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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 |
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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 |
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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 |
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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 |
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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 |
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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.. |