Timeline for Is there a metamathematical $V$?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2022 at 15:35 | comment | added | user21820 | @PaceNielsen: I don't think this post actually provides an answer, because the very term "ordinal" seems to be ill-defined from a pure meta-logical viewpoint. Just try to define it, and you will see that you cannot without invoking some (almost surely ZFC-based) notions of sets. Worse still, predicativists, not to say formalists, would likely stop around ATR0, but ATR0 is consistent with "there is no uncomputable well-ordering", so on what meta-logical grounds can you justify going further? | |
| Apr 13, 2021 at 5:12 | comment | added | Vladimir Reshetnikov | @AkivaWeinberger Despite that $\bf Ord$ is a proper class, you still can have well-ordered definable class relations that can be thought of as “pseudo-ordinals” that are longer than $\bf Ord$, and it’s an interesting question to ask how far they can extend. There is some analogy between them and recursive countable ordinals. See mathoverflow.net/q/116590/9550, math.stackexchange.com/q/3470982/19661, math.stackexchange.com/q/4072813/19661 | |
| Feb 16, 2021 at 14:56 | comment | added | Timothy Chow | See also Cantor's view that it is inconsistent to view the Absolute Infinite as a completed totality, which I mentioned in my answer. | |
| Feb 15, 2021 at 18:29 | comment | added | Joel David Hamkins | The theory of New Foundations does not seem to arise from any prereflective coherent vision of set theory; rather, this is a theory arising from a purely syntactic move attempting to avoid the Russell contradiction. There seems little reason to expect it to have any robust set-theoretic interpretation. | |
| Feb 15, 2021 at 18:09 | comment | added | Akiva Weinberger | @JoelDavidHamkins Thank you for the link. The New Foundations solution breaks my brain a little - it seems its ordinals violate my intuitive desires for what ordinals "should be". | |
| Feb 15, 2021 at 17:44 | comment | added | Joel David Hamkins | The main idea underlying this answer is known as the Burali-Forti paradox. en.wikipedia.org/wiki/Burali-Forti_paradox | |
| Feb 12, 2021 at 22:19 | vote | accept | Pace Nielsen | ||
| Feb 12, 2021 at 22:19 | comment | added | Pace Nielsen | Their answers are excellent, and provide some important resources. However, pointing out that Ord (if it existed as a completed object) would contradict itself cuts right to the heart of the philosophical issues. | |
| Feb 12, 2021 at 21:41 | comment | added | Akiva Weinberger | I should add that both Kameryn Williams and Timothy Chow are experts in this area and I am definitely not. | |
| Feb 12, 2021 at 21:40 | history | answered | Akiva Weinberger | CC BY-SA 4.0 |