Download AXIOMS AS DEFINITIONS 1. Introduction In the XIX century, the

AXIOMS AS DEFINITIONS
LAURA FONTANELLA
1. Introduction
In the XIX century, the mathematical community was affected by a crisis after the
discovery of the unprovability of Euclide’s fifth postulate. Many attempts to explain
this result were made, and surprisingly two illustrious mathematicians, Poincaré and
Hilbert, despite their very different views, arrived at the same conclusion: the axioms
of geometry were definitions in disguise. The main purpose of this paper is to endorse
the very idea that mathematical axioms in general could be regarded as definitions,
focusing in particular on the axioms of set theory.
While contemporary mathematicians would probably not be reluctant to view certain axioms such as the axioms for groups, topological spaces etc., as definitions, it
is generally believed that the axioms of foundational theories such as set theory or
arithmetic have a different status than the definitional axioms like the ones above,
and that foundational axioms should have a solid, non-conventional, possibly intrinsic, justification. In the words of Feferman [2]:
“When the working mathematician speaks of axioms, he or she usually means those
for some particular part of mathematics such as groups, rings, vector spaces, topological spaces, Hilbert spaces, and so on. These kinds of axioms have nothing to do
with self-evident propositions, nor are they arbitrary starting points. They are simply definitions of kinds of structures which have been recognized to recur in various
mathematical situations. I take it that the value of these kinds of structural axioms
for the organization of mathematical work is now indisputable. In contrast to the
working mathematician’s structural axioms, when the logician speaks of axioms, he
or she means, first of all, laws of valid reasoning that are supposed to apply to all parts
of mathematics, and, secondly, axioms for such fundamental concepts as number, set
and function that underlie all mathematical concepts; these are properly called foundational axioms.”
We will challenge this view and argue that even the axioms of set theory can be
regarded as definitions while still preserving their foundational role. The paper is
structured as follows. In Sections 2 and 3, we discuss Poincaré and Hilbert’s definitional view of geometry. In Section 4 we sharpen Hilbert’s view of axioms in general
to explain in what sense we endorse a definitional conception of axioms, one where
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the axioms cannot be taken individually as intrinsically true assertions, but together
with the other axioms of the theory they define a schema of concepts. In Section 5 we
propose a non-literal interpretation of the axioms of ZFC that leads us naturally to
regard them as a definition of what it means to be a ‘set’. Moreover, in this section
we suggests that the very nature of the concept of set should not be sought in the
idea of a collection of objects regarded as a totality in its own right, but rather in the
possibility of performing specific operations on such collections. Finally, in Section 6
we argue that our definitional account of set theory bring us more naturally to support
set theoretical pluralism, and we claim that, nevertheless, set theory can preserve its
foundational role, this can be sought in the ‘richness’ of the concept of ‘set’ underlying
the various theories.
2. Poincaré’s definitional account of geometry
Let us briefly recall the controversy about Euclide’s fifth postulate and the results
that lead to the development of non-euclidean geometries. Euclide’s Elements is the
first axiomatic presentation of a branch of mathematics; in this invaluable work, geometry is developed through five ‘axioms’ and five ‘postulates’. It is not clear what
meaning Euclide accorded to the terms ‘axiom’ and ‘postulate’, nor what would make
these two notions differ in Euclide’s view, in any case for centuries the Elements were
considered to a compendium of indisputable mathematical truths until the work of
Lobachevsky, Bolyai, Gauss and Riemann challenged the solidity of the fifth postulate.
The fifth postulate, often called ‘parallel postulate’, can be stated as follows:
Given a straight line and a point outside the line, there is one and only one straight
line passing through the point which is parallel to the given one.
In 1829, the Russian mathematician Nikolai Ivanovich Lobachevsky proved that
this statement could be violated with no contradiction while keeping the other axioms
of Euclide’s geometry. The postulate can be replaced by the statements asserting the
existence of more than one straight line (hyperbolic geometry) or no straight line (elliptic geometry) passing through the point and parallel to the given one. For instance,
elliptic geometry can be proven to be consistent by considering a sphere as a model,
where the plane is identified with the surface of the sphere, the straight lines are the
great circles, and points at each other’s antipodes are taken to be equal.
