4.7. Countability and Completeness

Countable sets

A set A is called countable if |A| ≤ |ℕ|, and countably infinite if |A| = |ℕ|.

Proposition. Any countable set is either finite or countably infinite (the converse is obvious).

Proof. Let A⊂ℕ, and assume A≠∅ (as ∅ is finite).
Let S' the successor function (3.12) of A.
If ∃x∈ℕ, A ⊂ <⃖(x) then |A| ≤ x thus A is finite.
Otherwise, A ⊂ Dom S' (as A is well-ordered), thus |ℕ| ≤ |A|.∎

Actually, in the infinite case, the bijection with ℕ is obtained without use of the Cantor-Bernstein theorem, for the following reason.
Let B = 〈{min A}〉S'. Then B=A.
Indeed otherwise x∈A\B implies B ⊂ <⃖(x) (from the definitions of S' and B), leading to the contradicting consequences that B is finite (|B| ≤ x), and B ⊂ Dom S'.∎

The definition of countability can be rephrased without the axiom of infinity, to mean either finite or countably infinite, the latter meaning the existence of a ground Σ-term algebra structure; thus, countability can be defined as the existence of a ground functional Σ-draft structure.

Without using AC, if there is a surjection f: ℕ ↠ A (or more generally a surjection from any countable set) then A is countable.
Indeed a section of f is given by y↦ min f•(y).

For the same reason, any product of nonempty subsets of ℕ is nonempty.

More statements of the axiom of infinity

Let us extend our list of equivalent statements of the axiom of infinity with the below 2. and 3. where
  1. Existence of ℕ
  2. Existence of a (countable) ground term algebra for any countable algebraic language
  3. Existence of a (countable) model for any consistent first-order theory with a countable language
  4. Existence of an injective Σ-algebra.
Obviously, 2. ⇒ 4. ⇒ 1.
3. ⇒ 4. depends on the claim of consistency of arithmetic, at least its very poor excerpt expressing the concept of injective Σ-algebra. A rigorous interpretation of the question is whether this consistency is provable in Finite Set Theory, which I do not know, but it does not seem very interesting anyway.

Let us refer instead to the intuitive obviousness of this consistency : we can interpret and accept the point philosophically, by a switch of perspective.
Namely, we are implicitly assuming the consistency of our set theory, fancied as describing a universe U (which just does not form "a set" from the same viewpoint). The excerpt of set theory which describes tuples forms a theory describing U as an injective algebra with language made of the tuples definers. Then 3. ⇒ 2. follows by noting, for any language L of algebraic symbols which belong to U, that U is an injective L-algebra. If U is standard, this takes the simplified conventional form LU ⊂ U, making (U, IdLU) an injective L-algebra.

We shall focus on proving 1. ⇒ 2. ⇒ 3. below.

y\x 01234
0 0 1 4 9 16
1 2 3 5 10 17
2 6 7 8 11 18
3 12 13 14 15 19

Countability of ℕ2

A natural bijection between ℕ×ℕ and ℕ is defined by (x,y) ↦ (y < x ? x⋅x + y : y⋅y + x + y).

y\x 0 1 2 3 4
0 0 1 3 6 10
1 2 4 7 11
2 5 8 12
3 9 13
Another bijection b2 : ℕ×ℕ → ℕ is defined as b2(x,y) = Tx+y+y from the sequence (Tn) of triangular numbers defined by 2⋅Tn = n⋅(n+1) (see properties of parity) or equivalently by T0 = 0 ∧ ∀n∈ℕ, Tn+1 = Tn+n+1. This makes ℕ×ℕ a ground term (0,S)-algebra where

0ℕ×ℕ = (0,0)
S(0,y) = (Sy,0)
S(Sx,y) = (x,Sy)

which recursively defines b2-1(z), where x+y is the only n such that Tn ≤ z < TSn.

It would follow that any countable sum of countable sets is countable if we accept the axiom of countable choice ACℕ to justify the existence of a sequence of injections to ℕ from each set. Otherwise the result still holds for finite sums (by the finite choice theorem), and when a rule can be given to pick a specific choice in a systematic way.

It will be left to the reader to define more natural bijections with ℕ for specific cases such as sums A⊔B when either one or both of the countable sets A and B is infinite.

Countability of ℕn

There are multiple ways of defining a bijection between ℕn and ℕ for any n>2, without the ones appearing much more natural than the others.
Some come by taking inspiration from each of both above bijections between ℕ2 and ℕ, to generalize it from n=2 to other n.
More ways are obtained by picking a bijection b2 : ℕ2 ↔ ℕ, and defining the next ones recursively, using either of

ℕSn ≈ ℕn×ℕ ≈ ℕ×ℕ ≈ ℕ
ℕSn ≈ ℕ×ℕn ≈ ℕ×ℕ ≈ ℕ

for each n∈ℕ*, which for n=1 gives back the chosen b2 in either case.
Formally, the first way recusively defines this sequence of bijections bn:ℕn ↔ ℕ for n∈ℕ* by

b1=Idℕ
∀n∈ℕ*,∀u∈ℕSn, bSn(u) = b2(bn(u|Vn),un)

Existence of countable term algebras

We saw that any ground term algebra whose language L has at least a constant and a non-constant, must be infinite. Now, it is more precisely countably infinite if L is countable. Indeed we can define a ground term L-algebra structure on ℕ, which is a bijection from Lℕ to ℕ, as follows.
Split L as C∪D where C is the set of constants, and D is the rest of symbols. Then Lℕ = C∪(Dℕ), while C and D are countable, and Dℕ is countably infinite (as a sum over D of countably infinite sets, across which a choice of bijections with ℕ can be defined). From there, a bijection from C∪(Dℕ) to ℕ can be easily found.
In practice, such a bijection from Lℕ to ℕ is likely to be already a ground term L-algebra structure provided that its inverse, as a sequence covering C∪(Dℕ), starts with an element of C.
Indeed, natural cases of such bijections lead to ∀(s,x)∈Dℕ, ∀i<ns, xi<s(x) which would be contradicted by the smallest element outside MinL ℕ. ∎

The Completeness Theorem

Let us finally prove the Completeness theorem previously commented in 1.1, 1.9, 1.10, 1.B and 1.C.

Theorem. First-order logic has a proving system both sound and complete in the following equivalent senses

Sketch of proof of the first statement (implicitly suggesting a possible form of a proving system, ignoring efficiency issues):

Reduce T to a single-type theory T1 simulating the use of types by relevant unary predicate symbols and axioms, but without requiring all objects to belong to some type. Thus, adding the axiom (∃x, 1) to T1 brings no contradiction (it does not prevent empty types).
This allows to re-write axioms of T1 in prenex form, that is chains of quantifiers applied to quantifier-free formulas, by equivalences adapted from 2.5 and 2.7, such as

(C ∧ ∃Ax, R(x))  ⇔  (∃x, C ∧ A(x) ∧ R(x))
(C ∨ ∃Ax, B(x)) ⇔  (∃x, C ∨ (A(x) ∧ B(x)))
(C ∧ ∀Ax, B(x)) ⇔ (∀x, C ∧ (A(x) ⇒ B(x)))

Interpret the symbol = in axioms as an ordinary predicate symbol, with the axioms of equality [1.11] (those only using the structures named in the language, which suffice, are already in prenex form and do not use ∃).

Replace each occurrence of ∃ in an axiom by the use of an additional operator symbol K, for example

(∀x,x', ∃y, ∀z, A(x,x',y,z)) ⇢ (∀x,x',∀z, A(x,x',K(x,x'), z)).

