Friedman on Material Set Theory (ZFC)

Here I repost a message from Harvey Friedman from the FOM mailing list, from 1997. (FOM is freely accessible, so I hope I don't violate any republication conventions here! The original message from Friedman is online here.)

It is connected to one of the criteria I mentioned before, namely Interpretability Strength.

I didn't intend to write this one, but the topic came to mind in reading the postings of those who are, in some way, unsatisfied with the usual set theoretic foundations for mathematics, and strive for a kind of "structuralist" viewpoint - particularly, Pratt and Barwise.

The usual set theoretic foundations is very powerful, coherent, concise, successful, explanatory, impressive, and totally dominating at this time. Taken as a whole, with the major supporting classical developments, it is certainly one of the few greatest acheivments of the human mind of all time.

However, it also does not come close to doing everything one might demand of a foundation for mathematics. At the present time, there is no full blown proposal for scrapping it and replacing it with anything substantially different that isn't far more trouble than its worth. Present cures are far far far worse than any perceived disease.

Now this does not mean that the usual set theoretic foundations might not give way to a better foundations, or might not be altered in some very significant and permanent way. In fact, I can tell you that I work on this from time to time. It just that people should recognize what's involved in doing such an overhaul, and not fool themselves into either

i) embracing something that is either essentially the same as the usual set theoretic foundations of mathematics; or ii) embracing something that doesn't even minimally accomplish what is so successfully accomplished by the usual set theoretic foundations of mathematics.

Now before I remind everybody of some of the most vital features of the usual set theoretic foundations for mathematics, let me state a great, great, great, theorem in the foundations of mathematics:

THEOREM. Sets under membership form the simplest foundationally complete system.

There is one trouble with this result: I don't know how to properly formulate it. In particular, I don't know how to properly formulate "foundational completeness" or "simplest."

Making sense of this "Theorem" and closely related matters are typical major issues in genuine foundations of mathematics. Now before coming back to this, let me summarize the greatest of the usual set theoretic foundations of mathematics.

First of all, set theory is unabashedly materialistic - a perhaps nonstandard word I use to describe the opposite of structuralistic. The viewpoint is that the empty set of set theory has a unique unequivocal meaning independently of context. There is the empty set, and that's that. It doesn't need any context. There is no talk of identifying distinct empty sets because they form the same function.

This materialistic concept of set seems to be very congenial to almost everybody for a while. Thus {emptyset} also has a unique unequivocal meaning independently of context. In fact, one can construct the so called hereditarily finite (HF) sets by the following process:

i) $\varnothing \in HF$; ii) if $x,y \in HF$ then $x \cup \{y\} \in HF$.

This has a clarity and congeniality for most people, without invoking any structuralist ideas.

Now I can already hear the following remark: see, you have used an inductive construction that has not only not been formalized in set theoretic terms yet, but is not even best formulated in set theoretic terms.

Yes, this is true. And yes, there is an idea of inductive construction - at least for the natural numbers - which is not directly faithfully conceived of in purely set theoretic terms. However, look at the costs of scrapping the set theoretic approach in favor of "inductive construction." Can this really be done? I have certainly thought about this, but without success. It is certainly an attractive idea, and we explicitly formulate this:

FOUNDATIONAL ISSUE. Is there an alternative adequate foundation for mathematics that is based on "inductive construction?" In particular, one wants to capture set theory viewed as an inductive construction. If not, one wants to construct a significant portion of set theory as an inductive construction.

Now, instead of scrapping the set theoretic approach in favor of "inductive construction," what about incorporating both? Yes, this can be done in various ways. However, so what? This is only really interesting if one can isolate a small handful of additional ideas that one wishes to directly faithfully incorporate into the prospective foundation for mathematics. Better yet - prove some sort of completeness of this handful.

However, consider the situation in mathematics that was one of the major precipitating factors that made people realize the urgency of foundations. Namely, people were creating all kinds of mathematical concepts - groups, rings, fields, integers, rationals, reals, complexes, division rings, functions of a real and complex variable, series, etcetera. There was no unifying principle as to what is or is not a legitimate construction. Mathematicians do not want to go down that road again, and are comforted by the fact that this matter has been resolved by set theory - even if it does not provide for a directly faithful formalization of the way they actually visualize and think. In summary, there is a danger of the cure being far far far worse than the disease.

Now, coming back to set theory and HF. Obviously, it is congenial and natural to most people to form the set HF. And then there is the natural idea of subset of HF. Then for each natural number n, one can form the $n$-th power set of HF; let's write this as $V_{\omega + n}$.

Let us give the name $V_{\omega + \omega}$ for the universe of all sets that are members of some $V_{\omega + n}$. There are a number of beautiful axioms one immediately writes down about this universe. A small number of them allow for the derivation of lots of others. This is a very coherent and workable system of objects, under epsilon, for a foundation of a very very large portion of mathematical practice. Now I have been very concerned with the following for nearly 30 years:

FOUNDATIONAL ISSUE. What interesting mathematics is missing if one uses $V_{\omega + \omega}$ (with the obvious associated axioms)? Obviously, one does not mean simply that $V_{\omega + \omega}$ itself is missing, since $V_{\omega + \omega}$ is meant to provide ontological overkill. Instead, one means that what mathematical information of an ordinary mathematical character cannot be derived in such a foundation?

Ex: Let E be a subset of the unit square in the plane, which is symmetric about the line $y = x$. Then E contains or is disjoint from the graph of a Borel measurable function.

This cannot be proved in such a Foundation, but can be proved in a somewhat more encompassing foundation. This result is a typical achievement in foundations of mathematics.