bio | website | math.harvard.edu/~nate |
---|---|---|
location | Cambridge, MA USA | |
age | 37 | |
visits | member for | 4 years, 8 months |
seen | 11 hours ago | |
stats | profile views | 431 |
I am a mathematical logician at Harvard.
Mar 28 |
asked | Reference for proof that consistency of $\omega_1$-Erdos cardinal implies Con(Chang's Conjecture) |
Mar 28 |
comment |
Which compact topological spaces are homeomorphic to their ultrapower?
Ah, of course, that is exactly what I wasn't seeing. Thanks Eric. If you want to write your comment as an answer I can accept it. |
Mar 28 |
asked | Which compact topological spaces are homeomorphic to their ultrapower? |
Mar 11 |
awarded | Autobiographer |
Mar 11 |
answered | Applications of forcing in model theory |
Mar 10 |
awarded | Yearling |
Mar 10 |
answered | Reverse Skolem's paradox |
Mar 8 |
awarded | Nice Answer |
Mar 7 |
comment |
Application of Fraïssé construction in set theory
The proof, in my opinion, is really model theoretic. However Hjorth, being a world class descriptive set theorist, was able to instead give a very short descriptive set theory version, which is what is the original paper. |
Mar 7 |
comment |
Application of Fraïssé construction in set theory
So the original paper is "A note on counterexamples to the Vaught conjecture" by Hjorth. The proof, while short, is very descriptive set theoretic. There is also a model theoretic version of the proof in the Baldwin/Friedman/Koerwien/Laskowski paper you mentioned. Also, to help me and some friends understand the Hjorth proof I wrote up a model theoretic version as well which you can find at Hjorth. |
Mar 6 |
comment |
Category of Gödel Codings? [Reference Request]
A further analogy is that just as continuous maps can be thought of as "computable relative to an oracle" Borel maps really can be thought of as "higher computable relative to an oracle" where here higher computable is in the sense of $\alpha$-recursion theory (and the many equivalent variants of the notions) |
Mar 6 |
comment |
Category of Gödel Codings? [Reference Request]
For example they both run into the issue that they are fundamentally only talking about equivalence relations on Baire space (as all Borel spaces of the same cardinality is isomorphic). This is something which isn't an issue with equilogical spaces. In particular I don't know if the category of represented spaces and continuous functions is Cartesian closed (although I imagine it wouldn't be to hard to check). |
Mar 6 |
comment |
Category of Gödel Codings? [Reference Request]
am far from an expert on realizability or TTE, but my guess is that the relationship between the collection of equivalence relations on Borel spaces (with Borel maps between them) and realizability will have a very similar feel to the relationship between represented spaces (and continuous maps) and realizability. |
Mar 6 |
comment |
Application of Fraïssé construction in set theory
@JoelDavidHamkins, yes, thanks. |
Mar 6 |
comment |
Application of Fraïssé construction in set theory
While even having a relative consistency result is a substantial step forward, to really have a proof of the categoricity conjecture (in my opinion) one needs to either prove it from ZFC, show it is independent from ZFC, or show that it is equivalent to some large cardinal assumption. |
Mar 6 |
comment |
Application of Fraïssé construction in set theory
First off, thanks for the link, that paper looks interesting! That said, while I am far from an expert in AEC, it seems from the abstract that Vasey is proving various structural properties from tameness and a categoricity assumption. And while this is very interesting (and I am sure if I understood more I would have an even better appreciation) his proof of the categoricity conjecture (like Boney's) uses a proper class of strongly compacts. |
Mar 6 |
comment |
Application of Fraïssé construction in set theory
Actually by work of Hjorth, if there is a counterexample to Vaught's conjecture there must be a ($L_{\omega_1,\omega}$) counterexample which has no models of size greater than $\aleph_1$. So in particular it need not have models of size $\aleph_2$ with arbitrary high Scott rank below $\aleph_3$. |
Mar 6 |
awarded | Yearling |
Mar 6 |
answered | Application of Fraïssé construction in set theory |
Mar 5 |
answered | Category of Gödel Codings? [Reference Request] |