Thanks for your clear and prompt reply (more below).
On Fri, 5 Sep 2014, Solomon Feferman wrote:
In answer to your questions below, it seems to me that my work has
philosophical significance in several ways. First, it shows that the reach
of Quine’s (and perhaps Putnam’s) indispensability argument is extremely
limited (for whatever that’s worth). Secondly, I believe it shows that one
can’t sustain the view from Galileo to Tegmark that mathematics (and the
continuum in particular) is somehow embedded in nature. Relatedly, it does
not sustain the view that the success of analysis in natural science must be
due to the independent reality of the real number system.
This explains the value of your reduction to the system W quite clearly. Thanks.
But my question was intended to be much broader. Leaving physics aside, what do we gain by calculating in which subsystems of ZFC various statements are provable, given that we already know that they can be proved in ZFC? Is there just some “principle of minimal resources” involved or is the point that one gains new mathematics and a new understanding of the statement by doing this? The latter seems quite reasonable to me, but the former strikes me as a remnant of the “paradox-scare” in set theory, where resources were minimised for the sake of consistency.
A related question: Is it possible that a statement has a perfectly natural proof with a clear conception behind it but when you succeed in proving it in a weaker system the naturality and clear conception are lost? Given a choice between a “natural” proof using say ZFC + Inaccessibles versus an unnatural one in 4th order arithmetic I surely would prefer the former, unless the latter generates a new understanding. Maybe you are familiar with examples that illustrate what I am talking about.
My results tell us nothing new about physics. And indeed, they do not tell us that physics is somehow conservative over PA. In fact it can’t because if Michael Beeson is right, quantum mechanics is inconsistent with general relativity; see his article, “Constructivity, computability, and the continuum”, in G. Sica (ed.) Essays on the Foundations of Mathematics and Logic, Volume 2 (2005), pp. 23-25. It just tells us that the mathematics used in the different parts of physics is conservative over PA.
Finally, to be “quite happy with ZFC” is not the same as saying that there is a good philosophical justification for it.
But there are good philosophical justifications for powerset independent of any platonistic view. The powerset axiom is intrinsic in the concept of set, as expressed via the maximum iterative conception. There is also overwhelming extrinsic evidence for it due to its fruitfulness in mathematics. So I cannot imagine disputing the truth of the powerset axiom, unless one wants to change the concept of set or limit what one takes to be “mathematics”. And nothing I have said demands any ontology (you can have a concept of set without it and many distinct “mental pictures” of the inherently vague continuum), so I don’t understand what you have in mind.
Perhaps you are indeed “limiting” what is meant by mathematics and by so doing somehow erasing the fruitfulness of powerset; how can that be done? Note that any “reductionist” approach for converting proofs using powerset to proofs without it falls victim to Goedel’s notion of “verifiable consequence”, thereby again providing extrinsic evidence for the powerset axiom!
All the best,