Poincaré’s analysis of these results is quite unassailable (see [8]): the axioms of
geometry do not establish experimental facts, because we do not have experience of
ideal straight lines and points, nor they express a priori knowledge for otherwise
it would not be possible to violate the fifth postulate without contradiction. What
are we to think, then, of these axioms? Poincaré’s answer is ‘they are definitions in
disguise’.
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“The axioms of geometry are therefore neither synthetic a priori intuitions nor
experimental facts. They are conventions. Our choice among all possible conventions
is guided by experimental facts; but it remains free and is only limited by the necessity
of avoiding every contradiction [...] In other words, the axioms of geometry (I do not
speak of those of arithmetic) are only definitions in disguise.” (Poincaré [8])
As definitions the axioms are neither true, nor false.
“What, then, are we to think of the question: Is Euclidean geometry true? It has
no meaning. One geometry cannot be more true than another; it can only be more
convenient [...] because it sufficiently agrees with the properties of natural solids,
those bodies which we can compare and measure by means of our senses.” (Poincaré
[8])
It is important to stress that here Poincaré is only referring to geometry and not
mathematics in general. Today it is not outrageous to consider geometry as a definition, in fact contemporary geometers study different geometries and this is not just a
great fun, it has also valuable applications to physics or even cryptography and coding
theory. On the other hand, geometry is not (or no longer) considered to be a foundational theory, and our goal is to argue that even the axioms of a foundational theory
such as set theory can be regarded as definitions and still preserve their foundational
role. Poincaré was one of the most famous opposers of modern set theory for its use
of impredicative concepts. We will discuss his criticisms in more details in Section
5 and we will appeal precisely on impredicative concepts to endorse our view that a
definitional approach is the best account for the axioms of set theory.
3. Hilbert’s definitional account of geometry
Hilbert’s first reference to axioms as a definition appears in the Grundlagen Der
Geometry [7] where he declares his axiomatization of geometry to be a definition of
the concepts of ‘point’, ‘line’ and ‘plane’. This view is further illustrated in a very
interesting correspondence with Frege, where he declares:
“In my opinion, a concept can be fixed logically only by its relations to other concepts. These relations formulated in certain statements, I call axioms, thus arriving
at the view that axioms (perhaps together with propositions assigning names to concepts) are the definitions of the concepts”. (Hilbert [4])
In order to understand this quote, imagine we were asked to define every geometrical
notion. Then, for instance, we would define the notion of ‘triangle’ as a ‘polygon’ with
three ‘edges’ and three ‘vertices’. The notion of ‘polygon’ is defined from the notion
of ‘plane’, the notion of ‘edges’ can be defined from the notion of ‘line’, and ‘vertice’
is defined from ‘point’. At this point we are supposed to define ‘plane’, ‘edges’ and
‘line’, but if we find a concept χ (or more concepts) that defines these notions, we
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will have to find a definition for χ and so on. Thus, as Hilbert says, we can define a
concept only if we put it in relation to other concepts. We can stop this process, if
we define the notions of ‘point’, ‘line’ and ‘plane’ by describing their relations to one
another through certain axioms.
“If one is looking for other definitions of a ‘point’, e.g. through paraphrase in terms
of extensionless, etc., then I must indeed oppose such attempts in the most decisive
way: one is looking for something one can never find because there is nothing there;
and everything gets lost and becomes vague and tangled and degenerates into a game
of hide-and-sick.” (Hilbert [4]).
Unlike Poincaré, Hilbert’s view of axioms as definitions seems to concern all axiomatic theories, not only geometry. In fact, the axioms of set theory can be explained
by a similar analysis: we can define every mathematical notion, including the notion
of ‘function’ and ‘number’ from the notions of ‘set’ and ‘membership’ which can then
be defined trough a series of axioms, as we will argue in Section 5.
Frege’s main objections to Hilbert’s approach to axioms can be phrased as follows:
(1) Axioms should be assertions, while definitions do not assert anything;
(2) Hilbert’s axiomatization of geometry does not fix unequivocally the basic concepts that are claimed to be defined by the axioms (i.e. ‘point’, ‘line’, etc.).
