Timeline for Uniqueness results that follow from CH
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 25 at 20:12 | comment | added | Joel David Hamkins | Namely, the internal sets you mention, for ultrapowers, are just those arising in the higher-order structure $P(\mathbb{R})^{\mathbb{N}}/\mathcal{F}$, but one will not generally achieve saturation for such a large structure using only an ultrapower indexed by $\mathbb{N}$, even with CH. But meanwhile, under the GCH there are indeed saturated models elementarily extending $P(\mathbb{R})$, and one can achieve this with larger ultrapowers. | |
| Jul 25 at 15:20 | comment | added | Joel David Hamkins | But let me add that with the GCH, one gets saturated models in every uncountable regular cardinality, this will provide the desired categoricity results for the higher-order structures. | |
| Jul 25 at 15:09 | comment | added | Joel David Hamkins | I don't really agree. When using infinitesimals in calculus, most of the benefit for the main theory arises already when concerned merely with functions and predicates on the reals $\mathbb{R}$, and the nonstandard analogues $f^*$ for the ordinary functions $f:\mathbb{R}\to\mathbb{R}$ and $A^*$ for $A\subseteq\mathbb{R}$. | |
| Jul 25 at 14:23 | comment | added | Mikhail Katz | Without preserving internal sets, there may not be much significance to the uniqueness of the isomorphism type of ${}^\ast\mathbb R$, because internal sets are the bread and butter of NSA, and they are the ones to which the transfer principle applies. If so, such "categoricity" would apparently not be very useful. | |
| Jul 25 at 14:19 | comment | added | Joel David Hamkins | If you are using higher sets in NSA, as I believe you usually do, then CH will not be enough to get saturation, and one doesn't generally even get categoricity for that case. | |
| Jul 25 at 14:09 | comment | added | Mikhail Katz | In model-theoretic approaches to NSA, there are three types of sets: (M1) *-transforms of "real" sets, (M2) internal sets; (M3) external sets. In axiomatic approaches to NSA, there are two types of sets: (A1) standard sets; (A2) nonstandard sets. Here (A1) and (A2) correspond to (M1) and (M2) (there is no counterpart for external sets). Thus, a singleton consisting of a nonstandard integer is of type (M2) or (A2) depending on which framework your are working in. An uniform infinitesimal partition is another. | |
| Jul 25 at 13:59 | comment | added | Mikhail Katz | I am not sure I follow. An infinite internal subset always contains nonstandard elements, so it can't be a subset of $\mathbb R$. | |
| Jul 25 at 13:54 | comment | added | Joel David Hamkins | Yes, because the models are saturated in the language that includes constants for every real, so the isomorphism will make a commutative diagram with the elementary embedding of $\mathbb{R}$ into $\mathbb{R}^{\mathbb{N}}/\mathcal{F}$. | |
| Jul 25 at 13:16 | comment | added | Mikhail Katz | Incidentally, when one has a CH-guaranteed isomorphism $\phi$ between two distinct ultrapowers of type $\mathbb R^{\mathbb N}/\mathcal F$, is there any chance $\phi$ might preserve the internal subsets? | |
| Jul 25 at 0:24 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Jul 24 at 16:21 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |