[FOM] Clarification on Higher Set Theory

[Harvey Friedman]

>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. ...
>
>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." ,,,
>
>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.

"Higher set-theoretic solutions of problems formulated by analysts 
within analysis, unsolvable without large resources" does exist. At 
least, they exist if you include among "analysts" those working with 
concepts from descriptive set theory. The first example, perhaps, is 
"uncountable co-analytic sets have perfect subsets". Later examples 
relate to the higher regions of the projective hierarchy.

More recently, I showed that certain statements involving Borel 
measurable sets/functions are such examples, and these statements 
were first formulated and written about by functional analysts at 
Paris VII (who actually proved them using large cardinals for 
analytic and coanalytic sets/functions). See Borel Selection under my 
name at the preprint server 
http://www.mathpreprints.com/math/Preprint/show/

CAVEAT: In the examples above, one can solve the problems without 
large cardinals, but with the axiom of constructibility, where the 
answers go the opposite way. This does not happen in the case of 
Boolean relation theory, discussed below. END.

In the long run, whether or not the problems shown to require large 
cardinals were formulated earlier is not decisive, and should become 
a side issue for reasons indicated below. Nevertheless, if one sticks 
to that criteria, then we have a few such examples involving Borel 
sets/functions, but we don't have such examples that are more 
concrete than that. Since mathematics is so preoccupied with contexts 
that are more concrete than that, it is crucial to see what can be 
done at the more concrete levels.
>
>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.

If a body of new mathematics such as Boolean relation theory acquires 
a certain level of beauty, depth, and breadth, then it will become 
accepted as a legitimate area of mathematics in its own right. This 
is what I anticipate will happen with Boolean relation theory. An 
additional factor is that Boolean relation theory can be developed in 
virtually any mathematical context, and so the development of Boolean 
relation theory is expected to draw from existing results in many 
areas of mathematics.

I would not say with any confidence that this would go the other way 
- i.e., Boolean relation theory being used to obtain results far 
removed from Boolean relation theory.

However, Boolean relation theory and related investigations can be 
seen to represent a new level of ambition for mathematical 
investigations - namely, to analyze all statements of a certain 
natural kind, in a tremendous variety of contexts. It is this kind of 
search for such complete information in concrete contexts that seems 
to require large cardinals.

The development of new mathematical theories of interest in their own 
right does happen from time to time, and the best of these come to be 
recognized as new fields.