Timeline for How is compactness related to countable saturation?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2016 at 12:04 | answer | added | Mikhail Katz | timeline score: 2 | |
| S Nov 21, 2016 at 14:50 | history | bounty ended | CommunityBot | ||
| S Nov 21, 2016 at 14:50 | history | notice removed | CommunityBot | ||
| Nov 17, 2016 at 18:44 | answer | added | François G. Dorais | timeline score: 6 | |
| Nov 13, 2016 at 13:51 | comment | added | Todd Trimble | Don't really know off-hand; sorry. | |
| Nov 13, 2016 at 13:49 | comment | added | Mikhail Katz | @Todd, thanks, that's great. Can one tie in nested sequences of internal sets somehow? | |
| Nov 13, 2016 at 13:44 | comment | added | Todd Trimble | Of course compactness theorems in logic are connected to topological compactness. To give a simple example: the main content of compactness for propositional logic is that any Boolean algebra $B$ (e.g., a Lindenbaum algebra of a propositional theory) that is nontrivial (i.e., $0 \neq 1$) admits a Boolean algebra map $\phi: B \to 2$; one can construct a model from this datum $\phi$. This can be inferred from compactness of the Cantor space $2^X$ where $X$ is a set of generators for $B$, and is a starting point of Stone duality. See Prop. 3.1 here: ncatlab.org/nlab/show/Tychonoff+theorem | |
| Nov 13, 2016 at 13:01 | comment | added | Mikhail Katz | @JoelDavid, but I don't see an immediate connection between compactness of sets and compactness theorem for first-order logic. Perhaps that's what I should be asking... | |
| Nov 13, 2016 at 12:59 | comment | added | Joel David Hamkins | One such unifying idea might be: we can use the compactness theorem of first-order logic to build nonstandard models. | |
| S Nov 13, 2016 at 12:58 | history | bounty started | Mikhail Katz | ||
| S Nov 13, 2016 at 12:58 | history | notice added | Mikhail Katz | Draw attention | |
| Nov 13, 2016 at 12:56 | comment | added | Mikhail Katz | @JoelDavid, obviously this can't be interpreted naively at the level of subsets of $F^\ast$ (for whatever $F$). I was wondering whether perhaps there is an overarching framework where both nesting phenomena would arise as particular manifestations of a common principle. It's really more a question about Los' lemma than nonstandard analysis perhaps, but we don't have a tag for that. | |
| Nov 13, 2016 at 12:51 | comment | added | Joel David Hamkins | Yes, of course. My point was that since $\mathbb{Q}^*$ can be saturated, but intervals in $\mathbb{Q}$ are not compact, it may somewhat undermine an expectation of a connection here. (Although your way of stating the principle for nested compact sets is fine for $\mathbb{Q}$.) | |
| Nov 13, 2016 at 12:34 | comment | added | Mikhail Katz | @JoelDavid, a nested sequence of internal sets in $\mathbb Z^\ast$ would in particular be a nested sequence of internal sets in $\mathbb R^\ast$ (and any common point would necessarily be a hyperinteger). | |
| Nov 13, 2016 at 11:13 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
added 34 characters in body
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| Nov 11, 2016 at 13:26 | comment | added | Joel David Hamkins | It's probably worth mentioning also that even the nonstandard rationals $\mathbb{Q}^*$ and the nonstandard integers $\mathbb{Z}^*$ can exhibit countable saturation (and it is possible for all these to be much more saturated than just countably saturated). Meanwhile, saturation is not a consequence of the transfer principle, since one can easily construct nonstandard models lacking saturation. In light of this kind of variation, it's not really correct to speak of the hyperreals. | |
| Nov 11, 2016 at 9:07 | history | asked | Mikhail Katz | CC BY-SA 3.0 |