(The axioms of equality over these are not needed)
Let M be the ground term algebra over the language of operator symbols enriched as just described.
Read the chain of ∀ in each axiom as declaring an axiom schema: one copy of the axiom per possible replacement of its variables by a tuple of elements of M. This gives a propositional theory T2 (its axioms are composed of logical connectives over Boolean variables), whose set of Boolean variables is PM where P is the set of predicate symbols in the theory plus the "equality" symbol.
Observe that T2 is still consistent, as any contradiction in T2 (= finite set of axioms not satisfied by any tuple of Boolean values of their variables) would provide a contradiction in T1, as follows:
From all sub-terms occurring in used axioms of T2, list without repetition all those whose root is a symbol S which was added to the language when converting some axiom, for example ∀x,∃y,∀z,∃u, A(x,y,z,u) of T1, into the axiom ∀x,z, A(x,K(x),z, S(x,z)) in T2.
Successively replace these symbols by variables in T1: for terms S(t0,t1) and S(t0,t2) (where t0, t1, t2 ∈ M), the reasoning in T1 will say
"Let y such that ∀z,∃u, A(t0,y,z,u);
let u1 such that A(t0,y,t1,u1);
let u2 such that A(t0,y,t2,u2);"
then use the contradiction in T2 where y replaces K(t0), u1 replaces S(t0,t1) and u2 replaces S(t0,t2).
Recursively by a chosen bijection from ℕ to PM (or anyway a ground functional Σ-draft structure on PM), add one by one to the axioms, each of these Boolean variables if it is consistent with previously accepted axioms, so that all get values without contradiction.
Then a model is formed by quotienting M by the "equality" relation, and removing the elements with no type. ∎

As this construction depends on multiple choices (between equivalent formalizations of a theory, possible bijections from ℕ to PM, and variations are possible on the method itself), it can give different models, which may be non-isomorphic and even have different truth values of the theory's undecidable statements.

Deduction of the second statement from the first :
T⊢ A
⇔ T ∪ {¬A} is inconsistent
⇔ T ∪ {¬A} has no countable model
⇔ A is true in all countable models of T.  ∎

Skolem's Paradox

Taking the completeness theorem with Cantor's theorem gives this paradox :
Any consistent set theory has a countable model U, including a set P defined by its term "℘(ℕ)" it sees as uncountable.

P being a part of U is meta-countable, but it is interpreted as uncountable, thus any of its meta-bijections with ℕ "does not exist" (has no representative) as an element of U.

Being meta-countable, P must differ from the meta-interpretation of ℘(ℕ) which is meta-uncountable: P contains all objects in U which play roles of subsets of ℕ, but not all meta-subsets of ℕ are so represented.

Indeed, when the powerset axiom declares the class of subsets of ℕ to be a set ℘(ℕ), it only determines ℘(ℕ) as given by the range of subsets of ℕ which happen to exist (be represented) in the current universe, thus remains relative to this universe.

In the other interpretation of what it means for a class to be a set ("strong" version in "Classes and sets in expanding universes"), this would mean that our universe already contains "really all" subsets of ℕ, forming the supposedly "true" set ℘(ℕ) which will stay fixed in any further expansion of the universe. However, this wish cannot be expressed in first-order logic nor any other conceivable mathematical formalism: there is no way to talk about meta-sets that "cannot be found" in this universe but would «exist elsewhere» (in any mysterious bigger universe) to exclude them from there. (Only one aspect of this idea can be formalized as the axiom schema of specification).

Set theory and foundations of mathematics
1. First foundations of mathematics
2. Set theory
3. Algebra 1
4. Arithmetic and first-order foundations
4.1. Algebraic terms
4.2. Quotient systems
4.3. Term algebras
4.4. Integers and recursion
4.5. Presburger Arithmetic
4.6. Finiteness
4.7. Countability and Completeness
4.8. More recursion tools
4.9. Non-standard models of Arithmetic
4.10. Developing theories : definitions
4.11. Constructions
4.A. The Berry paradox
5. Second-order foundations
6. Foundations of Geometry