Set Theory Jottings 12. Zermelo on “definiteness”

Prev TOC Next

In the last post, I mentioned Zermelo’s 1929 paper “On the concept of definiteness in axiomatics”. By this time, people had suggested replacing “definite” with “definable in first-order logic”. Zermelo did not agree with this.

Three points, I think, explain Zermelo’s views.

First, Zermelo never really “got” formal logic. He didn’t grasp the distinction between the meta-theory and the object theory, nor that between syntax and semantics. His correspondence with Gödel shows this. Another example: in the 1929 paper, he complains that the inductive definition of a first-order formula “depends on the concept of finite number whose clarification, after all, is supposed to be one of set theory’s principal tasks.” The inductions of course belong to the meta-theory.

Second, Zermelo approached axiomatization in the spirit of Euclid rather than with the philosophy of formalism. The axioms assert mathematical truths. They are not the arbitrary rules of a game. Hilbert’s famous “tables, chairs, and beer mugs” remark expresses the need to rid the development of any reliance on visual intuition. In much the same way, Zermelo stressed the purely objective character of the Axiom of Choice.

Third, Zermelo found Skolem’s countable model of ZF unacceptable. Recall the resolution of Skolem’s paradox: the power set of ω is not the “true” power set. In a 1930 paper, Zermelo gave his final version of the axioms of set theory. In a footnote to the Separation Axiom, he writes:

Like the replacement function in [the Replacement Axiom], the propositional function 𝔣(x) can be completely arbitrary here, and all consequences of restricting it to a particular class of functions cease to apply from the present point of view. I shall consider elsewhere more thoroughly “the question of definiteness” in connection with my last contribution to this journal and with the critical “remarks” by Mr. Th. Skolem.

The implicit criticism: if you hobble the Separation Axiom by allowing only first-order definable properties, no wonder you get a countable power set!

The 1929 paper offered the following definition of “definite property”. Zermelo writes Dp to say that p is a definite property. Then (changing notation and rewording somewhat, except for the quoted parts):

1. “First, all fundamental relations are definite.”
2. “Definiteness is passed on to composite assertions as follows”
   1. If Dp, then D(¬p).
   2. If Dp and Dq, then D(p∧q) and D(p∨q).
   3. If Df(x,y,z,…) “for all (permissible) combinations of values”, then D((∀x,y,z,…)f(x,y,z…)) and D((∃x,y,z,…)f(x,y,z…)).
   4. If DF(f) “for all definite functors f’’ then D(∀f F(f)) and D(∃f F(f)).
      “Definiteness is passed on to the quantifications.”
3. If P is the system of all definite properties, then “it has no proper subsystem P₁’’ that contains all the fundamental relations and is closed under the compositions listed above.

In clause (II.4), the f ranges over properties (or “propositional functions”), so we have a second-order quantification. Furthermore, we have an implicit circularity: the scope of ∀f is restricted to definite properties, just what we’re in the midst of defining. But clause (III) is perhaps even worse: without a robust set theory, how are we to interpret the quantification over all subsystems of P? Skolem made both these points in his reply.

Zermelo’s 1930 paper “On boundary numbers and domains of sets: New investigations in the foundations of set theory”, gave (as I mentioned above) his final version of the axioms. This includes both Replacement and Foundation, but curiously not Choice—as an axiom. Zermelo writes:

We have not explicitly formulated the “axiom of choice” here because it differs in character from the other axioms […] However, we use it as a general logical principle upon which our entire investigation is based; in particular, it is on the basis of this principle that we shall assume in the following that every set is capable of being well-ordered.

Prev TOC Next