Timeline for Why believe in the existence of large cardinals rather than just their consistency?
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46 events
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| Sep 27 at 23:52 | comment | added | Joseph Van Name | @JesseElliott I gave an answer where we look at 15215 near inconsistencies. But others here are trying really hard to get that answer deleted, so I do not know what the people here want. | |
| Sep 27 at 23:32 | review | Close votes | |||
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| Sep 27 at 21:25 | answer | added | Joseph Van Name | timeline score: 3 | |
| Sep 26 at 22:27 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Sep 26 at 7:40 | vote | accept | Jesse Elliott | ||
| Sep 26 at 7:39 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Sep 24 at 23:30 | vote | accept | Jesse Elliott | ||
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| Sep 24 at 23:30 | vote | accept | Jesse Elliott | ||
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| Sep 24 at 0:05 | comment | added | Jesse Elliott | @JosephVanName Apologies, but this sounds very cryptic to me. Can you provide an explicit example among those ~10^5 to ~10^6 examples? | |
| Sep 21 at 12:38 | comment | added | Joseph Van Name | @JesseElliott These examples that I know of are non-obvious, but I have computed about a hundred thousand to a million examples of specific self-distributive algebras arising from rank-into-rank cardinals. The existential statements are of the form "there exists a self-distributive algebra satisfying Property P" and I have often found just one such algebra after an exhaustive backtracking search. | |
| Sep 19 at 16:38 | comment | added | Hanul Jeon | For a stronger large cardinals, well, it depends on your definition of set theory to see if there is any non-set-theoretic justification for stronger large cardinals. Projective determinacy or consequences of the proper forcing axiom may give some hint on why LCAs beyond $L$ (roughly corresponding to infinitely many Woodins and supercompact) should be consistent and exist since Woodin cardinals imply PD and forcing PFA supposedly requires a supercompact, but they might be too set-theoretic in a non-set-theoretic POV. | |
| Sep 19 at 16:34 | comment | added | Hanul Jeon | @Jesse I do not deny that the confidence on theories decreases as theories goes stronger, and it is also true that some (but not all) mathematicians tend to avoid LCAs. However I do believe that LCAs that are not too far from $\mathsf{ZFC}$ (like inaccessibles) is widely accepted by category theorists in a form of Grothendieck universe, but LCAs in $L$ should have better justification, like, their $\mathsf{CZF}$-analogues have a pretty low consistency strength. (cont'd) | |
| Sep 19 at 14:23 | comment | added | Mikhail Katz | You may want to make it explicit that your question is only meaningful on the assumption of a realist (platonist?) position with regard to, say, ZFC. This will discourage answers challenging such realist assumptions, since this is not your concern. | |
| Sep 19 at 1:33 | comment | added | Jesse Elliott | @HanulJeon Also, my intention is not to argue against measurables. I'm merely asking for reasons why someone who accepts their consistency to some degree should also accept the claim that they "exist." Degrees of assurances of consistency, and our beliefs thereof, generally decrease as one climbs the consistency hierarchy. This is why mathematicians tend to avoid large cardinal hypotheses. Insisting that very large cardinals exist invalidates such variations in assurances and beliefs across people and across various large cardinal hypotheses. | |
| Sep 19 at 0:48 | comment | added | Jesse Elliott | @HanulJeon Regarding your third point, I've taken my background theory to be ZFC, which is justified because the questions examined now are whether large cardinals axioms are consistent relative to ZFC and whether they are "true" in addition to the axioms of ZFC. I'm happy to consider many subtheories of ZFC as natural. But general mathematical practice has accepted ZFC as acceptable, and results in algebraic set theory etc provide support for that practice. Though very large cardinals have support by set theorists, they do not share the same support as ZFC does by mathematicians at large. | |
| Sep 18 at 23:49 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
adds some comments in a comment from Chow and an answer from Hamkins
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| Sep 18 at 23:43 | comment | added | Jesse Elliott | @TimothyChow Thanks for pointing that out. I'm aware of those posts and will add them to my question. My own question has subtle differences between them. As Hamkins points out in his answer, it's related to the question of whether or not the instrumentalist dodge or variants thereof are enough to "dodge" large cardinals or not. | |
