the axiom
of infinity, prove the existence of an infinite set. Otherwise we would also prove this
existence within [M] which has the same axioms.

Therefore, a fortiori, we cannot prove the existence of an infinite set for [M] outside
of [M].

However, for [M], outside of [M], we prove on the contrary that there is no infinite
set.

2) We know that in classical set theory (with or without the axiom of infinity), there
is no set of all objects (Cantor's paradox).

In the [M]-model, a set of all objects would be $\mathbb{N}$ itself. But precisely, $\mathbb{N}$ is not an
object of [M].

In conclusion, Laurent Schwartz's construction of the [M]-model of set theory clearly
highlights the necessary character of the axiom of infinity in set theory: we must
necessarily posit infinity as an axiom, since it is not deducible from theory and,
nevertheless, we have the greatest need for it in many important areas of mathemat-
ics<fnref n="5" />.

<footnote n="5">⁵Infinity is one of the concepts most often used in mathematics. For example, it is found in</footnote>
# 5 Philosophical infinite in a nutshell

Merleau-Ponty once castigated the innocence with which the philosophers of the Classical Age had manipulated infinity (see [Merleau-Ponty 53]). This irony is perhaps unfounded if we consider that these ancient philosophers obeyed the same necessity as today's mathematicians.

## 5.1 Descartes and the infinite

Descartes, in a very famous text from the third of his "Meditations on Metaphysics", had defended, in fact, an entirely positive conception of infinity. "I must not imagine that I do not conceive infinity by a true idea, but only by the negation of what is finite, just as I understand rest and darkness by the negation of movement and light: since on the contrary I clearly see that more reality is found in infinite substance than in finite substance, and therefore that I have in some way first in me the notion of infinity, than of finite" (See [Descartes 47], 49-52).<fnref n="6" />

In the following pages, the argument developed is that we could not perceive the finite as finite, notice our imperfections or experience any lack if we did not have the idea of the infinite, which cannot be the effect of a defect in our nature like the sensations, sometimes fallacious, of heat or cold. What is more, it is an actual infinity, whereas everything in us can only be at best in potentiality. Now this idea, which has not been received by the senses and which cannot be a pure fiction of the mind (because nothing can be added or subtracted from it) must be recognized in some way as "innate".

Schwartz's [M]-model of set theory makes it clear, in the same way, that we cannot conceive of the infinite without positing it as an axiom. If we do not do so, we cannot, within a model, prove or refute the existence of the infinite. And if we are in a model like [M], where only finite objects exist, we are condemned to move in the finite, without any idea of possible infinity. Since such an idea exists, we must necessarily introduce it as an axiom.

differential and integral calculus to define limits, continuity, convergence, etc. It is also present in numbers, in geometry and in many other branches of mathematics.

<footnote n="6">⁶Even today, a French philosopher like François Dagognet (see [Dagognet 00], 138) has taken up this idea according to which the infinite cannot be derived from the finite: "In principle, the limited being only thinks of being sure of himself, of being satisfied with him and of enjoying his status." We will therefore not say that it is the finite which produced the idea of infinity because, from where would it know that it is finite?</footnote>
However, as we know, it is not certain that this idea is as "clear and distinct"
as Descartes claims, since we can conceive of infinity today in different ways, and
the relationship between these forms of infinity is still far from being completely
elucidated.

## 5.2 Finite and infinite in the texts from Malebranche

In Malebranche, we find the same characterization of God by the infinite, or, as
a further hyperbole, the infinitely infinite. According to the *Conversations Chré-
tiennes*, "infinity is an attribute of the Divinity, and its essential attribute" (see
[Malebranche 92¹], 1163). The *Entretiens sur la Métaphysique et la Religion*
des-ignate God as "the infinite, the Being without restriction" (see [Malebranche 92²],
VIII, I, 802), or even as an "abbreviated expression of the infinitely perfect Being"
(see [Malebranche 92²], VIII, I, *ibid*.). What characterizes this infinity is its capacity
to be both infinite and One, which the human mind probably cannot think of and
which therefore remains confused for it, so that God cannot be a simple being for
man (*ibid*., 692). Thus, this infinitely infinite, which can be everywhere and nowhere,
cannot be understood by the human mind, even if infinity, as the absence of limits,
is a familiar and even intimate idea, of which one can have a positive knowledge,
notably in arithmetic and geometry.

