Timeline for There are two slightly different notions of ultraproduct. Why is one said to be better than the other?
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16 events
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| Aug 25, 2012 at 19:24 | comment | added | Yemon Choi | small tangential remark: Type I is more common in the Banach-space version of these things, probably because when the likes of Krivine were setting up the basics they were thinking along the lines described in Francois's comments | |
| Aug 25, 2012 at 16:52 | comment | added | François G. Dorais | (Typo: $\beta I$ instead of $\beta X_i$.) | |
| Aug 25, 2012 at 16:50 | comment | added | François G. Dorais | Yes, that is exactly right. | |
| Aug 25, 2012 at 16:41 | comment | added | Benjamin Steinberg | Is type II not what you get by viewing the X_i as a sheaf on the discrete set I, pushing the sheaf forward to \beta X_i and then taking germs at your ultrafilter? | |
| Aug 25, 2012 at 15:24 | comment | added | François G. Dorais | The Type 2 definition makes more sense in a lot of contexts, as your answer illustrates. It also arises naturally in a lot of places, as I illustrated in my answer to Joel's question. mathoverflow.net/questions/11261/… However, the Type 1 definition is easier to work with on the element level since there is a lot of unnecessary fuss when working with the Type 2 definition. | |
| Aug 25, 2012 at 15:11 | comment | added | Tom Leinster | OK, "exclude" would have been a better word: I just meant it factually, not judgementally. | |
| Aug 25, 2012 at 15:08 | comment | added | François G. Dorais | I don't think 'forbid' is the right word. There are plenty of contexts where empty domains are not relevant. It is risk-free to use the Type 1 definition in these contexts. Since the Type 1 definition is simpler, a lot of people prefer it. Note that the difference between the two types is only visible when getting your hands dirty playing with ultraproduct elements. If that's not something you plan on doing, the Type 2 definition is not that inconvenient. | |
| Aug 25, 2012 at 12:37 | comment | added | Tom Leinster | So am I right in understanding that the only people who use the type 1 definition are those who forbid the empty domain? | |
| Aug 25, 2012 at 8:38 | comment | added | Andrej Bauer | The criterion is very simple: the correct definition is the one that generalizes to other settings (filter-quotient construction in toposes comes to mind), and it is the one that enjoys a universal property which does not involve ad-hoc side conditions and special cases. | |
| Aug 25, 2012 at 8:34 | comment | added | Andrej Bauer | <flame>I strongly disagree with "avoidance of the empty domain is a good thing". It is a very bad thing, it promotes ad-hoc hacks and definitions, it makes it harder to generalize constructions and proofs. Along the same lines, requiring non-empty metric spaces, non-empty topological spaces, non-trivial rings, etc., is a big mistake. It builds in classical logic, so people have to then rework everything when they pass to sheaves, computability and more generally intuitionistic logic. Bad idea.</flame> | |
| Aug 24, 2012 at 19:13 | comment | added | François G. Dorais | Yes, this is exactly what the Type 2 ultraproduct is. (Tom's 'explicit' definition is the usual one, I think.) | |
| Aug 24, 2012 at 18:56 | comment | added | Goldstern | You could of course get a modified version 1' by allowing partial functions, as long as their domain is in the ultrafilter. This type 1' ultraproduct is then canonically isomorphic to the type 1 ultraproduct iff all structures are non-empty, but it will never be empty (unless most of the base structures are empty, of course). | |
| Aug 24, 2012 at 18:07 | comment | added | Tom Leinster | Thanks, François and Benjamin. That's all very helpful. | |
| Aug 24, 2012 at 17:52 | comment | added | François G. Dorais | The Wikipedia article linked above links to the entry for free logic which describes the axiomatization of first-order logic that allows empty domains. See also the Stanford Encyclopedia of Philosophy - plato.stanford.edu/entries/logic-free | |
| Aug 24, 2012 at 17:49 | comment | added | Benjamin Dickman | Related: See the Appendix (pdf p. 14/16) in archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1986__27_2/… | |
| Aug 24, 2012 at 17:46 | history | answered | François G. Dorais | CC BY-SA 3.0 |