Timeline for Interpreting Conway's remark about using the surreals for non-standard analysis
Current License: CC BY-SA 4.0
11 events
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| Mar 23, 2022 at 3:39 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Added links to papers
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| Mar 23, 2022 at 3:28 | comment | added | Timothy Chow | To elaborate a bit on point #1: the obvious candidate for an analogue of the integers is Oz (the omnific integers), but Oz does not have good properties for the purposes of nonstandard analysis. For example, every number is the ratio of two omnific integers, so in particular you cannot prove that $\sqrt 2$ is irrational since that's false. | |
| Mar 22, 2022 at 21:07 | comment | added | Mike Battaglia | Maybe worth making a different question, if the answer is very involved... | |
| Mar 22, 2022 at 21:01 | comment | added | Mike Battaglia | @SamSanders I think I should ask a clarifying question. When you write "transfer is only available for the surreals via isomorphism at class level with the hyperreals," is there some kind of theorem we have that that really is the only way forward? Or to make the question rigorous, suppose we are in a set theory that has no hyperreals or for which the negation of ultrafilter lemma is true; this would suggest that it is provable that the surreals have no transfer principle in such a set theory. Is that a theorem we have? | |
| Mar 22, 2022 at 20:54 | comment | added | Mike Battaglia | @DavidRoberts Ah ok, good to note... | |
| Mar 21, 2022 at 23:31 | comment | added | David Roberts♦ | @MikeBattaglia here's a better link: doi.org/10.1016/j.apal.2012.07.003 (it's preferable to paste doi links here, and in fact anywhere on the internet, because publishers cough Springer cough have been known to refactor their entire URL system and break many many links) | |
| Mar 21, 2022 at 20:52 | comment | added | Sam Sanders | @ Alec Rhea: I should point out that E.H. Moore's ca 1915 "general analysis" was also intended to be the "ultimate analysis". The only remnant of that approach is nets (aka Moore-Smith sequences), which had been independently discovered by Vietoris. Moore's approach is neigh unreadable and written in a rather strange symbolism. It was taught by his successor at Chicago, and then fizzled out. | |
| Mar 21, 2022 at 20:14 | comment | added | Alec Rhea | As a member of ‘the surreal analysis cult’ who enjoyed reading this post, I just wanted to point out that existence in ZF was never part of the appeal for me. The surreals immediately struck me as ‘the ultimate place to do analysis’, in the sense that a complete theory of surreal analysis ‘should’ subsume all other theories of analysis as a subset of the things it can talk about/prove. I am also an extremist Platonic, though, so I ultimately believe that we’ll find the surreals intimately involved with the ‘final theory of physics’ in our universe/the multiverse. | |
| Mar 21, 2022 at 19:14 | comment | added | Mike Battaglia | The APAL paper: sciencedirect.com/science/article/pii/… | |
| Mar 21, 2022 at 19:13 | comment | added | Mike Battaglia | Thanks @Sam, that APAL paper looks very interesting and I will read it. These are all very good points. So is the takeaway that we think Conway was simply mistaken in what he originally wrote, then? It's a very very strong claim, and he kind of just throws it in there with no proof. But as you note, the claim is much stronger than just being a real-closed field and would require some kind of transfer principle. I'm pretty sure that the results re: isomorphism w/ class-sized hyperreals weren't published until after his book, so I am curious where this idea even came from. | |
| Mar 21, 2022 at 12:48 | history | answered | Sam Sanders | CC BY-SA 4.0 |