We then encounter, in Malebranche, but already, as we have seen, in Descartes,
something of the situation of the two models (the [M]-model of Schwartz and the
ZF model of classical set theory) that we mentioned above, and in which it was
impossible to prove infinity (or to prove its absence).

The infinite, in Malebranche, is defined only in relation to its attributes and does
not function as an operator of the presence of a first existence external to that of my
thought, contrary to the device of Descartes' Third Meditation, insofar as the union
of the finite with God is immediately conceived as such. In this sense, the infinite
distance, which is found between the sovereign Being and the spirit of man, does not
prevent it from being united to it immediately, and in a very intimate way (see [?],
preface). Since infinity is no longer external, the question of infinity in a particular
genre does not pose a problem insofar as it can only be related to the infinite divine
being to which we are united. This is how Malebranche will be able to affirm the
positive infinity of intelligible or material extension, without conceding extension to
God.

But the key point is that the thought of infinity, for Malebranche, then constitutes
the index of self-revelation of the finitude of the mind, which, from his point of view,
can obviously only lead us to God.

We therefore find ourselves here in a situation symmetrical to that mentioned above. In the case of the models of set theory, the finitary aspect of [M] required the introduction of an axiom of infinity in ZF and justified it. In the case of Malebranche, even more than in the case of Descartes, the internalization of infinity in the model leads to posing this time, a sort of axiom of finitude.

## 5.3 Cantor, before and after

Spinoza already used the difference between continuous and countable, of which neither Pascal nor Leibniz were aware, but his God, identical to Nature itself, was in fact essentially a continuum. The discovery of the existence of many other infinities by Cantor, in the 19th century, raised the question of their relationship (continuum hypothesis and generalized continuum hypothesis) and this mathematical use of infinity forced the Church to take a position (see [Dauben 90]).

As has already been shown on several occasions, the Church wanted to clearly differentiate a mathematical continuum created by man (created or transfinite infinity) from the uncreated or absolute divine infinity, the only admissible theological infinity, the distinction between the one and the other being irreducible (see [Parrochia 18], chap. 9)<fnref n="7" />.

For a long time, according to the work of Cohen (see [Cohen 08]), the continuum hypothesis proved undecidable, the uncertainty concerning the nature of the alephs was able to confirm this choice. Mathematicians could manipulate infinities, but they had nothing to do with the only truly existing absolute infinity, the divine infinity, which by its very nature escaped mathematics.

But if the hypothesis of the continuum were to prove true, as some recent works might suggest (see [Woodin 01a], [Woodin 01b] and [Parrochia 23], chap. 7), infinity could thus acquire a more univocal and objective definition, which would doubtless raise anew the question of the real nature of the "absolute infinite".

In such a context, Laurent Schwartz's [M]-model could make visible the inevitability of the concept of infinity and, at the same time as its necessary position as an axiom of set theory, its status of primacy in relation to the finite. This would not only be

<footnote n="7">⁷Cantor, in a letter to the Cardinal Franzelin dated January 22 1896, finally also agreed to distinguish between an *infinitum aeternum increatum sive absolutum* (an eternal, uncreated and absolute infinity) and an *infinitum creatum sive transfinitum* (a created or transfinite infinity), the first belonging only to God and the second being solely that of creatures.</footnote>
another way of verifying that it is often infinity, in mathematics, which allows us to obtain information on the finite. Maybe it is also a way of giving an ontological meaning to this notion without falling into known paradoxes such as that of the set of all sets or others of the same kind.

# 6 The Axiom of Infinity and the philosophical-theological question

To know whether the inescapable character of the position of infinity in mathematics has any relation to the theological question, it is necessary to examine carefully this axiom which has, in fact, different variants.

First published by Ernst Zermelo as part of his set theory in 1908 (see [Zermelo 08], 266), the axiom of infinity can be expressed as follows:

$$ \exists I (\exists o (o \in I \land \neg \exists n (n \in o)) \land \forall x (x \in I \Rightarrow \exists y (y \in I \land \forall a (a \in y \Leftrightarrow (a \in x \lor a = x))))). $$

In English, this sentence means: "there exists a set I such that the empty set is an element of it, and for every element x of I, such that x is an element of y, the elements of x are also elements of y and nothing else is an element of y.

This sentence can be abbreviated in the following inductive set:

$$ \exists I (\emptyset \in I \land \forall x (x \in I \Rightarrow (x \cup \{x\}) \in I)). $$

This axiom is closely related to the von Neumann construction of the natural numbers in set theory, in which the successor of x is defined as $x \cup \{x\}$. If x is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the natural numbers. In this encoding, zero is the empty set:

0 = {}. The number 1 is the successor of 0:

$$ 1 = 0 \cup \{0\} = \{\} \cup \{0\} = \{0\} = \{\{\}\}. $$

Likewise, 2 is the successor of 1:

$$ 2 = 1 \cup \{1\} = \{0\} \cup \{1\} = \{0, 1\} = \{\{\}, \{\}\}, $$

and so on:

$$ 3 = \{0, 1, 2\} = \{\}, \{\}, \{\}, \{\} $$
$$4 = \{0, 1, 2, 3\} = \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}.$$

A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers. This construction forms the natural numbers. As the other axioms are insufficient to prove the existence of the set of all natural numbers $N_0$, its existence is taken as an axiom – the axiom of infinity. This axiom asserts that there is a set $I$ that contains $0$ and is closed under the operation of taking the successor; that is, for each element of $I$, the successor of that element is also in $I$.

Thus the essence of the axiom says: "there is a set, $I$, that includes all the natural numbers".

But to extract the natural numbers, we need a definition of which sets are natural numbers. The natural numbers can be defined in a way that does not assume any axioms except the axiom of extensionality and the axiom of induction. That is what we do when we say that a natural number is either zero or a successor and each of its elements is either zero or a successor of another of its elements. There are also an alternative method taking the intersection of all inductive sets. But both these methods produce systems that satisfy the axioms of second-order arithmetic, since the axiom of power set allows us to quantify over the power set of the first ordinal $\omega$, as in second-order logic. Thus they both completely determine isomorphic systems, and since they are isomorphic under the identity map, they must in fact be equal.

Some old texts use an apparently weaker version of the axiom of infinity, to wit:

$$\exists x (\exists y (y \in x) \wedge \forall y (y \in x \rightarrow \exists z (z \in x \wedge y \subsetneq z))).$$

This says that $x$ is nonempty and for every element $y$ of $x$ there is another element $z$ of $x$ such that $y$ is a subset of $z$ and $y$ is not equal to $z$. This implies that $x$ is an infinite set without saying much about its structure. However, with the help of the other axioms of ZF, we can show that this implies the existence of $\omega$. First, if we take the powerset of any infinite set $x$, then that powerset will contain elements that are subsets of $x$ of every finite cardinality (among other subsets of $x$). Proving the existence of those finite subsets may require either the axiom of separation or the axioms of couple and union. Then we can apply the axiom of replacement to replace each element of that powerset of $x$ by the initial ordinal number of the same cardinality (or zero, if there is no such ordinal). The result will be an infinite set of ordinals. Then we can apply the axiom of union to that to get an ordinal greater than or equal to $\omega$.
As we know, the axiom of infinity cannot be proved from the other axioms of ZFC if they are consistent (to see that, we can note that ZFC $\vdash$ Con(ZFC - Infinity) and we have to use Gödel's Second incompleteness theorem). But, conversely, the negation of the axiom of infinity cannot be derived from the rest of the axioms of ZFC, if they are consistent.

Thus, ZFC implies neither the axiom of infinity nor its negation and is compatible with either.

Indeed, using the von Neumann universe, we can build a model of ZFC - $Infinity + (\neg Infinity)$. It is $V_ω$, the class of hereditarily finite sets, with the inherited membership relation. Note that if the axiom of the empty set is not taken as a part of this system (since it can be derived from $ZF + Infinity$), then the empty domain also satisfies ZFC - $Infinity + \neg Infinity$, as all of its axioms are universally quantified, and thus trivially satisfied if no set exists.