Hilbert’s response to these criticisms enlighten us about his general view of axioms
and theories in mathematics. Let us discuss each point individually. Frege writes:
“Every definition contains a sign (an expression, a word) which had no meaning
before and which is first given a meaning by the definition. Once this has been
done, the definition can be turned into a self-evident proposition which can be used
like an axiom. But we must not loose sight of the fact that a definition does not
assert anything but lays down something. Thus we must never present as a definition
something that is in need of proof or some other confirmation of its truth. [...] I
call axioms propositions that are true but are not proven because our knowledge of
them falls from a source very different from the logical source, a source which might
be called spatial intuition . From the truth of the axioms it follows that they do not
contradict one another. There is therefore no need for a further proof.” (Frege [4])
By defending a definitional account of axioms, we are precisely making the point
that axioms (even those of foundational theories) are not assertions, and as such they
are neither true nor false per se. This view, however, does not necessarily lead one
to abandon entirely mathematical truth and declare mathematical theories as mere
conventions as Poincaré does for geometry. For instance, Hilbert did not question the
soundness of mathematics, but he postponed the proof of its correctness to a posterior
proof of consistency. Hilbert’s response to Frege is the following:
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“I have been saying the exact reverse: if the arbitrarily given axioms do not contradict one another with all their consequences, then they are true and the things defined
by the axioms exist. This is for me the criterion of truth and existence.” (Hilbert [4])
As C. Frank brilliantly observes in [3], Hilbert was not skeptical about the correctness of mathematics, he simply had higher standard for what counts as a proof of it.
While he was convinced that mathematical experience speaks for the consistency of
axioms, his goal was to show that mathematics could stand on its own and prove its
own consistency without appealing to meta-mathematical justifications.
Nevertheless, today we know from Gödel’s incompleteness results that Hilbert’s
project cannot be accomplished. Therefore, a confirmation of the truth of mathematical theories would require a different source. Whatever could work as a valid
justification for the correctness of a theory, or whether mathematics is simply not a
body of truths, is a very intriguing philosophical problem, yet this is not the object of
this paper. The question inspiring us is simply whether each axiom individually is an
assertion absolutely true or false, or (as we claim) the axioms of a theory altogether
constitute a definition. What we take from both Poincaré and Hilbert view is the idea
that, when writing down the axioms of a theory, we suspend any judgement on the
truth of the axioms, which are simply neither true nor false per se as they are not
assertions, but altogether those axioms define something.
In the next section we discuss Frege’s second criticism to Hilbert’s view of geometry
as a definition and we draw some important conclusions about the very nature of the
definition associated with axioms.
4. Axioms as the definition of a schema of concepts
Frege’s second main objection to Hilbert is that the basic concepts that are claimed
to be defined by his axioms (‘point’, ‘line’, etc.) are not unequivocally fixed. For
instance, a ‘point’ could be a pair or numbers, a triple, a tuple etc. Hilbert replies:
“You say that my concepts, e.g. ‘point’, ‘between’ are not unequivocally fixed; e.g.
‘between’ is understood differently on p.20, and a point is there a pair or numbers.
But it is surely obvious that every theory is only a scaffolding or schema of concepts
together with their necessary relations to one another, and that the basic elements can
be thought in any way one likes. If in speaking of my points I think of some system of
things, e.g. the system: love, law, chimney-sweep... and then assume all my axioms
as relations between these things, then my propositions, e.g. Pythagoras’ theorem,
are also valid for these things. [...] At the same time, the further a theory has been
developed and the more finely articulated its structure, the more obvious the kind of
application it has in the world of appearances and it takes a very large amount of ill
will to want to apply the more subtle propositions of plane geometry or of Maxwell’s
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theory of electricity to other appearances than the ones for which they were meant...”
(Hilbert [4])
Thus in Hilbert’s view, as well as in our view, the axioms do not point out concrete
systems of things, altogether they define a schema of concepts. Roughly, it is the same
difference that runs between an individual and its shape, in the sense that distinct
individuals can have the same shape; the axioms define the ‘shape’ so to speak, not
the individuals. When facing a theory, we propose a certain system of things as a
candidate that may or may not ‘fit’ into the schema of concept defined by the theory,
the axioms determine the conditions for matching with the schema.
