I am a finitist mainly because the most simple problems when you introduce infinities seem completely unsolvable and even meaningless (particularly CH but also many other problems) but this article by CH intrigued me.
https://www.ams.org/notices/200106/fea-woodin.pdf
In it he explains how second order arithmetic can also be almost as real as first-order arithmetic as long as you assume projective determinacy.
>The next structure, H(?1), is also a familiar one. It is essentially just the structure P(N), N, +, ·, ? , which is the standard structure for Second Order Number Theory. Of course, neither N, +, · nor P(N), N, +, ·, ? is a structure for the language L(ˆ=, ?ˆ), but each is naturally a structure for a specific formal language which is easily defined. There are natural questions about H(?1) which are not solvable from ZFC. However, there are axioms for H(?1) which resolve these questions, providing a theory as canonical as that of number theory, and which are clearly true. But the truth of these axioms became evident only after a great deal of work. For me, a remarkable aspect of this is that it demonstrates that the discovery of mathematical truth is not a purely formal endeavor.
I don't know bros, but if I read up on this and what he says is actually true, I might become a (second-order but no higher) infinitist. It would be a good thing because I love mathematics and most of it I cannot justify from my measly first-order arithmetic framework. The reverse mathematics program suddenly becomes very relevant again.
What does /sci/ think about this?
And where can I find a nice exposition on projective sets which would convince me of Woodin's thesis?
https://www.ams.org/notices/200106/fea-woodin.pdf
In it he explains how second order arithmetic can also be almost as real as first-order arithmetic as long as you assume projective determinacy.
>The next structure, H(?1), is also a familiar one. It is essentially just the structure P(N), N, +, ·, ? , which is the standard structure for Second Order Number Theory. Of course, neither N, +, · nor P(N), N, +, ·, ? is a structure for the language L(ˆ=, ?ˆ), but each is naturally a structure for a specific formal language which is easily defined. There are natural questions about H(?1) which are not solvable from ZFC. However, there are axioms for H(?1) which resolve these questions, providing a theory as canonical as that of number theory, and which are clearly true. But the truth of these axioms became evident only after a great deal of work. For me, a remarkable aspect of this is that it demonstrates that the discovery of mathematical truth is not a purely formal endeavor.
I don't know bros, but if I read up on this and what he says is actually true, I might become a (second-order but no higher) infinitist. It would be a good thing because I love mathematics and most of it I cannot justify from my measly first-order arithmetic framework. The reverse mathematics program suddenly becomes very relevant again.
What does /sci/ think about this?
And where can I find a nice exposition on projective sets which would convince me of Woodin's thesis?
