1,967 reputation
1921
bio website math.ucla.edu/~justinpa
location UCLA
age
visits member for 4 years, 8 months
seen Jun 25 '13 at 0:32
UCLA grad student studying set theory.

Jul
2
awarded  Curious
May
23
awarded  Enlightened
May
23
awarded  Nice Answer
May
19
awarded  Nice Question
Mar
20
awarded  Nice Answer
Dec
8
awarded  Yearling
May
24
awarded  Nice Answer
Mar
6
awarded  Nice Question
Dec
8
awarded  Yearling
Dec
3
comment Why does the Solovay-Tennenbaum theorem work?
With that in mind I'd suggest that a good 'next' iterated forcing argument to look at is Baumgartner's construction of a model where all $\aleph_1$-dense set of reals are isomorphic, since the iteration only uses ccc forcings, but the result doesn't itself doesn't follow from MA (The original paper is freely available at the FM archive matwbn.icm.edu.pl/ksiazki/fm/fm79/fm79111.pdf, and here's an expository note presenting the same result: scholarworks.sjsu.edu/etd_theses/3834)
Dec
3
comment Why does the Solovay-Tennenbaum theorem work?
When I'm in situations where I feel like I understand the mechanism of a proof but don't 'grok' them I am reminded of the quote of von Neumann: "In mathematics you don't understand things. You just get used to them."
Nov
26
comment Which of these relations on partial orders allows us to identify forcing equivalence?
I see, so it looks like the map $q$ to $[q\in\tau]$ isn't even well-defined in that case, since forcing-wise RO($\mathbb{Q}$) consists only of nonzero elements. If it is well-defined it's a complete embedding, and that will only happen if (and only if) for any $q$ we can find a $p$ forcing $q$ into $\tau$.
Nov
26
comment Which of these relations on partial orders allows us to identify forcing equivalence?
The paper "On the Alaoglu-Birkhoff equivalence of posets" by Todorcevic and Zapletal seems relevant, since it discusses the relationship among several natural preorderings on posets, including $\lhd_1$, Tukey reducibility and others. The paper is available open access at projecteuclid.org/…
Nov
26
comment Which of these relations on partial orders allows us to identify forcing equivalence?
If I'm interpreting the definitions correctly, I think $\lhd_1$ and $\lhd_4$ are equivalent as long as the forcing notions in question are separative. Certainly $\lhd_4$ implies $\lhd_1$ since a separative forcing $\mathbb{P}$ is forcing equivalent to its Boolean completion RO($\mathbb{P}$). And if $\mathbb{Q}\lhd_1\mathbb{P}$ then in particular there is $\tau$ a RO($\mathbb{P}$)-name for a RO($\mathbb{Q}$)-generic. The map sending q in RO($\mathbb{Q}$) to the Boolean value $[q\in\tau]$ (calculated in RO($\mathbb{P}$) is a complete embedding.
Nov
24
awarded  Nice Question
Sep
17
comment Theory of (definable) ideals on a multi-dimensional countable set
Any/all of the above. All the structural results in the literature on 'definable' ideals I know of would follow from determinacy. I'm willing to make large cardinal assumptions here, so I have a fairly large umbrella in mind. But if something about multidimensional ideals can be extracted from stronger definability assumptions I'd be happy to hear about them..
Sep
17
asked Theory of (definable) ideals on a multi-dimensional countable set
Jun
27
comment Forcing over set theory versus forcing over arithmetic
Can you recommend a good general reference for forcing over models of arithmetic? (for someone who knows the set theory side, but not the arithmetic side very well)
May
28
revised Mathias forcing with Ramsey ultrafilters, and Cohen reals
added 86 characters in body; deleted 25 characters in body
May
28
comment Mathias forcing with Ramsey ultrafilters, and Cohen reals
Ah right, yes I was mentally conflating bounded and finite. (The Cohen real I described only needs the $A$ in $\mathcal{U}$ to have unbounded intersection with the $A_k$). Thanks