At this point we can clarify a couple of aspects that may easily lead to misunderstandings. In mathematical logic, the expression axiomatic definition may refer to
the definition of a symbol that is not included in a given language through a series of
sentences in the language: suppose we have a theory T in a language L and we want
to define a new symbol R (a singular term, a predicate or a function) that is not in
the language L , we may be able to give a definition of R through sentences in the
expanded language L + {R}. This sort of definitions are called implicit definitions.
Beth definability theorem establishes that, in first order logic, every implicit definition
is equivalent to an explicit definition, namely one that depends on a formula of the
original language L . But it should be clear from the passage above, that the kind of
definitions that Hilbert associated to axioms has nothing to do with such syntactical
definitions. With Hilbert, we hold as well that what the axioms are defining is not a
symbol but a schema of concept together with their relation to one another. In fact,
implicit definitions in the sense above would require a background theory that is not
available, for instance, in the case of set theory.
We could also be tempted to consider the axioms as the definitions of a structure in
the sense of Shapiro who, in facts, calls Hilbert’s kind of axiomatic definitions, ‘structural definitions’. Nevertheless, one of the main concerns of the Ante Rem structuralism is categoricity, which doesn’t seem to be any of Hilbert’s worries: in fact, in the
above passage, Hilbert seems to suggest that no system of things is the unique possible
interpretation of the axioms. Thus an analogy between Hilbert and modern Ante Rem
structuralism seems a bit of a stretch. In the view that we want to articulate in this
paper, the axioms do not entail any ontological commitment to a structure. In fact, in
the case of set theory, a structuralist approach (the Ante rem kind) to axioms as definitions would be especially problematic, because neither ZFC, nor second-order ZFC
are categorical1. More importantly, Shapiro replaces the requirement of consistency
1Shapiro,
however, remarks that second-order ZFC is quasi-categorical, namely if M and M ? are
two standard models of second order ZFC, then either M is isomorphic to M ? or else one of them is
isomorphic to an initial segment of the other. Quasi-categoricity is enough for determinate reference,
e.g. the empty set, ω, the continuum etc. are unique up to isomorphism
AXIOMS AS DEFINITIONS
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of a theory with the one of ‘satisfiability’, namely the existence of a model2, but as
Shapiro himself points out, satisfiability requires a background set theory where such a
model can be found, hence it cannot work as a reasonable criterion for set theory itself.
In our view, the axioms of a theory do not entail any ontological commitment to the
schema of concept so defined. The mathematician who faces the axioms of a theory
has his own ontological background and selects from a domain of legitimate objects or
legitimate mathematical notions what he thinks are valid candidates for fitting into
the schema defined by the axioms. By ‘legitimate objects/mathematical notions’ we
mean either mathematical objects that the mathematician truly believe exist in some
abstract reality, or subjective mental images, or more simply well-defined mathematical notions that are not immediate sources of paradoxes (like ‘the set of all sets that do
not belong to them-selfs’), or anything that the mathematician believe is a legitimate
object of the mathematical discourse. The purpose of the axioms is not to establish
‘what is there’ and ‘what isn’t’, nor what is the nature of the legitimate objects in
the mathematician’s domain; what the axioms provide is simply the conditions for
the candidates to match the schema. It is important to stress this aspect because it is
often a source of ambiguity in set theory. In fact, when discussing the legitimacy of
certain axioms such as the axiom of choice or large cardinals axioms, we ask whether
or not a choice function exists, inaccessible cardinals exist and so on. This leaves
us under the impression that the foundational goal of set theory is to detect which
mathematical entities do or do not exist, are or aren’t legitimate mathematical notions. But it would be naı̈ve to think that set theory can dictate the terms for an
ontology of mathematics, and as we will argue in Section 5 the existential quantifiers
in set theoretic formulas should not be intended literarily as affirmations of existence
of mathematical entities.
Analogously, the axioms of a theory cannot enlighten us about the nature of mathematical objects. The mathematician is free to consider his domain of legitimate
notions as part of a platonic reality immutable and independent from human thinking, or as a body of subjective, mental constructions, or he may believe that a system
of concrete physical objects is a legitimate candidate for the theory; to discuss the
nature of mathematical objects is not the object of this paper. In our view, the axioms
can neither corroborate, nor confute his convictions about the ontological or epistemological status of his legitimate mathematical notions, they can only give necessary
conditions for the candidates to satisfy or not satisfy the schema.
