Martin's "Philosophical Issues about the Hierarchy of Sets"

Question

Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the hierarchy of sets".

Abstract: I will discuss some philosophical questions about the cumulative hierarchy of sets, its levels, and their theories. Some examples:

(1) It is sometimes asserted one cannot quantify over everything. A related assertion is that each of our statements about the universe of sets can from a different perspective be seen as a statement about some Va. Thus the class-set distinction is really a relative one. Does this make sense? Is it right?

(2) Is the first order theory of V determinate? Does every sentence have a truth value? Are there levels of the hierarchy whose first order theories are indeterminate? If so, what is the lowest such level? What about L and the constructibility hierarchy?

(3) There are many examples of proofs of a statement about one level of the hierachy that use principles about a higher level. Under what conditions and in what sense do these count as establishing the lower level statement?

I will discuss these questions mainly from a viewpoint that takes mathematics to be about basic mathematical concepts, e.g., those of natural number, real number, and set.

I am highly interested in learning how these questions might be answered (as you may problably know from previous questions of mine here in MO), so I would be grateful if anyone could give any information in this respect, especially for those questions of 1 and 3 (I am afraid it is almost impossible to do justice to 2 in a few lines).

Answer 1

Of course there are no universally agreed-upon answers to these philosophical questions, and if you are interested in Martin's views specifically, then I suggest that you read his articles. Meanwhile, allow me simply to explain a few of the issues arising in the specific questions you mention.

• "One cannot quantify over everything." This is a reference to the predicative/impredicative debate in the philosophy of set theory. One of the objections to the replacement and collection axioms is that they are used to describe sets by means of properties of a totality of which they themselves are a member. That is, you define a subset of $B=\{a\in A\mid \varphi(a)\}$, but $\phi(a)$ may be a very complicated property that quantifies over the entire universe, referring to objects and properties of objects, including $B$ itself. But also, it can be a reference to the cumulative view of set theory as building up more and more sets in a process that is never completed, and in this case, it may not be sensible to form sets by means of properties holding in the entire universe, as though it were completed.

• "Each of our statements about the universe of sets can from a different perspective be seen as a statement about some $V_\alpha$." The Levy reflection theorem shows that for any assertion $\sigma(x)$, there is an ordinal $\alpha$ such that $\sigma(x)$ is true if and only if it is true in $V_\alpha$, for any $x\in V_\alpha$. That is, $\sigma$ is absolute between $V_\alpha$ and $V$. Going a bit beyond this, consider the theory denoted "$V_\delta\prec V$", which asserts, in the language with a constant for $\delta$, that $\forall x\in V_\delta\, [\varphi(x)\iff \varphi^{V_\delta}(x)]$. This is the scheme asserting that $V_\delta$ is an elementary substructure of the universe. Although some set theorists are surprised to hear it, this scheme is equiconsistent with ZFC, and any model $M$ of set theory can be elementarily embedded into a model of this theory. (This is done by a simple compactness argument; one writes down the theory $V_\delta\prec V$ plus the elementary diagram of $M$, and observes that the reflection theory shows that it is finitely consistent.) Finally, note that in a model of $V_\delta\prec V$, every sentence can be viewed as an assertion about $V_\delta$, rather than about $V$, since they have exactly the same theory.

• "Is the first order theory of $V$ determinate?". This question is asking whether there is a fact of the matter in regard to our set-theoretic questions. For example, does it make sense to say that there is ultimately an answer to the question of whether the Continuum Hypothesis is really true? Or whether large cardinals exist? This question is connected in my mind with issues about whether there is a unique structure that we are investigating when we do set theory---the universe $V$ of all sets---or is there instead a multiverse of possibilities? In other words, is there a final truth of the matter in set theory, or is set theory instead something more like geometry, having a plethora of diverse Euclidean and non-Euclidean worlds? In the slides for my talk at the same conference, I explore the multiverse view in detail.

• "Are there levels of the hierarchy whose first order theories are indeterminate? What is the lowest such level?" Some set theorists may view questions about the Continuum Hypothesis to be a source of indeterminateness, in the sense that there is no fact of the matter about CH. But CH is a statement expressible in $H_{\omega_2}$, or alternatively in $V_{\omega+2}$. Martin is asking whether we might expect indeterminateness at lower levels. In his talk at the workshop you mention, I recall him saying that he found it unacceptable to think that there would be indeterminateness arising at the level of $V_\omega$, and that arithmetic truth was absolute in some very strong sense.

• "There are many examples of proofs of a statement about one level of the hierachy that use principles about a higher level." This is referring to the fact that mathematicians routinely use higher level objects in order to make conclusions about lower level objects. For example, one might use infinite objects (such as automorphisms of field extensions) in order to make conclusions about finite objects, or very large function spaces or ultrafilters in order to make conclusions about a lower level object. Part of Martin's point was the philosophical concern that if there is indeterminism about features of the higher level objects, then they might seem unsuited for this purpose.