Timeline for answer to The most outrageous (or ridiculous) conjectures in mathematics by Joël
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15 events
| when toggle format | what | by | license | comment | |
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| Mar 19, 2019 at 10:03 | comment | added | მამუკა ჯიბლაძე | @Joël "convicted" is probably a misprint (instead of "convinced")? Looks quite significant in any case :) | |
| Jan 22, 2017 at 17:46 | comment | added | Joël | Gentzen's proof cannot give us the certainty that Peano is consistent since it is based on the unproved consistency of "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal $\epsilon_0$", on which I know much less than on Peano. It is nice, a little bit like Lagarias's theorem that RH is equivalent to $\sigma(n)<H_n+e^{H_n} \log H_n$ is nice, but not a proof of RH. Moreover, we have hope to get a proof of RH sometimes (anyway now!) but no hope to ever get more than results of Gentzen's type concerning the consistency of Peano. | |
| Jan 22, 2017 at 17:38 | comment | added | Akiva Weinberger | What about Gentzen's proof? | |
| Jan 22, 2017 at 8:11 | comment | added | Nemo | @Joël, sure, but you should have seen the other professors' faces at the conference. :) | |
| Jan 20, 2017 at 14:00 | comment | added | Joël | @Nemo. I wish I had. Though his conjecture can be called "outrageous" I would not call it "ridiculous", nor his work on the subject. In fact, while my opinion on this issue is that Peano is consistent, it is not strongly held, more something like when Socrates says in substance "I don't known anything about the gods, so why not believe in those of my city?". Actually, reading some texts of Nelson I remember having been almost convicted by them, though now I can't remember my line of thoughts then. | |
| Jan 20, 2017 at 10:53 | comment | added | Nemo | I attended a conference by Edward Nelson and I agree it's a good example. | |
| Jan 18, 2017 at 22:17 | comment | added | Joel David Hamkins | No, unfortunately, I don't know of any article of his where he states this definitively. Lacking an actual proof of inconsistency, I have understood that he was naturally reticent to state the view explicitly. And so my information is merely second-hand, and may be wrong, so please take it with a grain of salt, although I heard the statements from people at Berkeley who were in a position to know his true plan. I do know for a fact from personal experience that in his set theory lectures he described the various philosophical justifications of large cardinals as, "full of hot air." | |
| Jan 18, 2017 at 22:07 | comment | added | Burak | @JoelDavidHamkins: Dear Joel, I was not aware of the story behind the Silver indiscernibles. Do you know of some article of Silver where he explains his thoughts on the possible inconsistency of ZFC? | |
| Jan 18, 2017 at 16:09 | comment | added | Joel David Hamkins | In a similar vein, Jack Silver long conjectured that ZFC was inconsistent, and tried over a period of several decades to prove it, although unsuccessfully. His attempt to refute the existence of measurable cardinals led to the theory of Silver indiscernibles, now a fundamental part of the subject. | |
| Jan 18, 2017 at 5:24 | comment | added | მამუკა ჯიბლაძე | Fantastic example! Let me just add that it possibly reveals either one more feature that might be included in the list of requirements, or the very basic explanation of why exactly such conjectures are found outrageous. This is the property of baring cruel limitations to reconciling our abilities and disabilities. I mean if you think of it there is absolutely no way to ever know whether PA is consistent or not. A finite derivation of contradiction from the induction principle indeed feels impossible but at the same time it feels so precisely because it seems to require something non-finite... | |
| Jan 18, 2017 at 4:19 | comment | added | Joël | Yes, that what I meant by "(in as system etc.)", that is "(in a system that does not contain Peano's itself)". I acknowledge it was not really clear. | |
| Jan 18, 2017 at 4:18 | history | edited | Joël | CC BY-SA 3.0 |
added 189 characters in body
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| Jan 18, 2017 at 3:58 | comment | added | Todd Trimble | The consistency of Peano arithmetic is provable in ZF(C). Which is a stronger system, of course. | |
| S Jan 18, 2017 at 3:51 | history | answered | Joël | CC BY-SA 3.0 | |
| S Jan 18, 2017 at 3:51 | history | made wiki | Post Made Community Wiki by Joël |