Timeline for Why are real-valued measurable cardinals never explicitly mentioned in Gödel's "What is Cantor's Continuum Problem"?
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Oct 2, 2017 at 4:31 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
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| Oct 2, 2017 at 4:24 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
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| Sep 30, 2017 at 13:48 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
Added addendum for further clarification
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| Sep 30, 2017 at 3:01 | review | Close votes | |||
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| Sep 26, 2017 at 4:59 | comment | added | Thomas Benjamin | @AndreasBlass: You are, of course, correct. As I am sure you are aware, Godel mentions these in both versions of his paper (I quote the relevant passage in my question). Since the large cardinals you mention are, in fact, consistent with $V$=$L$, Godel notes that for these types of cardinals, "the continuum hypothesis, e.g., goes through for all of them without any change". So to get a large-cardinal axiom which implies the falsity of $CH$, such a large cardinal axiom when added to $ZFC$ must, of necessity, imply $V$ $\ne$ $L$. | |
| Sep 25, 2017 at 16:33 | comment | added | Andreas Blass | You mention one definition of "large cardinal" as "ZFC + 'this cardinal exists' proves $V\neq L$". This definition looks strange to me. There are lots of properties that are usually thought of as "large cardinal" properties yet are consistent with $V=L$: inaccessible, Mahlo, weakly compact, other levels of indescribability, subtle, ineffable. | |
| Sep 25, 2017 at 16:07 | comment | added | Thomas Benjamin | @NoahSchweber: I, of course meant "weakly inaccessible" rather than "weakly compact". I guess I have 'weakly compact' on the brain (for some unknown reason...). Also, thanks for the previous comment--very helpful. | |
| Sep 25, 2017 at 15:59 | comment | added | Noah Schweber | @ThomasBenjamin "So then the cardinal collapse I envisioned would leave $\kappa$ weakly inaccessible" No! Why would it? As has been pointed out already, $\aleph_1$ cannot be weakly inaccessible. A weak inaccessible is, by definition, three things: uncountable, regular, and a limit cardinal. $\aleph_1$ isn't a limit. Large cardinal properties, and combinatorial properties in general, aren't preserved by arbitrary forcing; if you want to claim that $\kappa$ remains [thing] after forcing, you have to argue that. And in this case it's wildly false. (CH would, however, hold in that model.) | |
| Sep 25, 2017 at 15:54 | comment | added | Thomas Benjamin | @NoahSchweber: Because I don't claim to have perfect knowledge of anything (but thanks for confirming what I thought to be true). So then the cardinal collapse I envisioned would leave $\kappa$ weakly inaccessible and $CH$ would hold in that model of $ZFC$? | |
| Sep 25, 2017 at 15:46 | comment | added | Noah Schweber | @ThomasBenjamin It wouldn't result in a contradiction. Why do you think it would? And where did weakly compact cardinals enter the picture (or did you mean weakly inaccessible)? I think Godel was well aware of these possibilities, but didn't view them as fitting his goal since they're a priori about the continuum already; I think he was hoping for something like "if there exists a real-valued measurable, then CH holds" since the mere existence of a real-valued measurable a priori has nothing to do with the continuum; "the continuum is real-valued measurable" would not have been as natural. | |
| Sep 25, 2017 at 15:43 | comment | added | Thomas Benjamin | (cont.) a contradiction? Also, thanks for pointing out the ommission of regularity in my truncated definition of strongly inaccessible cardinal. If I could correct my previous comment now, I would... | |
| Sep 25, 2017 at 15:42 | comment | added | Thomas Benjamin | @NateEldredge: In which case, why didn't Godel mention the possibility of $2^{\aleph_0}$ being weakly inaccessible which he could have known through Ulam's (1930) paper, Banach's (1930) paper, and Banach and Kuratowski's (1929) paper? Also, why couldn't one collapse all the cardinals $\lt$ the weakly compact cardinal $\kappa$=$2^{\aleph_0}$ but $\gt$ $\aleph_0$ to $\aleph_0$, leaving the only cardinalities left $\kappa$ and $\aleph_0$ . In that model of $ZFC$, $CH$ would hold and $\kappa$=$2^{\aleph_0}$. Why would asuming such a collapse could occur result in | |
| Sep 25, 2017 at 14:28 | comment | added | Nate Eldredge | I don't know this stuff very well, but isn't "limit cardinal" part of the definition of "weakly inaccessible"? So $\aleph_1$ can't be weakly inaccessible, and hence if $2^{\aleph_0}$ is, then $2^{\aleph_0} \ne \aleph_1$, i.e. $\lnot CH$. (Also note that in your definition of "strongly inaccessible", you have left out the condition that it be regular; otherwise $\beth_\omega$ would be strongly inaccessible. And of course $\aleph_0$ should be excluded.) | |
| Sep 25, 2017 at 14:27 | comment | added | Todd Trimble | Well, a weakly inaccessible cardinal is a limit cardinal, and $\aleph_1$ is a successor cardinal, so... | |
| Sep 25, 2017 at 13:12 | comment | added | Thomas Benjamin | (cont.) '$2^{\aleph_0}$ is weakly inaccessible' implies $\lnot$$CH$? I don't know. How would you argue for the validity of the above claim? | |
| Sep 25, 2017 at 13:06 | comment | added | Thomas Benjamin | @bof: All real-valued measurable cardinals are weakly inacessible. Note the definition of strongly inaccessible cardinal: a cardinal $\kappa$ is strongly inaccessible iff for all $\lambda$ $\lt$ $\kappa$, $2^{\lambda}$ $\lt$ $\kappa$. All this is saying, in essence, is that a strongly inaccessible cardinal must be larger than the powerset of any cardinal less than it (i.e., it can't be the cardinality of a power set). However, this is not to say that a real-valued measurable cardinal cannot be (to use D.H. Fremlin's term) 'enormous'. Can one prove that | |
| Sep 25, 2017 at 13:03 | review | Close votes | |||
| Sep 25, 2017 at 14:59 | |||||
| Sep 25, 2017 at 12:17 | comment | added | bof | This stuff is way over my head, but if "$2^{\aleph_0}$ is real-valued measurable" is an example of a "large cardinal axiom" that contradicts the continuum hypothesis, isn't "$2^{\aleph_0}$ is weakly inaccessible" another one? | |
| Sep 25, 2017 at 12:04 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
added quotation mark in title
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| Sep 25, 2017 at 12:00 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
corrected spelling of Gödel
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| Sep 25, 2017 at 11:53 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |