Re: Paper and slides on indefiniteness of CH

Dear Peter,

You scold Sy:

You take it as a given that the principles of Zermelo set theory
follow from (that is, are intrinsically justified on the basis of) the
iterative conception of set. This is highly contested. See, for
example, the classic paper of Charles Parsons on the topic.

Another, earlier, classic paper on the subject is of course Goedel’s “What is Cantor’s continuum problem” (1947) in which he certainly expresses the view that ZF and more is implicit in the iterative conception of set theory. I would say that Bill Reinhardt’s paper on reflection principles, etc. (1974) is another earlier example supporting that view. Gödel doesn’t really present much argument, but it seems to me that Reinhardt’s discussion is strongly persuasive.

In the phrase “iterative concept of set” the sneaky word is “iterative” when we are speaking of transfinite iteration. In the presence of the ordinals, there is no problem: iterate a given operation along the ordinals. But, in the case of foundations of set theory, the ordinals are not given: they are elements of the universe being generated. So what is the engine of iteration? Maybe better: what are the engines of iteration?

A  reasonable suggestion was Reinhardt’s: given a transitive class $X$ of ordinals, it should be a set if it has some mark that distinguishes it from all of its elements. This leads to the indescribability of the universe: if $V_X$ satisfies $\varphi(A)$ for some formula $\varphi$ and $X \subseteq V_X$ and $\varphi(A \cap V_\alpha)$ fails for all $\alpha \in X$, then $X$ is an ordinal.

As long as $\varphi$ is first-order, this seems unobjectionable as an engine of iteration—and that is more than enough to found ZF.  I grant that when $\varphi$ is higher order, there are problems (viz. concerning the meaning of quantification over subclasses of $X$ when $X$ might turn out to be $V$).

The reflection principal behind Gödel’s examples (Mahlos, hyperMahlos, etc.) is even more modest: an operation on $\text{Ord}$ is total on some $\alpha$.

I think that somewhere in the vast literature in this thread piling up in my email box is the claim by Sy that the existence of cardinals up to Erdős $\kappa_{\omega_1}$ (or something like that) is implicit in the notion of the iterative hierarchy. That does seem to me to be questionable; at least from the “bottom-up” point of view that what can be intrinsically justified are engines for iterating. Sy, as I recall, believes that axioms are intrinsically justified by reference to the universe V that they describe—a top-down point of view. No slander intended, Sy!

Bill