| Sep 18 at 23:34 | comment | added | Jesse Elliott | @VincentR.B.Blazy If I understand your question correctly, then, from your point of view, if you believe in the existence of a given large cardinal in an absolute sense, then you believe that it is true in all models of your background theory, whereas, if you believe just in the consistency of the existence of such a large cardinal, then you believe that is true in some model of your background theory. The former vantage point erases all distinctions between theories of lower consistency strength, and so such theories have to be examined with background theories other than your chosen one. | |
| Sep 18 at 23:25 | comment | added | Jesse Elliott | @HanulJeon I've tried to digest Woodn's papers on Ultimate-$L$, but most aspects of it still remain mysterious to me, and I do not know which aspects of it are still conjectural. Moreover, isn't the same is true for many set theorists, no less mathematicians at large? Let's grant that it places an upper bound on large cardinals in a way similar to how $V = L$ does. Can $V$ = Ultimate-$L$ rule out cardinals $\kappa$ such that $V_\kappa$ is a model of $V$ = Ultimate-$L$? Can it prove even that it has no transitive models, or even that it has no models at all? | |
| Sep 18 at 22:46 | comment | added | Jesse Elliott | @JosephVanName Perhaps you can give some good examples of near inconsistencies? Pardon if they should be obvious. As for $V = L$, it can prove the axiom of choice and the axiom of foundation over ZF-foundation, if that's at all worth noting. I assume your examples of near inconsistencies are more compelling. | |
| Sep 18 at 17:18 | comment | added | Joseph Van Name | @HanulJeon They do not necessarily have to be in any particular level of the Levy hierarchy. I currently do not see a way of formalizing the notion of a near inconsistency, so these results need to be looked at individually. | |
| Sep 18 at 16:55 | comment | added | Hanul Jeon | @JosephVanName Do you mean $\Sigma^0_1$ or $\Sigma_1$ in Levy hierarchy? | |
| Sep 18 at 16:47 | comment | added | Hanul Jeon | the type-theoretic interpretation usually invalidates the law of excluded middle or Full Separation that are important component of $\mathsf{ZFC}$. Thus your argument against measurable cardinal is not fulfilling to me since we may bring another canonical framework for set theory even refuting some axioms of $\mathsf{ZFC}$. | |
| Sep 18 at 16:46 | comment | added | Hanul Jeon | Even worse, things may be quite different at the level of choiceless large cardinals as most canonical inner models (either $L$ like or $\mathsf{HOD}$ like) satisfy the axiom of choice while choiceless large cardinals refute this axiom. On the other side of your argument, we could consider a hypothetical opinion "anything beyond constructive $\mathsf{ZF}$ (aka $\mathsf{CZF}$) plus some choice principles are dangerous since they may not admit type-theoretic interpretation of set theory." This hypothetical POV may refute $\mathsf{ZFC}$ since... | |
| Sep 18 at 16:43 | comment | added | Hanul Jeon | @JesseElliott It is what inner model theorists work on. Although I am not an expert on this field, I heard that Woodin's work on suitable extender models indicate inner models for supercompacts should be very different from the currently known inner models for measurables or Woodin cardinals. Thus your Pandora's box argument may break down at the level of supercompact. | |
| Sep 18 at 14:54 | comment | added | Timothy Chow | Related: What "forces" us to accept large cardinal axioms? Philosophical arguments in defense (or against) large cardinals, Arguments against large cardinals. | |
| Sep 18 at 11:38 | comment | added | Joseph Van Name | @JesseElliott I consider a near inconsistency as a consequence of large cardinals of the form $\exists x \phi(x)$ but where we later find that a weaker axiomatic system such as ZFC (or PA) implies that there is a unique $x$ where $\phi(x)$ is true and where such a unique $x$ is non-obvious beforehand. This applies up to the level of rank-into-rank embeddings. | |
| Sep 18 at 9:17 | answer | added | Monroe Eskew | timeline score: 14 | |
| Sep 18 at 8:47 | comment | converted from answer | Vincent R.B. Blazy | How the existence or not of anything in some "absolute" sense is something not counting among features of what your background set theory is, i.e. of which set-theoretic axioms (at least) are actually true in a same some "absolute" sense? Or, leaving any set-theoretic absoluteness aside, how is such an existence of large cardinals more or less such a feature of your chosen set theory, than their mere consistency/existence in models? 🤔 | |