2
This is motivated by the observation that in second-order logic a theory can be consistent and yet
not have a model. In fact, there is no completeness theorem, so if T is the conjunction of the second
order axioms of Peano arithmetic and G is a standard Gödel sentence that states the consistency of
T, then by the incompleteness theorem P ∧ ¬G is consistent, but it has no models because every
model of P is isomorphic to the natural numbers, hence G is true in all such models. Therefore,
despite its consistency, P ∧ ¬G fails at describing a possible structure.
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5. On the meaning of existential quantifiers in set theory
We explained in what sense we can take the axioms of a theory to be definitions.
We claimed that this involves a separation between the domain of legitimate mathematical notions and the schema of concept described by the axiom, and that the
real content of the axioms concerns the schema rather than the domain of legitimate
notions (the ‘shape’ rather than the individuals). Let us justify this claim further; our
main focus is set theory, so we will reason within this theory.
We mentioned Poincaré’s skepticism about set theory due to the use of impredicative
concepts. An example is given by the Powerset axiom which is formally stated as
∀x∃y∀z(z ∈ y ⇐⇒ z ⊆ x). This sentence, if interpreted as ‘given a set x, the
set of all subsets of x exists’, it involves a circularity, because for this statement to
make sense, one needs to assume the very possibility of a totality of all subsets of x,
precisely the Powerset of x. For this reason, if one think of this axiom as legitimating
the totality of all subsets of x, one runs into a circle. On the other hand, if one thinks
at that statement as a way of ‘singling out’ something which is already taken to be
legitimate, then the axiom is not problematic. In this sense, the most appropriate
interpretation of the Powerset axiom would be ‘given a set x, the Powerset of x is
a set’. Thus, proving or stating the formal ‘existence’ of something in set theory
means establishing that it has the property of being a set, or that we can label it as
‘set’. In other words, for the axiom to make sense, we need to assume that in the
domain of legitimate mathematical notion that we propose for set theory we already
have something that plays the role of the Powerset of x, but the real content of the
axiom is not to establish the existence of such thing3, but rather to label it as ‘a set’.
Thus, the existential quantifiers act as filters of sets, namely as a way of selecting
sets from other collections that should not be regarded as sets. Zermelo’s original
axiomatisation of set theory seemed to be conceived in the same spirit:
“Set theory is concerned with a ‘domain’ of individuals, which we shall call simply
‘objects’ and among which are the ‘sets’.” (Zermelo [10] 1908)
That is the main reason why we consider the axioms not to be assertions to be
intended in their literal sense. Based on these considerations, we propose the following
non-literal interpretation of the axioms of ZFC.
• (Extensionality) Two sets are equal if they contain the same sets as elements.
• (Pairing) Given two sets a and b, the pair {a, b} (i.e. the collection containing
exactly a and b as elements) is a set;
• (Separation) Given a formula ϕ(x, p~) with sets parameters p~, and given a set
a, the collection of all x in a that satisfy ϕ(x, p~) is a set;
3We
stress again that the axiom tells us nothing about the ontological status of this ‘thing’ and
that this problem is not the object of this paper.
AXIOMS AS DEFINITIONS
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• (Union) Given a set a, the union of a (i.e. the collection of all sets that belong
to some element of a) is a set;
• (Power Set) Given a set a, the collection of all subsets of a is a set;
• (Replacement) Given a function f (x) (defined with set parameters) and a set
a, the collection of all f (x) with x ∈ a is a set;
• (Foundation) A collection of sets that does not have an ∈-minimal element is
not a set;
• (Infinity) Consider all the collections of sets that contain the empty set and
are closed by the operation x 7→ x ∪ {x}, at least one of such collections is a
set;
• (Choice) For every family of nonempty sets which is itself a set, the image of
the choice function is a set.