The cardinality of the set of natural numbers, aleph null ($\aleph_0$), has many of the properties of a large cardinal. Thus the axiom of infinity is sometimes regarded as the first large cardinal axiom, and conversely large cardinal axioms are sometimes called stronger axioms of infinity. Obviously, large cardinal axioms introduce new versions of the infinite which call into question Cantorian relative univocity and, at the same time, could multiply its possible theological meanings and generate, for a facetious mind, so many different concepts of little gods.

# 7 Some mathematical objections to infinity: Finitism and ultrafinitism

Maybe the [M]-model from Laurent Schwartz proves the necessity of the axiom of infinity in set theory in the sense that we cannot construct infinite sets without this axiom.

However, it remains that nothing obliges us to accept the existence of infinite sets in mathematics, just as nothing obliges us to accept the existence of an infinite God in metaphysics<fnref n="8" />.

<footnote n="8">⁸Indeed, there exists a theistic finitism, also known as finitistic theism or finite godism, which is the belief in a deity that is limited (see [Geisler-Watkins 89]). It has been proposed by some philosophers and theologians to solve the problem of evil. Most finitists accept the absolute goodness of God but reject, for example, omnipotence.</footnote>
Indeed, from Aristotle to Kronecker, and beyond, a whole finitist mathematics has
developed.

Obviously, before Cantor and the discovery of the transfinite, all classical mathemat-
ics was finitist. But beyond Cantor, the intuitionists wanted to ensure the foundation
of mathematics by demanding that only finitist methods be used in demonstrations.
From Hilbert to Wittgenstein and beyond, mathematical finitism was expressed in
different ways and today there are several versions, of different degrees.

## 7.1 Classical and strict finitism

Roughly speaking, finitism is a philosophy of mathematics that accepts the existence
only of finite mathematical objects and does not accept the existence of infinite
objects such as infinite sets. While all natural numbers are generally accepted as
existing by finitists, the set of all natural numbers is not considered to exist as a
mathematical object. Therefore quantification over infinite domains is not considered
meaningful. So the mathematical theory often associated with finitism is Thoralf
Skolem's primitive recursive arithmetic.

Historically, the paradoxes encountered by "naive" set theory in its Cantorian devel-
opment (Russell's paradox, Berry's paradox and the Burali-Forti paradox...) were
certainly at the origin of mathematicians' distrust of infinity and their desire to
retreat to a more secure domain.

All agreed about finite mathematical objects such as natural numbers. However
there were disagreements regarding infinite mathematical objects. One position was
the intuitionistic mathematics that was advocated by L. E. J. Brouwer (1881-1966),
which rejected the existence of infinite objects until they are constructed.

Another position was endorsed by David Hilbert (1862-1943): finite mathematical
objects are concrete objects, infinite mathematical objects are ideal objects, and ac-
cepting ideal mathematical objects does not cause a problem regarding finite mathe-
matical objects. More formally, Hilbert believed that it is possible to show that any
theorem about finite mathematical objects that can be obtained using ideal infinite
objects can be also obtained without them. Therefore allowing infinite mathemati-
cal objects would not cause a problem regarding finite objects. This led to Hilbert's
program of proving consistency of set theory using finitistic means as this would
imply that adding ideal mathematical objects is conservative over the finitistic part.
But this. Hilbert's goal turned out to be an impossible task due to Kurt Gödel's
incompleteness theorems. However, by Harvey Friedman's grand conjecture most
mathematical results should be provable using finitistic means and through a work
with Paul Bernays some experts such as William Tait have argued that the primitive
recursive arithmetic can be considered an upper bound on what Hilbert considered
finitistic mathematics.

In her book *Philosophy of set theory*, Mary Tiles (see [Tiles 04]) characterized those who allow countably infinite objects as "classical finitists", and those who do not allow countably infinite objects as "strict finitists". Historically, the written history of mathematics was thus classically finitist until Cantor discovered the hierarchy of transfinite cardinals in the end of the 19th century. In those years, Leopold Kronecker was a strident opponent to Cantor's set theory. A little later, although the author denied it, much of Ludwig Wittgenstein's writing on mathematics has a strong affinity with finitism.

Closer to us, Reuben Goodstein (1912-1985) was another proponent of finitism. Some of his work involved building up to analysis from finitist foundations.

