Cardinalities

To express in second-order logic that a mathematical concept, e.g. natural number, has an infinite extension the notion of Dedekind infinity can be used.39 _{In the more} common, set-theoretical formulation, it can be defined that a set α is Dedekind infinite if, and only if, there is an injection (one-to-one function) fromαto a proper subset of it. To express in second-order logic with standard semantics a sentence ‘INF(F)’, stating that the extension of a predicate ‘F’ is infinite, all we have to do

38_{(Shapiro, 1985), pp. 734–739; (Shapiro, 1991), pp. 110–116.}

39_{To show that a Dedekind infinity is indeed infinite the Axiom of Choice is needed. As this is}

usually assumed in the model theory this does not pose a technical problem here; see (Shapiro, 1991), p. 130, n. 7. A philosophical discussion of the Axiom of Choice cannot be delivered here.

is find an open sentence that expresses the condition for Dedekind infinity. It can be done like this:40

INF(X):

∃f[∀x∀y(f x=f y ⊃x=y)∧ ∀x(Xx⊃Xf x)∧ ∃x(Xx∧ ∀y(Xy⊃f y 6=x))]

‘INF(X)’ is satisfied by a model if, and only if, the class assigned to ‘X’ by the standard model theory is infinite. Now it is easy, of course, to characterise finitude using this:

FIN(X): ¬INF(X)

Note that neither of the formulae contains any non-logical vocabulary. The in- fluence of the assumption of the standard semantics is also particularly apparent here. ‘FIN(X)’ states that there is no such function. The second-order quantifiers have to be assumed to haveall functions in their range. The standard semantics is considered guaranteeing this.

Using an individual constant for zero, ‘0’, and a constant function letter for successor, ‘s’, one can construct first-order formulae that are satisfiable only on infinite domains. The following is such a formula:

∀x∀y(sx6= 0∧(sx=sy ⊃x=y))

So we can, in a sense, also express that a predicateϕhas an infinite extension:

∀x∀y(sx6= 0∧(sx=sy ⊃x=y))∧ϕ(0)∧ ∀x(ϕ(x)⊃ϕ(sx))

These can thus be taken to express the infinity of the domain or of the extension of a predicate in a first-order language (with non-logical constants). This characterisa- tion has the flaw, however, that the negations of the above formulae do notexpress finitude, as will be shown below.

It is well know that first-order logic with identity can express any specific fi- nite cardinality. ‘There are at most three things that are F’, for instance, can be expressed as:

∃x∃y∃z∀w(F w ≡(w=x∨w=y∨w=z))

It cannot, however, be expressed in any first-order language that a predicate has a finite extension without specifying a finite upper bound on its size. This is a corollary of the compactness theorem:

Theorem: Let S be a set of sentences formalised in a first-order language. Let

pϕ(x)_qbe an arbitrary open sentence of that language, containing ‘x’ free. If, for each natural number n, there is a model of S in which the extension of ϕ

has at least n members, then there is a model of S in which the extension of

ϕ is infinite.

Proof: Let ‘c0’, ‘c1’, ‘c2’, ... be individual constants that do not occur in ϕor in any of the sentences that are member of S. Let S0 be the set such that

S0 =df S∪ {ci 6=cj |i < j} ∪ {ϕ(ci)|i∈ω}

If a subset T of S0 has n members then T is satisfiable in any model of S in which the extension of ϕ has at least 2n elements – simply assign different

elements of the extension of ϕ to the new constants that occur in T. T was chosen arbitrarily, so every subset of S0 is satisfiable, and, by compactness,S0

is satisfiable. But any model of S0 is a model of S in which the extension of ϕ

is infinite.41 2

As already mentioned, the second-order expressions of finitude and infinity are by far not the only mathematical concepts that can be defined in pure second-order logic. George Boolos, for example, has shown that even comparatively simple notions like ‘is equinumerous with’ or ‘has an extension that is at least as large as the extension of’ cannot be expressed in a first-order language, but require second-order resources.42

It is possible to express the equinumerousity of two predicates in second-order logic, because a bijection (a function that is one-to-one and onto) between the ex- tensions of two predicates can be expressed:

EQUIN(X, Y):

∃f(∀x∀y((Xx∧Xy∧f x=f y)⊃x=y)∧ ∀x(Xx⊃Y f x)∧

∀x(Y x⊃ ∃y(Xy∧x=f y))) With the help of this we can also easily express being countable (that is, being either countably infinite, i.e. equinumerous with the natural numbers, or finite) in second-order logic. If ‘X’ has a countable extension, then every predicate ‘Y’, such that everything that is ‘Y’ is also ‘X’, has an extension that is either finite, or equinumerous with ‘X’. Thus:

41_{Compare (Shapiro, 1991), pp. 101–102.}

COUNT(X): ∀Y(∀x(Y x⊃Xx)⊃(FIN(Y)∨EQUIN(X, Y))

To see that this expresses countability, suppose that ‘X’ was uncountable, say, ‘is a real number’. There would be a predicate ‘Y’ that applied to countably many real numbers, say, ‘is a real number expressible as a whole number’. Clearly, everything in the extension of ‘Y’ is also in the extension of ‘X’, but ‘Y’ is neither finite nor equinumerous with ‘X’.

Is countably infinite, or is of cardinality ℵ0, i.e. the cardinality of the natural numbers, is thus easily expressed as:

ALEPH-0(X): COUNT(X)∧INF(X)

Having thus characterised the smallest infinite cardinality, it is straightforward to express the next smallest,ℵ1, and the next smallest after that, ℵ2, and so on, using the same idea:

ALEPH-1(X): INF(X)∧ ¬ALEPH-0(X)∧ ∀Y(∀x[Y x⊃Xx)

⊃(FIN(Y)∨ALEPH-0(Y)∨EQUIN(X, Y)] ALEPH-2(X): INF(X)∧ ¬ALEPH-0(X)∧ ¬ALEPH-1(X)

∧ ∀Y(∀x[Y x⊃Xx)⊃(FIN(Y)∨ALEPH-0(Y)

∨ALEPH-1(X)∨EQUIN(X, Y)]

This process can obviously be continued, to characterise any cardinality ℵn, for all

finiten. Indeed, it does not stop there: being of cardinalityℵω (and beyond) can be

expressed, too,43 _{but I will not go into the details here. Instead of chasing higher} up in the hierarchy of cardinalities, let us turn to the most prominent example in

the more recent discussion of second-order logic: the continuum problem.