Timeline for What can be preserved in mathematics if all constructions are carried out in ZF?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2022 at 13:50 | comment | added | Sergei Akbarov | Tim, I understand what you mean, but before doing what you say, the first step for me would be an attempt to see what happens if I pull out the brick that everyone is trying to pull out. | |
| Sep 21, 2022 at 13:18 | comment | added | Timothy Chow | @SergeiAkbarov Correct. But I think the point is that if you're serious about getting rid of AC, then the way to go about it is not to begin with classical mathematics, hunt around for uses of AC, and see if you can pull out AC like a Jenga piece without causing the whole tower to collapse. The way to do it is to rebuild mathematics from the ground up. Since nobody seems to have done exactly what you were hoping for (building math using ZF), the closest thing is to see how various people have developed large portions of mathematics without using AC. | |
| Sep 21, 2022 at 13:09 | comment | added | Sergei Akbarov | Tim, yes, I am reading this, but you know, this is not what I expected. This is not the same as removing AC from ZFC. | |
| Sep 21, 2022 at 12:08 | comment | added | Timothy Chow | @SergeiAkbarov The terminology "second-order arithmetic" may be slightly misleading; it's really a two-sorted theory that is still based on first-order logic. One sort is natural numbers and the other sort is sets of natural numbers. It's called second-order because you can quantify over sets of natural numbers and hence over real numbers, but $\mathbb{R}$, thought of as the set of all subsets of $\mathbb{N}$, is not something you can refer to directly. Chapter 1 of Simpson's book is available online; I refer you to it for more details. | |
| Sep 21, 2022 at 6:12 | comment | added | Sergei Akbarov | Tim, I guess the universe must be different, not the usual construction of $\mathbb N$. And real numbers must be also different, not the usual $\mathbb R$ as it is normally constructed in ZF, because otherwise we would get that Bolzano-Weierstrass holds in the usual $\mathbb R$ without the axiom of choice. | |
| Sep 21, 2022 at 5:03 | comment | added | Sergei Akbarov | A second order theory is interpretable in a first order theory? The universe of this interpretation, is it the usual $\mathbb N$? | |
| Sep 20, 2022 at 22:30 | comment | added | Timothy Chow | I would say it this way: second-order arithmetic is interpretable in ZF. The theorems of second-order arithmetic, so interpreted, are theorems of ZF. | |
| Sep 20, 2022 at 19:23 | comment | added | Sergei Akbarov | Tim, "on the basis of ZF" - does this mean that Simpson turns ZF somehow into a second order theory? | |
| Sep 20, 2022 at 18:15 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Added quote from Bishop
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| Sep 20, 2022 at 18:00 | history | answered | Timothy Chow | CC BY-SA 4.0 |