The Continuum Hypothesis

One of the most notorious problems in set theory is that of the continuum hypothesis. It is known that the continuum, i.e. the cardinality of the real numbers, is the cardinality of the powerset of the natural numbers, i.e. of a countable infinite set. Recall that the powerset of a set α is the set that contains all and only the subsets of α. Generally, if a set has the cardinality κ, its powerset has the cardinality 2κ. We say that countable infinite sets are of cardinality ℵ0, and so the continuum has the cardinality 2ℵ0_{. The question thus arises: is the continuum the smallest infinite}

cardinality that is larger thanℵ0, or are there uncountable sets that are smaller than the continuum? This question is known as the continuum problem. Georg Cantor conjectured that the size of the continuum was indeed the next cardinality afterℵ0: this is his continuum hypothesis.

Formally, it can be expressed as:

ℵ1 = 2ℵ0

Cantor’s formulation of the continuum hypothesis is that every subset of the set of the real numbers is either countable or can be bijected with the reals. Kurt G¨odel showed in 1938 that the continuum hypothesis is consistent with first-order ZFC.44 Paul Cohen managed to prove in 1963 that its negation is consistent with first-order ZFC too.45 _{It was thus established that the continuum hypothesis is independent of}

44_{(G¨odel, 1938)}

first-order ZFC. Indeed, Chuaqui has shown that it is also independent of second- order ZFC.46

The continuum hypothesis can be generalised to the thesis that for every cardi- nality the smallest larger one is reached by applying the powerset operation to it. Formally:

ℵα+1 = 2ℵα, for any ordinal α

This is the generalised continuum hypothesis. Note that this conjecture is not re- stricted to finite ordinals. It is meant to apply all the way up. The generalised continuum hypothesis is independent of ZFC, too, as is also shown by G¨odel and Cohen.

We have seen above that the different cardinalitiesℵn, forn∈ωcan be expressed

in pure second-order logic; and it was said that indeed all cardinalities ℵα, for any

ordinalα, can so be expressed. We can, indeed, also express being of the size of the continuum.47 _{In order to do so, notice that for a continuum-sized set X there is a} countably infinite set S and a binary relationR with X as its domain and S as its range, such that (i) for eachY that is a subset of S there is an x inX such thatY

is{y|Rxy}, and (ii) for everyx andy inX, if{z|Rxz}={z|Ryz} thenx=y. R

is hence a bijection betweenX and the powerset of S. The following second-order formula expresses being continuum-sized in accordance with these considerations:

CONTINUUM(X):

∃S∃R[ALEPH-0(S)∧ ∀x∀y(Rxy ⊃(Xx∧Sy))

∧ ∀Y(∀y(Y y⊃Sy)⊃ ∃x∀y(Rxy ≡Y y))

46_{(Chuaqui, 1972); see also (Weston, 1977).}

∧ ∀x∀y([Xx∧Xy∧ ∀z(Rxz≡Ryz)]⊃x=y)]

This definition in effect says that being continuum-sized is being bijectable with the powerset of a countably infinite set. So now we can express the continuum hypothesis in pure second-order logic. Call this second-order sentence ‘C’.

(C) ∀X[ALEPH-1(X)≡CONTINUUM(X)]

C not only expresses the continuum hypothesis, but is also linked with it in a much stronger way. In the standard model theory, the range of the second-order variables is the powerset of the domain. If the domain is countably infinite, the range of the second-order variables is continuum-sized. What the cardinality of this is, depends, thus, on what way the continuum hypothesis is decided. Moreover, whenever the domain is infinite (countably infinite or larger), it will depend on the continuum hypothesis whether any given predicate whose extension is continuum-sized has the cardinalityℵ1or not. Should the continuum hypothesis turn out to be true, however, any extension of a predicate that is continuum-sized is of size ℵ1 in every model – there just cannot be any model in which a predicate is assigned a continuum-sized set that is notℵ1, as no such sets exist if the continuum hypothesis is true. Hence,

C will be true in all models, i.e. a validity of the standard semantics of second-order logic. To put it in a different way: the standard semantics of second-order logic declares C to be a logical truth if, and only if, the continuum hypothesis is true. This seems hard to swallow to many.

Before we turn to the criticism connected to the case of the continuum problem, however, it is worth adding some more details surrounding the issue. One point is that it is notthe case that ‘¬C’ is a true in all models if the continuum hypothesis

is false. ‘¬C’ is trivially true in every countable model, i.e. it is true irrespective of the truth or falsity of the continuum hypothesis. There is another sentence, call it ‘D’, however, that is a validity of the standard semantics of second-order logic if, and only if, the continuum hypothesis is false:

(D) ∀X[ALEPH-1(X)⊃ ¬CONTINUUM(X)]

Neither C, nor D, nor their negations, are theorems of the deductive system of second-order logic.48 Thus, we have here an example for the incompleteness of second-order logic with respect to the standard semantics: one of C and D is a logical truth according to the standard semantics that cannot be proven in any standard deductive system of second-order logic.

It does not stop there. There is a similar pair of sentences, one of which is declared a logical truth if, and only if, the generalised continuum hypothesis is true. Another second-order sentence – and this also is a sentence that contains only logical vocabulary – asserts the existence of inaccessible cardinals, and is declared a logical truth by the standard semantics, if, and only if, there are inaccessible cardinals.

The examples mentioned so far all use the expressive resources of second-order logic to express these statements directly. There is a trick, however, whereby in generalalltruths of set theory49 _{can be turned into second-order sentences that in a} way correspond to them. The recipe is the following: take the second-order axioms of ZF (or ZFC, if you will) in their canonical formulation with ‘∈’ as only non-logical constant; take their conjunction – let us call it Z. Let P be the true ZF sentence in question; form the conditional ‘Z ⊃ P’. Replace every occurrence of ‘∈’ by a

48_{(Shapiro, 1991), p. 105.}

two-place predicate variable in this conditional and bind that variable with a prenex universal quantifer. The resulting generalised sentence will in all cases be true in all models, as follows from the quasi-categoricity50 of second-order ZF.

This method works for second-order arithmetic and second-order real analysis just as well as for set theory. All of the truths will, when conditionalised to the conjunction of their axioms and generalised, turn into validities of the standard se- mantics of second-order logic. To borrow Ignacio Jan´e’s metaphorical description of the situation, second-order logic, in a sense, “has all the solutions to the problems of ordinary mathematics, but it keeps them to itself”.51 _{None of the mathemati-} cal truths that are unprovable in their respective theories will be provable in their conditionalised and universally generalised form in second-order logic, on pain of contradiction (provided only that the deductive system of second-order logic is con- sistent). If there were a sentence amongst them that is a theorem of second-order logic, one could within the second-order theory instantiate this theorem, and dis- charge the antecedent with the axioms. This would prove the respective statement in the second-order theory, contradicting the assumption that this is not possible. For the case of arithmetic the mentioned technique will play an interesting role later on, in chapter 8. Let us now turn to a critical discussion of the mentioned results.