| Sep 18 at 7:44 | comment | added | Jesse Elliott | @HanulJeon Point taken. But, by the same token, ZFC+a measurable exists satisfies "most" of what ZFC+$0^\dagger$ satisfies, which satisfies most of what ZFC+two measurables satisfies...And so on, ad infinitum. Once you allow large cardinals, you open a Pandora's box of axioms with larger and larger consistency strengths, and it becomes impossible to maintain with any assurance that they "all" exist. Where, if anywhere, do you draw the line? Can you say they exist if they're consistent, and they're consistent because they exist? Reminds me of the ontological argument for the existence of God. | |
| Sep 18 at 6:29 | history | became hot network question | |||
| Sep 18 at 5:46 | comment | added | Hanul Jeon | Jesse, your argument with $V=L$ may be countered by arguments with more sophisticated inner models (like, $L[U]$ for a normal ultrafilter $U$) as it also satisfies most of what $L$ satisfies like diamond principle while it allows a measurable cardinal. | |
| Sep 18 at 3:05 | comment | added | Jesse Elliott | @JosephVanName Interesting argument. Are you in favor of all large cardinals for which there haven't been found an inconsistency? Can you elaborate on what you mean by near inconsistencies? Ignoring that aspect of your explanation for why very large cardinals exist, I would counter that $V=L$ (and the existence and absoluteness of $L$) produces diverse conclusions without producing any inconsistency, and $V=L$ arguably is the only explanation for why its seemingly unrelated consequences (like its answer to Whitehead problem) ae consistent (except maybe the diamond principle). | |
| Sep 18 at 0:41 | comment | added | Joseph Van Name | I see no way of explaining the consistency (so far) of large cardinals other than their existence when these large cardinal axioms exhibit many near inconsistencies so far that have no other explanation. | |
| Sep 18 at 0:15 | comment | added | Jesse Elliott | @Wojowu For a long time, even after reading Maddy's books on the subject, I really wanted to believe that $V = L$. I have many arguments, as Hamkins does, as to why it is not as limiting an axiom as many believe. Others believe $V \neq L$ because measurable cardinals exist. But aren't both axioms limiting? Why should we believe that measurable cardinals exist, even if we believe strongly that their existence is consistent with ZFC? | |
| Sep 18 at 0:15 | answer | added | Joel David Hamkins | timeline score: 23 | |
| Sep 17 at 23:59 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Sep 17 at 23:55 | comment | added | Hanul Jeon | @TheoJohnson-Freyd I do not have a solid opinion about it (although I think large cardinal axioms should be true) and I do not want to make a philosophical discussion on this comment line. I do not disagree with your opinion, though, and various logicians (like Gödel or Woodin) also have a similar stance with you as you may know. | |
| Sep 17 at 23:51 | comment | added | Theo Johnson-Freyd | @HanulJeon I believe that there are lots of statements which are consistent but not true. In particular, I definitely believe that there is a "true" universe out there, and I also believe that our exploration of it has not uncovered all essential truths. | |
| Sep 17 at 23:51 | comment | added | Hanul Jeon | Good question. I do not think there is an easy answer to say what being true means. The truth of LCAs has a wide agreement from set theorists, but the truth of the continuum hypothesis is still on (and possibly in an endless) debate. I also have no idea what truth should mean, and it is worthwhile to note that Woodin changed his opinion about the truth of CH. | |
| Sep 17 at 23:47 | comment | added | Jesse Elliott | @HanulJeon RE your first comment: What does it even mean to say they are "true"? RE your second comment. I didn't mean to imply that all are consistent. That of course varies according to your background theory, while degrees of beliefs in consistency, say, relative to ZFC, vary from person to person, and from axiom to axiom. | |
| Sep 17 at 23:42 | comment | added | Hanul Jeon | Also, set theorists' opinion about very strong large cardinal axioms (especially around at the level of rank-into-rank cardinals or choiceless large cardinal axioms) varies by people, so not every set theorists believe every large cardinal axiom is consistent. (Oh, this comment counters the first line of my first comment...) | |
| Sep 17 at 23:38 | comment | added | Hanul Jeon | If we can believe in their consistency, then is there any reason to mind to refuse their truth? But Hamkins argued there are universes of set theory in which some large cardinal axioms are not true. I hope he provides a better comment or answer about his position. | |
| Sep 17 at 23:36 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Sep 17 at 23:12 | comment | added | Wojowu | Penelope Maddy, Believing the Axioms. II | |
| Sep 17 at 22:23 | history | asked | Jesse Elliott | CC BY-SA 4.0 |