[FOM] Clarification on Higher Set Theory
sean.stidd at juno.com
Fri Feb 14 09:13:43 EST 2003
Professor Friedman - thank you very much for your response.
It seems that there are three different ways in which one could motivate belief in higher set theory:
1) Use it to show things which mathematicians believe but can't prove and which can't obviously be proven any other way.
2) Use it to prove results which are useful or fruitful in ongoing mathematical reserach. This was the justification I thought you were offering in "Does Normal Mathematics Need New Axioms?" when you wrote: "There are a variety of mathematical results that can only be obtained by using more than the usual axioms for mathematics. For several decades there has been a gradual accumulation of such results that are more and more concrete [which you gloss as 'within the Borel measurable universe', but which (as indeed some of your examples in that paper show) are also connected to things like the real numbers about which mathematicians who are somewhat dubious about set theory in general clearly are already committed to - SS] more and more connected with standard mathematical contexts, and more and more relevant to ongoing mathematical activity." Here I suppse the idea is that higher set theory can "pay its way" in analysis, number theory, topology, etc. by providing theorems which are in some sense useful to that practice.
3) In your FOM response you offer a third possibiilty for such justification: "When such discrete consequences of large cardinals, not provable without them, reach a certain level of "beauty, depth, and breadth", the mathematical community will come to accept them as an important tool for obtaining mathematical results." Putting it this way seems to suggest that even if one doesn't ever get results from higher set theory that satisfy (1) or (2) above, such results could come to be accepted, and higher set theory along with them, as long as these results and/or the proofs which lead to them are sufficiently beautiful or interesting (in the mathematical senses) in their own right. Mathematicians will be attracted to work on higher set theory because of the beauty of the structures, problems, and proofs it makes possible, irrespective of their relevance to existing problems in other mathematical subfields.
What I was hoping for in my original query was a specific result of type (1) or (2) - a higher set-theoretic proof of a result which was already widely believed but unprovable in any other way, or else a higher set-theoretic solution of a problem formulated by analysts within analysis, topologists within topology, etc., which was unsolvable without such resources. I don't know if such a result exists.
Assuming I have not mischaracterized your position you're probably right that results of type (3) will be enough to get higher set theory accepted among mathematicians though; and I think many people also believe that there is a kind of 'unity' in mathematics which makes it inevitable that 'beautiful' results of type (3) will eventually lead to results of type (2), or maybe even (1). I actually believe this myself, but it is an article of faith rather than something I could defend rationally.
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