Timeline for Anti-large cardinal principles
Current License: CC BY-SA 3.0
6 events
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| Apr 26, 2017 at 15:46 | comment | added | Andrés E. Caicedo | @NeilBarton Finally, Martin's maximum (which appeared shortly thereafter), seems to have been at least partially motivated by an attempt to prove the consistency of the principle (see the statement of the second theorem in page 40 of the 1986 paper). | |
| Apr 26, 2017 at 15:44 | comment | added | Andrés E. Caicedo | @NeilBarton There's also a more subtle fact. It is a consequence of the proper forcing axiom (actually, $\mathsf{BPFA}$ suffices) that any forcing that adds a subset of $\omega_1$ must either add a real or collapse $\omega_2$. The principle is an attempt at an ultimate generalization of this result. In their 1986 paper on the maximality principle, Foreman, Magidor and Shelah show that $0^\sharp$ implies that any nontrivial forcing that belongs to $L$ adds a real when used to force over $V$. The principle claims a similar consequence without restricting where the poset used to force comes from. | |
| Apr 26, 2017 at 3:38 | comment | added | Mohammad Golshani | @NeilBarton It is called a maximality principle since it implies the existence of many sets, for example it implies that the GCH fails everywhere, or there are no $\kappa$-Souslin trees for all uncountable regular cardinals $\kappa$ (so all trees are in some sense fat). | |
| Apr 25, 2017 at 9:37 | comment | added | Neil Barton | Thanks for this. In what sense is Foreman's Maximality Principle a ``maximality'' principle? It strikes me as more of a `dichotomy' principle (or similar), but I may be missing something here. | |
| Apr 24, 2017 at 18:19 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
added 367 characters in body
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| Apr 24, 2017 at 15:46 | history | answered | Mohammad Golshani | CC BY-SA 3.0 |