Under this interpretation, the axioms of ZFC are not regarded as instructions for
constructing mathematical objects or for legitimating those notions, but rather as
a definition of what does it mean to be a ‘set’. What ‘exists’ is really a matter of
what the axioms, taken as a whole, determine to be a ‘set’. The basic set theoretic
operations such as the union, the power set of a set etc. are not really legitimated
by the axioms, instead the possibility of those operation is already given; the axioms
state that the resulting collections can be considered to be ‘sets’. This means that
the collections obtained from these operations become available for the other axioms.
In fact, when we use the existential quantifier over a certain collection, we make the
collection available for the other axioms. Thus being a set means being available for
the other axioms. In this sense, the whole theory ZF (or ZFC) is a definition of ‘set’.
Now, we argue that the very nature of the concept of set should not be sought in
the idea of a collection of objects regarded as a totality in its own right, but rather
in the possibility of performing specific operations on such collections. To support
this claim, let us go back to the very origin of the concept of set, that historians of
mathematics date back to Cantor’s ‘derived point-set’.
“It is a well determined relation between any point in the line and a given set P to
be either a limit point of it or no such point, and therefore with the point-set P the
set of its limit point is conceptually co-determined; this I will denote P’ and call the
first derived point-set of P. (Cantor, as quoted in Ferreirós [1], p.143)”
At this point, Cantor applied the operation of derivation to the derived point-set
P , obtaining ‘the second derived point-set P 00 and reiterated the process. What is
crucial here is that, before Cantor, there was already talk of sets, collections of points
etc. Therefore, Cantor’s original contribution should not be sought in the concept of
collections intended as totalities in their own tight, but rather in the very idea that
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such collections were available for certain mathematical operations such as the derivation. In this spirit, we consider the concept of ‘set’ to be related to the possibility of
applying specific operations to a given collection of objects.
Consider for instance the Axiom of Choice. The Choice function can be regarded
as a perfectly legitimate operation –the fact that AC was implicitly used in many
mathematical proofs even before Zermelo’s explicit formulation suggests that this was
indeed viewed in general as a natural principle (except for a constructivist)– the real
debate is not over the legitimacy of this function, but rather over the idea that the
collection of objects derived from it (i.e. the image of the choice function) can be taken
to be a ‘set’, namely whether it is available for the set theoretic operations definable
from the other axioms.
In this perspective, the question of the legitimacy of new axioms such as large
cardinals axioms should not be phrased in terms of existence or non-existence of
large cardinals, but rather as the problem of whether a certain ‘large’ collection of
ordinals can be made available for the operations defined by the axioms of ZFC (or ZF)
without contradiction. In fact, in second-order logic one can define being ‘(strongly)
inaccessibility’ as a property of classes, then, for instance, the class of ordinals is
inaccessible in this sense (provided this class is accepted as a legitimate mathematical
notion). Thus the problem of the legitimacy of large cardinals axioms concerns the
question of whether or not any inaccessible collection can be taken to be a set, namely
whether one can apply the other set-theoretic operations to such a collection without
contradiction.
6. A multitude of concepts of set
We have outlined our definitional view of the axioms of set theory. In this perspective, a single axiom is not an assertion inherently true or false, but the whole
system of axioms is a definition of the concept of ‘set’. It follows that by adding or removing one or more axioms we change the concept defined. Accordingly Hilbert wrote:
“[...] to give a definition of a point in three lines is to my mind an impossibility, for
only the whole structure of axioms yields a compete definition and hence every new
axiom changes the concept.” (Hilbert 29 Dec. 1899, as translated in [4]).
So, in particular, the concept of set defined by the theory ZF is different than the
one defined by the theory ZFC, or ZF+V=L, or ZF + ∃κ(κ is measurable) etc. Now,
a definitional view point on axioms does not necessarily leads us to embrace a wild
formalist view of mathematics considered to be a meaningless game where random
theories are investigated together with their logical consequences. We cannot rule
out, for instance, the possibility of an inter-subjective or innate concept of set that
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would make some set theories ‘truer’ than others to the extent that they conform to
the intended concept.
