Cantorian Abstraction: Cardinal Numbers as Arbitrary Sets

Cantorian Abstraction:

Cardinal Numbers as Arbitrary Sets

Nicola Bonatti

Munich Center for Mathematical Philosophy Ludwig Maximilan University

Outline

1 Bourbaki Theory of Sets

The ε-operator

The Axiomatic System Cardinal Numbers

2 Definitions of Cardinal Numbers

Zermelo-von Neumann Account Frege-Russell Account

3 Cantorian Abstraction

Frege’s Objection I Frege’s Objection II

1. Bourbaki Theory of Sets

What’s new? An explicit definition of cardinal numbers adopting the resources of the ε-operator − as proposed by Ackermann (1938) and Bourbaki (1968).

The axiomatic system of Bourbaki (1968) Theory of Sets (BK) is a first-order set theory replacing the classical quantifiers ∀ and ∃ with the ε-operator.

The language of BK is divided into terms − which represent the objects of the theory, i.e. sets − and relations − which represent statements about these objects, i.e. propositions.

The ε-operator

The ε-operator is a variable-binding operator which forms terms from open sentences, like εxϕ(x ), which is interpreted as ‘an arbitrary x

such that ϕ(x), if any’. Two axioms:

Ax.1 ϕ(t) → ϕ(εxϕ(x )) (Critical Formula)

Ax.2 ∀x [ϕ(x ) ↔ ψ(x )] → [εxϕ(x ) = εxψ(x )] (Extensionality)

Based on the ε-operator, the classical quantifiers are defined as:

∃xϕ(x) ≡ ϕ(εxϕ(x ))

The Axiomatic System

In BK, sets are introduced through the notion of collectivizing

relation. If R is a relation, and ∃y ∀x (x ∈ y ) ↔ R is a theorem of BK, then we say that R is collectivizing in x . In that case, we introduce an ε-term εy(∀x (x ∈ y ) ↔ R), which names the so defined set.

The set theoretic axioms of BK are:

(Ax.1) ∀x ∀y ((x ⊂ y ∧ y ⊂ x ) → (x = y )) (Extensionality) (Ax.2) ∀x ∀y ∃z(z = x ∨ z = y ) (Pairing) (Ax.3) ∀x ∀y ∃z(y ∈ z ↔ y ⊂ x ) (Power set) (Ax.4) There exists an infinite set N. (Infinity) (Sc.1) ∀v ∃y ∀z(R → (z ∈ y )) → ∀x ∃y ∀z(z ∈ y ↔ ∃v (v ∈ x ∧ R))

The Axiomatic System

For any set X, the relation ∀x (x /∈ X ) is functional in X. Therefore, the term εX∀x(x /∈ X ) corresponding to this functional relation is represented

by the symbol ∅, called the empty-set.

Proof.

Bourbaki (1968) p.72.

Th. II:

The relation ∃y ∀x (x ∈ y ) ↔ x /∈ x is not collectivizing in x.

Proof.

The Axiomatic System

Two main features of BK:

1 The ε-operator is equivalent to the Axiom of Global Choice.

Moreover, in BK the Axiom of Selection and Union is not restricted, and so the Axiom of Choice is derivable in BK (Leisenring, 1969).

2 _{Secondly, BK lacks the Axiom of Foundation, which is independent}

from the other axioms of ZFC, denoted as ZFC−. While the explicit axioms Ax.1-4 clearly belong to ZFC−, Anacona et al. (2014) prove that also the axiom schema Sc.1 is verified in ZFC−.

Cardinal Numbers

Then, Bourbaki (1968) associates with each set t an object, called the cardinal-set of t and denoted |t|.

Def.1

Let t be a set and x a variable not occurring free in t then the cardinal of t is defined as:

|t| =_Df εx(x ≈ t)

Where ≈ is the equivalence relation of equinumerosity (one-to-one correspondence). The idea is that Def.1 can be used to specify a representative element from each equivalence class ≈.