Towards the end of the 20th century, the American mathematical philosopher John
Penn Mayberry (1939-2016) (see [Mayberry 01] seems even more rigorous and devel-
oped a system of finitary mathematics which he called "Euclidean Arithmetic". The
most striking tenet of his system is a complete and rigorous rejection of the special
foundational status normally accorded to iterative processes, including in particu-
lar the construction of the natural numbers by the iteration "+1". Consequently
Mayberry is in sharp dissent from those who would seek to equate finitary mathe-
matics with Peano Arithmetic or any of its fragments such as Primitive Recursive
Arithmetic.

This whole current of thought seems in fact in line with Aristotle, who can be char-
acterized as one of the first finitists, in the sense that he especially promoted the
potential infinity as a middle option between strict finitism and actual infinity. the
last one meaning simply for him an actualization of something never-ending in nature
(in contrast the Cantorist actual infinity where the transfinite cardinal and ordinal
numbers have nothing to do with the things in nature). On the other hand, however,
Aristote (*Physics*, Book 3, Chapter 6) was seeing that the inexistence of the infinite
leads obviously to many problematic consequences: there will be a beginning and
end of time, a magnitude will not be divisible into magnitudes, numbers will not be
infinite... So he maintained the possibility of potential infinity.
## 7.2 Ultrafinitism

In the philosophy of mathematics, *ultrafinitism* (also known as *ultraintuitionism*), is an extreme form of finitism or intuitionism. Though there are various philosophies of mathematics that are called ultrafinitism, a major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers.

Like other finitists, ultrafinitists deny the actual existence of the infinite set $N$ of natural numbers, on the basis that it can never be completed.

In addition, some ultrafinitists are concerned with acceptance of objects in math-
ematics that no one can construct in practice because of physical restrictions in
constructing large finite mathematical objects. Thus some ultrafinitists will deny
or refrain from accepting the existence of large numbers, for example, the floor of
the first Skewes's number, which is a huge number defined using the exponential
function as $\exp(\exp(\exp(79)))$, or $e^{e^{e^{79}}}$. The reason is that nobody has yet calculated
what natural number is the floor of this real number, and it may not even be physi-
cally possible to do so. Similarly, $2 \uparrow\uparrow\uparrow 6$ (in Knuth's up-arrow notation<fnref n="9" />) would be
considered only a formal expression that does not correspond to a natural number.
The brand of ultrafinitism concerned with physical realizability of mathematics is
often called "actualism".

Edward Nelson (1932-2014) criticized the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as 0 and numbers obtained by the iterative applications of the successor function to 0. But the concept of natural number is already assumed for the iteration. In other words, to obtain a number like $2 \uparrow\uparrow\uparrow 6$ one needs to perform the successor function iteratively (in fact, exactly $2 \uparrow\uparrow\uparrow 6$ times) to 0.

Some versions of ultrafinitism are forms of constructivism, but most constructivists view the philosophy as unworkably extreme. The logical foundation of ultrafinitism is unclear; in his comprehensive survey *Constructivism in Mathematics* (1988), the constructive logician A. S. Troelstra (see [Troelstra 88]) dismissed it by saying "no satisfactory development exists at present". This was not so much a philosophical

<footnote n="9">⁹Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth (see [Knuth 76]) in 1976. The single arrow (↑) represents exponentiation (iterated multiplication). For example, $2↑4 = 2^4 = 16$. The double arrow (↑↑) represents tetration (iterated exponentiation). For example, $2↑↑4 = 2^{2^6} = 65,536$. The triple arrow ↑↑↑ represents penta-tion (iterated tetration), etc. The general definition of the up-arrow notation is as follows (for $a \ge 0, n \ge 1, b \ge 0$: $a \uparrow^n b = a[n+2]b$.</footnote>
objection as it was an admission that, in a rigorous work of mathematical logic, there
was simply nothing precise enough to include.

