Timeline for Minimal Generalized Continuum Hypothesis & Axiom of Choice
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2013 at 4:04 | comment | added | user42090 | @TrevorWilson: Of course theories with class many sentences are not meaningful. I used this special notation just because I want to emphasize on looking at single statement $\text{GCH}$ as a theory with class many sentences. Obviously when we want to write it formally we should use legitimated notation like what I used in (a) and (b). | |
| Nov 30, 2013 at 18:02 | comment | added | Joel David Hamkins | Yes, I agree that it is worse than that. I had in mind that even if one might augment the theory asserting all the $\text{CH}_\alpha$ with the assertions that every such $\alpha$ is an ordinal, and that they are all distinct, and in the right order, etc., but nevertheless the compactness argument shows that all those assertions plus the $\text{CH}_\alpha$ do not together imply the GCH. | |
| Nov 30, 2013 at 17:20 | comment | added | Trevor Wilson | It seems like everywhere you say $\{\text{CH}_\alpha \mid \alpha \in \text{Ord}\}$ or $\{\text{CH}_\alpha \mid \alpha \in C\}$ you mean $\forall \alpha \in \text{Ord}\,\text{CH}_\alpha$ or $\forall \alpha \in C\,\text{CH}_\alpha$ respectively. The former two notions don't make any sense, as Lawrence Wong has pointed out below, and the latter two notions are what you are using in (a) and (b) anyway (after the "i.e.") | |
| Nov 30, 2013 at 17:13 | comment | added | Trevor Wilson | @Joel Isn't the problem even worse than that? I don't see how "the scheme of assertions about each $\alpha$ separately" makes any sense (although if it did, then what you said about compactness surely would apply.) What are ordinals in the meta-language? | |
| Nov 30, 2013 at 16:11 | comment | added | Adam Epstein | @Asaf Okay, that's all clear now. | |
| Nov 30, 2013 at 12:53 | comment | added | Joel David Hamkins | I agree with Lawrence that the OP has not specified his theory properly. He surely wants the sentence $\forall\alpha\in C \text{CH}_\alpha$, rather then the scheme of assertions about each $\alpha$ separately, for even when every ordinal is in $C$, the latter does not prove GCH by a simple compactness argument. | |
| Nov 30, 2013 at 12:27 | comment | added | Lawrence Wong | @Asaf: Thank you for the response. It gets clearer to me now: apparently $\{\mathrm{CH}_\alpha:\alpha\in C\}$ just means the sentence $\forall\alpha\in C\ \mathrm{CH}_\alpha$ when $C$ is a class definable without parameters. So there is no issue of having too many formulas/sentences. | |
| Nov 30, 2013 at 12:24 | comment | added | Asaf Karagila♦ | @Adam: I think that I understand it well enough to understand it, but not well enough to explain it. I guess it's just something you have to get used to in mathematics, and in this particular instance, in set theory. | |
| Nov 30, 2013 at 12:23 | comment | added | Asaf Karagila♦ | @Adam: I can say that "There exists $\alpha$ such that for every $\beta>\omega$ of cofinality $\omega$, $\sf CH_\beta$ holds". In this statement I didn't quantify over all the ordinals, or all the sets. I instead limited to some subclass thereof, and I claimed that there is some parameter $\alpha$ that this holds above it. | |
| Nov 30, 2013 at 11:52 | comment | added | Adam Epstein | @Asaf I think I am still confused. Could you give a syntactically explicit example illustrating this difference? | |
| Nov 30, 2013 at 11:18 | comment | added | Asaf Karagila♦ | @Adam: It's not, really. But if you want to start talking about subclasses then it makes a difference. Moreover, you can say something like "There exists an ordinal such that ..." which specifies existence of ordinals satisfying the formula; but it doesn't explicitly points out which ordinals these are. | |
| Nov 30, 2013 at 10:49 | comment | added | Adam Epstein | @Asaf How is this any different from specifying a single universally quantified formula? | |
| Nov 30, 2013 at 10:05 | comment | added | Asaf Karagila♦ | While the language is countable, we can talk about parameterizied formulas. Here $\alpha$ is a parameter. | |
| Nov 30, 2013 at 9:34 | comment | added | Lawrence Wong | What is $\{\mathrm{CH}_\alpha:\alpha\in\mathrm{Ord}\}$ actually? The language for set theory is countable, and so there cannot be ordinally many different sentences. Or are we assuming there is an intended (set) model of set theory at the back of our mind? | |
| Nov 30, 2013 at 7:50 | answer | added | Asaf Karagila♦ | timeline score: 5 | |
| Nov 30, 2013 at 2:53 | vote | accept | CommunityBot | ||
| Nov 30, 2013 at 1:55 | answer | added | Joel David Hamkins | timeline score: 8 | |
| Nov 30, 2013 at 1:52 | history | edited | user42090 | CC BY-SA 3.0 |
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| Nov 30, 2013 at 1:43 | history | asked | user42090 | CC BY-SA 3.0 |