Cardinal Numbers

“There is a certain vagueness in the assignment of a set and a cardinal number, for it is not explained how cardinal numbers and sets are to be understood. In order to fix this vagueness, one can take different ways. [...] The other possibility is that one under-stands for a cardinal number the arbitrary set equinumerous to the given set. [...] The advantage then is that you do not need any special axioms of abstraction, but the relevant formulas become provable. [...] From the axiomatic point of view, the mentioned indefiniteness is not disturbing, since all properties of the cardinal numbers can also be derived in this way.” (Ackermann, 1938)

Cardinal Numbers

Th.1

Given that ≈ is an equivalence relation, Hume Principle follows from Def.1.

Proof.

Let s and t be any two terms (i.e. sets). Since t ≈ t, then ∃z(z ≈ t), and consequently, εz(z ≈ t) ≈ t, i.e. |t| ≈ t (1). Similarly, we get |s| ≈ s (2).

From (1) and (2) and the fact that ≈ is an equivalence relation we obtain |s| = |t| → s ≈ t (3). On the other hand, the fact that ≈ is an

equivalence relation implies s ≈ t → ∀z(z ≈ s ↔ z ≈ t) (4). By the Ax.2 of Extensionality, we obtain ∀z(z ≈ s ↔ z ≈ t) → εz(z ≈ s) = εz(z ≈ t)

(5). Therefore, (4) and (5) yield s ≈ t → |s| = |t| (6). Consequently, from (3) and (6) we get s ≈ t ↔ |s| = |t| (7). HP clearly follows from (7). This proof is adapted from (Leisenring, 1969, pp. 104-105)

2. Definitions of Cardinal Numbers

BK definition of cardinal numbers (Def.1): |t| =Df εx(x ≈ t)

Claim: The BK definition formalizes Cantor (1915) insights on cardinal numbers because:

1 _{It does not depend on a prior definition of ordinal numbers − unlike}

the Zermelo-von Neumann account.

2 _{It is representational, in the sense that |t| is itself a set − unlike the}

Zermelo-von Neumann Account

The Zermelo-von Neumann account is carried out in ZFC and defines the notion of cardinal number based on that of ordinal numbers.

An ordinal numbers is a transitive set well-ordered by membership. Indeed, it is a theorem of ZFC that every set can be well-ordered.

Then, von Neumann cardinal assignment defines |t| to be the smallest ordinal equinumerous with t.

Zermelo-von Neumann Account

What ordinal notions are adopted in Def.1?

I On the one hand, Def.1 is an explicit definition which does not rest on a prior definition of ordinal numbers.

I On the other hand, the Axiom of Global Choice is equivalent to the statement that there is a global well-ordering of the universe V.

Zermelo-von Neumann Account

“The concept of well-ordered set reveals itself as fundamental for the theory of manifolds. That it is always possible to arrange any well-defined set in the form of a well-ordered set is, it seems to me, a very basic law of thought, rich in consequences, and particularly remarkable in virtue of its general validity.” (Cantor, 1915)

Frege-Russell Account

The Frege-Russell account is carried out in either Second order Logic (SOL) or Type theory and defines the notion of ordinal number based on that of cardinal number.

Concerning cardinal numbers, we say that sets t and s are

equinumerous if there is a one-to-one correspondence between them, namely Hume Principle.

Based on the background logical theory of the Frege-Russell proposal, a cardinal is an equivalence class of equinumerous sets. That is to say, the cardinal of a set t is the set of all sets equinumerous with t.

Frege-Russell Account

In order to compare Def.1 with the Frege-Russell proposal, it is useful to consider Hallett (1984) distinction between representational and non-representational accounts of cardinal numbers.

This difference can be better appreciated by pinning down the conditions for the definition of cardinal numbers. There are two minimal requirements:

i) The operation |t| is defined for all sets t.

ii) ∀x∀y (x ≈ y ) ↔ (|x| = |y |).

Yet, Cantor (1915) adds two further conditions:

iii) For every set t, |t| is a set.

Frege-Russell Account

“The cardinal number of a set M is itself a set obtained when one abstract from the nature and order of the elements of M and reflects only on what is common to all sets which are equivalent to M.” (Cantor, 1915)

3. Cantorian Abstraction

Problem: if |t| is a set equinumerous to t, then what are the elements of |t|?

Cantor (1915) abstractionist theory of units:

“Since each individual element m if we disregard its nature be-comes a ‘one’, the cardinal number is itself a definite set composed of nothing but ones which exists in our mind as the intellectual image or projection of the given set M.”

Claim: Def.1 holds the abstractionist thesis of Cantor (1915) while avoiding the psychological objections of Frege (1980).

Frege’s Objection I

Frege (1980) first objection: if cardinal-sets are composed by

indistinguishable units which correspond to the members of the given set, and if two identical cardinal-sets have identical units, then the units of any cardinal-set bigger than 2 would be equal to that of 1!

Therefore, we must explain how a cardinal-set can result from abstracting on the elements of set even though no units of it can be uniquely associated with each element of the base set.

As suggested by Tait (1996), there are two ways by which a Cantorian abstraction process can be carried out:

1 _{The units composing the cardinal-set |t| are obtained from t by the}

Frege’s Objection I

In Def.1 the arbitrary set picked out by the ε-term is specified by the only property of being equinumerous to the base set t. Therefore, according to the BK definition of cardinal numbers the abstraction process should be taken to apply to the elements of a set as a whole. This is remarked by Cantor (1915) himself:

“The addition of ones, however, can never serve for a definition of a number, since here the specification of the main thing, namely how often the ones must be added, cannot be achieved without using the number itself. This proves that the number is to be explained only as an organic unity of ones achieved by a single act of abstraction.”

Frege’s Objection II

Frege (1980) second objection: we must explain how the units within a cardinal-set are indistinguishable from one another even though the units from different cardinal-sets are not.

The units of |t| are indistinguishable because they are member of an arbitrary set picked out by the ε-term, which is equinumerous to t.

Then, based on Def.1 and HP, an arbitrary set x equinumerous to t is co-extensional with an arbitrary set y equinumerous to s, if and only if t and s are equinumerous. But this is not the case for different cardinal sets, which explain why the units of their respective arbitrary sets are distinguished.

4. Explicit vs Implicit Definitions

Neologicist program: adding HP to SOL is sufficient to infer the Peano Axioms, which grounds arithmetical knowledge on logical basis.

HP is an implicit definition: the principle does not explicitly define |t|, but contextually defines it by defining contexts in which it occurs.

Wright (1997) argues that there are abstraction principle which are individually consistent, but not jointly consistent, like:

Explicit vs Implicit Definitions

Claim: The Good Company objection does not arise if someone endorse the BK definition of cardinal numbers, because:

1 _{Def. 1 is an explicit definition by which HP can be deduced. The}

cardinal-set |t| is the definiendum specified by the definiens containing the ε-term.

2 _{The axiomatic system of BK comprehends the Axiom of Infinity, which}

implies that also the set-sized domain interpreting BK is countably infinite.

Ackermann, W. (1938). Mengentheoretische begr¨undung der logik. Journal of Symbolic Logic, 3(2):85–85.

Anacona, M., Arboleda, L. C., and P´erez-Fern´andez, F. J. (2014). On bourbaki’s axiomatic system for set theory. Synthese,

191(17):4069–4098.

Bourbaki, N. (1968). Theory of sets. Springer.

Cantor, G. (1915). Contributions to the Founding of the Theory of Transfinite Numbers. Number 1. Open Court Publishing Company. Fine, K. (1998). Cantorian abstraction: A reconstruction and defense. The

journal of philosophy, 95(12):599–634.

Frege, G. (1980). The foundations of arithmetic: A logico-mathematical enquiry into the concept of number. Northwestern University Press. Hallett, M. (1984). Cantorian Set Theory and Limitation of Size.

Clarendon Press.

Leisenring, A. C. (1969). Mathematical Logic and Hilbert’s & Symbol. London: Macdonald Technical & Scientific.