Serious work on ultrafinitism was led, from 1959 until his death in 2016, by Alexander
Esenin-Volpin (1924-2016), who in 1961 sketched a program for proving the consis-
tency of Zermelo-Fraenkel set theory in ultrafinite mathematics. Other mathemati-
cians who have worked in the topic include Doron Zeilberger, Edward Nelson, Rohit
Jivanlal Parikh, and Jean Paul Van Bendegem. The philosophy is also sometimes
associated with the beliefs of Ludwig Wittgenstein, Robin Gandy, Petr Vopfónka,
and Johannes Hjelmslev.

Shaughan Lavine has developed a form of set-theoretical ultrafinitism that is con-
sistent with classical mathematics. Lavine has shown that the basic principles of
arithmetic such as "there is no largest natural number" can be upheld, as Lavine
allows for the inclusion of "indefinitely large" numbers (see [Lavine 94]).

Other considerations of the possibility of avoiding unwieldy large numbers can be
based on computational complexity theory, as in András Kornai's work on explicit
finitism (which does not deny the existence of large numbers)(see [Kornai 03]) and
Vladimir Sazonov's notion of "feasible" numbers.

There has also been considerable formal development on versions of ultrafinitism that
are based on complexity theory, like Samuel Buss's bounded arithmetic theories,
which capture mathematics associated with various complexity classes like P and
PSPACE. Buss's work can be considered the continuation of Edward Nelson's work
on predicative arithmetic as bounded arithmetic theories like S12 are interpretable
in Raphael Robinson's theory Q and therefore are predicative in Nelson's sense. The
power of these theories for developing mathematics is studied in bounded reverse
mathematics as can be found in the works of Stephen A. Cook and Phuong The
Nguyen. However these are not philosophies of mathematics but rather the study of
restricted forms of reasoning similar to reverse mathematics.

## 7.3 Finitism and toposic interpretation

In recent mathematics, we can say that finitism (in the weak sense of not accepting an axiom of infinity) is essentially the mathematics that can be done internal to an arbitrary Boolean topos (at least if one is not also being predicative or construc-tive). For constructive mathematics as usually practised, one goes beyond finitism by positing a natural numbers object.

Although often considered an extreme form of constructivism, finitism in the strong
sense (actually denying the axiom of infinity) can make excluded middle and even the axiom of choice constructively acceptable (and similarly make power sets predictively acceptable). This is because even constructivists agree that these are true in the category FinSet, of finite sets.

As we have seen before, Ultrafinitism is an even more extreme form of finitism, in which one doubts the existence of certain very large natural numbers. The theory of ultrafinite mathematics is well developed by Edward Nelson in Nelson arithmetic and also carried out by the provocative opinions of the Israeli mathematician Doron Zeiberger. Foundational systems such as soft linear logic can also be argued to have an ultrafinitist flavor, and so is paraconsistent logic, since there is a paraconsistent version of the Löwenheim-Skolem theorem, which states that "any mathematical theory presented in first order logic has a finite paraconsistent model."

One definition of finite mathematics is as the mathematics of the category FinSet. This includes the basic arithmetic of natural numbers, since these are the cardinalities of finite sets. This may seem contradictory to the entire conception of "finite mathematics", since the natural numbers form an infinite set. The way out is to state that the natural numbers do not form a set (or class) at all; instead, they are formally defined outside of the set theory via an auxiliary theory like primitive recursive arithmetic. This phenomenon is similar to defining the universe hierarchy of Russell universes or Coquand universes in dependent type theory without a separate type judgment, where the natural numbers used in the universe indices are not elements of a type, but rather formally defined outside of the type theory using some other theory.

An alternative definition of finite mathematics is neutral finite mathematics, i.e. mathematics done internally to an elementary topos (constructively) or Boolean topos (classically), but which does not assume either the axiom of infinity or the axiom of finiteness. There are no natural numbers; instead one works directly with the finite sets in neutral finite mathematics.

Alternatively, one can also attempt to approximate arithmetic in the natural numbers by considering increasing long t-uples of digits - elements of a finite set $D$, via the inclusions:

$$1 \to D \to D \times D \to D \times D \times D \to \dots$$

In particular, in the internal type theory of neutral finite mathematics, since types in the internal type theory are not free variables, one cannot quantify over finite sets,
which means that many concepts in elementary number theory cannot be internally defined in neutral finite mathematics for finite sets.

