Timeline for How dangerous are set-size assumptions?
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| Jun 22, 2019 at 0:39 | comment | added | Timothy Chow | @TimCampion : 1-consistency is a standard technical term meaning that all $\Sigma_1^0$ consequences are true, or if you like, that every Turing machine that it claims halts really does halt. So it's a weak form of your "arithmetic consequences are true." Most of if not all of Harvey Friedman's "necessary uses of large cardinals" are combinatorial statements that are equivalent to the 1-consistency of a large cardinal axiom (and are provable by a slightly stronger large cardinal axiom). I know people who are willing to accept the combinatorial statement but not the large cardinal axiom itself. | |
| Jun 21, 2019 at 20:27 | comment | added | Tim Campion | @TimothyChow I'm not sure what you mean by "1-consistent" -- do you mean that the theory doesn't prove all arithmetic sentences? But then it follows that the theory is consistent, so 1-consistency comes out the same as consistency! I can see how there would be space to believe that large cardinals are consistent without believing that they're true. What I would find exotic would be the view that the arithmetic consequences of large cardinals are true, but the large cardinals themselves are not. | |
| Jun 21, 2019 at 20:17 | comment | added | Timothy Chow | @TimCampion : I've met people who are comfortable accepting the 1-consistency of a large cardinal axiom but not the large cardinal axiom itself. Roughly speaking, they're suspicious of set theory and so don't want to endorse an actual set-theoretic axiom, but they're comfortable with arithmetic and are willing to accept "inductive evidence" that a set-theoretic axiom is 1-consistent. I don't know that anyone has tried to make this argument formally, though. | |
| Jun 20, 2019 at 13:08 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jun 20, 2019 at 3:41 | comment | added | Tim Campion | That statement makes sense if you don't read too much into it: if a statement is provable in ZFC, then it's standard mathematical practice to regard it as true. So the statement isn't that different from saying "Goodstein's theorem is a theorem of ZFC but not of PA". (Of course, Goodstein's theorem can presumably be proven using less than the full strength of ZFC, but that's beside the point.) | |
| Jun 20, 2019 at 0:37 | comment | added | Pace Nielsen | Regarding your second-to-last paragraph, on "truth" in mathematics, how would you interpret the wikipedia article on Goodstein's Theorem, as it asserts that this is "a true statement that is unprovable in Peano arithmetic". What do they mean by "true" in this context? I think my comment about "safe" means "consistent with anything that we naturally would take to be true"---but I'm also a little confused at what things we take to be true! | |
| Jun 19, 2019 at 22:02 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jun 19, 2019 at 21:56 | history | answered | Tim Campion | CC BY-SA 4.0 |