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At effects of foundations on “real” mathematics I’ve put in the example of Fermat’s last theorem as being potentially derivable from PA, and pointed to two articles by McLarty on this topic.
(Edit: the naive wikilink to the given page breaks, due to the ” ” pair)
The problem with minimalistic approaches to foundations is always that they require much harder paths for proving anything more involved. Of course, it can be fun for understanding such points for easy/simple things, but when something is really complex, than the subject usually grows more interesting and one considers it a content beyond any particular choice of foundations. It is hard to expect that redoing something as complex as Fermat would be really proportionally insightful for a person in foundations.
I don’t think that the particular example of FLT is useful except as an exercise before the general ideas are understood, or as an example for sceptics unwilling to learn the general ideas. The point should be (if it is in fact true) that the methods used to prove FLT are obviously valid in these weak foundations.
I guess my motivation was to show that as far as foundations go, you can do a lot with little. Like Friedman (wow, I didn’t think I’d ever say that), me picking on a particular theorem in a particular paper in a particular journal just pins down the conjecture. The fact it uses a lot of seemingly complicated machinery helps to show that the logical strength of that machinery is far less than what the set theorists like to promote as standard. Perhaps the example should have been called something like ’the non-effects of foundations’. At least, I would like to keep McLarty’s (meta)result that all SGA works in MacLane set theory with one universe, if not the application of this program of paring down foundations to find what is necessary for FLT.
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