Nevertheless, the panorama of theories defended nowadays leaves little hope for a
universal agreement on a unique concept of set. Conflicting intuitions are behind the
most promising enrichments of ZF: on the one hand, we have the idea that sets must
be obtained by a cumulative process, namely at each stage we throw in ‘the basket of
sets’ only those collections that are obtained from the previous stages with operations
that are definable in a somehow ‘canonical’ way – it is the case, for instance, of V=L
or V=Ultimate L–, on the other hand we have more ‘liberal’ theories, such as Forcing
Axioms, where roughly anything that can be forced by some ‘nice’ forcing is in V, i.e.
is a ‘set’.
A natural consequence of this pluralism of concepts of sets is that, from our definitional perspective, the continuum hypothesis is an inherently vague question, as
the very meaning of the continuum c changes over the theories. Indeed, we may even
agree on what is a real number (or a collection of natural numbers), but which of
those reals are ‘sets’ depends on the concept of ‘set’ considered. Thus for instance,
in the framework of Forcing Axioms, all the reals that can be forced by nice forcing
notions are ‘sets’, thus the continuum is quite large and, not surprisingly, CH fails
(the strongest forcing axioms imply c = ℵ2 ). On the contrary, the concept of ‘set’
defined by the theory ZF+V=L is more restrictive, thus only few reals are sets in this
theory, and in fact CH holds.
Now, if set theory is not a body of absolute truths, but a mere definition of some
concept, it is natural to wonder in what relies its foundational role. The answer may
be sought in the richness or abstractness of the concept defined, as all the basic mathematical notions such as groups, vector spaces, even numbers can be regarded as sets
and are passible of the set theoretic operations definable from the axioms of the theory
ZF.
This line of thoughts brings us to consider a natural motivation for large cardinals
axioms. As pointed out by Steel,
“The language of set theory as used by the believer in V=L can certainly be translated into the language of set theory as used by the believer in measurable cardinals,
via the translation ϕ 7→ ϕL . There is no translation in the other direction.”(Steel [2])
In our terminology, the concept of set carried by V=L can be expressed in the theory of measurable cardinals, while the converse seems to be prima facie impossible.
Thus, the notions of ‘set’ underlying the theory of large cardinals is more expressive
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than V=L and, in this sense, it better accomplishes its foundational goal.
On the other hand, Hamkins presented a serious challenge to this argument. In [6]
he shows that
“even if we have very strong large cardinal axioms in our current set-theoretic universe V, there is a much larger universe V + in which the former universe V is a
countable transitive set and the axiom of constructibility holds.”
Thus, even the axiom of constructibility is reach enough to allow us to talk about
the concept of sets in the sense of large cardinals within a model of V = L.
Therefore, we should conclude that the natural outcome of our definitional perspective is pluralism, which in contemporary set theory is represented by the ‘multiverse
conception’ of which Hamkins is the main supporter [5]. This can be described as
the view that there is no absolute background set-theoretic universe in which every
set-theoretic question has a definite answer (the opposite view is usually called ‘the
universe view’). However, it is important to clarify the following aspect: some set
theorists endorse the multiverse view as an extreme form of platonism where not just
one, but many universes exist as an independent reality; in our view, on the contrary,
this is intended simply as a pluralism of concepts of sets (or schema or concepts of sets)
to which we do not associate any metaphysical (nor physical) realm, or universality.
7. Conclusion
In conclusion, we have sharpened Hilbert’s claim that axioms are the definitions
of a concept. Then, we have proposed an interpretation of the axioms of ZFC where
the existential quantifiers are intended as filters of sets, namely as ways of singling
out sets from collections that are not to be considered to be sets. This naturally
leads us to regard set theory as a definition. Furthermore, we have argued that the
very nature of the concept of set should not be sought in the concept of a collection
of objects regarded as totality in its own right, but rather in the idea that certain
collections are passible of specific operations definable from the other axioms. We
have observed that the concept of set so defined changes if one adds or remove one
or more axioms from the theory. This leads to a pluralism of concepts of sets varying
with the theories. After considering some of the most popular new theories of sets, we
have outlined the basic ideas underlying these theories and we have argued that there
is little hope for the possibility of a universal concept of sets. Finally, we claimed
that despite pluralism, set theory can still cover a foundational role, and this is to be
sought in the ‘richness’ of the concept of set underlying the theory, which embraces
all mathematical notions.
AXIOMS AS DEFINITIONS
13
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