Unlike the case for division of natural numbers, which can be defined via induction or recursion on the natural numbers, the division of two finite sets cannot be defined via the usual set-theoretic operations on finite sets in neutral finite mathematics. Instead, one needs to add additional set constructors to neutral finite mathematics which state that given finite set $A$ and finite pointed set $B$ with an element $p : B$, one can construct finite sets $A \div B$ and $A\%B$ with bijections:

$$A \simeq ((B \times (A \div B)) + (A\%B)) \quad (B \hookrightarrow A\%B) = \emptyset$$

meaning that it is no longer neutral finite mathematics.

As a result, the analogues of the notions of divisibility relation, divisor, greatest common divisor, least common multiple, prime number, prime factorization, and so forth, cannot be internally defined for finite sets in neutral finite mathematics.

Furthermore, the definition of prime number requires quantification over finite sets, which is impossible in the internal logic since there are no finite set free variables. (The collection of finite sets are infinite, and so internally quantifying over that collection implies that one has an infinite collection in the foundations, contrary to finitism.) From this internal perspective, Doron Zeilberger, despite his opinions, is not exactly finitist since he is able to define prime numbers and work with them (cf. Zeilberger 01), which requires some notion of quantification over finite sets to define finite sets with prime number cardinality, or otherwise the infinite set of natural numbers so that one can internally quantify over them for the definition of prime number.

One can prove that given individual finite sets $A$ and $B$ that there is a bijection $A + B \simeq B + A$, but one cannot prove that there is a bijection $A + B = B + A$ for all $A$ and $B$, because there are no finite set free variables in the internal logic. The same goes for any arithmetic or order-theoretic property for the set-theoretic operations on finite sets. Syntactically, the properties for finite sets are given by inference rules with at least one hypothesis, rather than axioms as would be for the natural numbers:

$$\frac{\text{Aset } p: \text{isFinite}(A) \quad \text{Bset } q: \text{isFinite}(A)}{\text{comm}_{A+B}(p, q) : A + B \simeq B + A}$$
In predicative constructive neutral finite mathematics; i.e. internal to a Π-pretopos
without the natural numbers or a set of truth values, one cannot even define when
a set is a finite set, and thus one cannot do any natural numbers arithmetic in the
guise of finite set arithmetic at all. Even basic arithmetic properties of the natural
numbers like the cancellativity of multiplication do not apply to arbitrary sets in the
absence of excluded middle (see [Swan 18]), and arbitrary sets only form a preorder,
rather than a total preorder.

In homotopy type theory, another possibility is available: what is called π-finite
mathematics, which can be said to be the study of only the *n*-truncated π-finite
types for *n*: N, which have finite homotopy groups, and type operations that preserve
finiteness. Finite sets are just finite 0-truncated tame homotopy types in homotopy
type theory. Certain type operations do not preserve tameness, such as suspensions
and homotopy pushouts, and so would not be part of finite mathematics. This
extends traditional finite mathematics based in set theoretic foundations to more
modern fields such as higher category theory and homotopy theory. Just as the cardinalities
of finite sets are natural numbers (finite set cardinalities), the cardinalities
of these *n*-truncated homotopy types are positive rational numbers.

# 8 Classifying topoi and infinity

Andreas Blass, in an article written in memory of Evelyn Nelson (see [Blass 89]), has shown a connection between classifying topoi and the axiom of infinity. His remarks can be summarized as follows. In the known constructions of classifying topoi for geometric theories, it is always assumed that the base topos *S* satisfies the axiom of infinity. In this article, Blass show that this assumption is necessary. After him, if a certain very simple geometric theory *T* has a classifying topos over *S*, then *S* has a natural numbers object. He also shows, using well-known methods, that the existence of a classifying topos for *T* follows from the existence of an object classifier, i.e., a classifying topos for the one-sorted geometric theory having no non-logical symbols and no axioms, the theory whose models in any category are just the objects of that category. Thus, his main result, which answers a question posed before, can be stated as follows;

**Theorem 8.1.** *Suppose that there exists an object classifier over the elementary topos *S*. Then *S* has a natural numbers object.*

This connection shows that if we admit classifying topoi, then we must inevitably
admit at least the actual infinity of N.
So it seems that we do not get rid of the infinite any more easily in mathematics
than in metaphysics.

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