Timeline for On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$
Current License: CC BY-SA 3.0
15 events
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| Oct 18, 2013 at 23:08 | comment | added | Asaf Karagila♦ | @Joel: Well, in the context of set theory and forcing, sure. But in the general context of Boolean-valued models? I rarely met those, and I never met the term "name" outside forcing. I'll take your word for it, though. | |
| Oct 18, 2013 at 23:00 | comment | added | Joel David Hamkins | I take them as synonomous in this context, but 'name' is probably more common. | |
| Oct 18, 2013 at 18:39 | comment | added | Asaf Karagila♦ | @Joel: Huh, good to know. I usually remember the naming conventions quite well (and not the content). Thanks for the corrections. Also, in the last comment I presume that by "name" you meant "term". Right? | |
| Oct 18, 2013 at 17:27 | comment | added | Joel David Hamkins | More generally, any Boolean-valued structure (not just models of set theory) is said to be full, if for every $\varphi$ there is a name $\tau$ such that $[[\varphi(\tau)]]=[[\exists x\ \varphi(x)]]$. | |
| Oct 18, 2013 at 17:23 | comment | added | Joel David Hamkins | @Asaf, that principle is widely known as the "fullness" principle. The mixing lemma is the assertion that: if $A$ is a maximal antichain and $\tau_a$ is a name for each $a\in A$, then there is a name $\tau$ such that $[[\tau=\tau_a]]\geq a$ for each $a$. One usually proves fullness by appeal to mixing. The Maximality Principle, in contrast, is the principle that says: if $\varphi$ is forceable in such a way that it remains true in all further forcing extensions, then it was already true in the first place. | |
| Oct 18, 2013 at 7:30 | comment | added | Asaf Karagila♦ | @Noah: Yeah, I was also slightly baffled by the name when I first heard it. But it sort of fits. It works out nicely as well: in the context of $\sf ZF$ one can say that a forcing poset has the $\sf MP$ property if we can apply the mixing lemma; but also if it satisfies the maximality principle. :-) | |
| Oct 18, 2013 at 7:26 | comment | added | Noah Schweber | Good to know - I'd never heard it called that before. | |
| Oct 18, 2013 at 7:25 | comment | added | Asaf Karagila♦ | @Noah: It is also known as "the maximality principle" in forcing, stating that if $p\Vdash\exists\tau\varphi(\tau)$, then there is a name $\tau$ such that $p\Vdash\varphi(\tau)$. | |
| Oct 18, 2013 at 6:57 | comment | added | Noah Schweber | Asaf, what is the "mixing lemma?" | |
| Oct 17, 2013 at 23:14 | comment | added | Asaf Karagila♦ | Generally, the structure of cardinals becomes quite interesting in $\sf ZF$. It can get so... strange. | |
| Oct 17, 2013 at 23:01 | comment | added | Asaf Karagila♦ | Well, it is consistent to have the failure of $\sf AC$, but with $\sf DC_\kappa$, for an arbitrary $\kappa$, while having descending chains of cardinals of every ordinal length; and a proper class of incomparable cardinals while at it. I also like the fact that the mixing lemma (from forcing) is equivalent to the axiom of choice. | |
| Oct 17, 2013 at 22:58 | comment | added | user36136 | Dear friend, say something that can "force" me to enter your realm! :-) | |
| Oct 17, 2013 at 22:55 | comment | added | Asaf Karagila♦ | Haha, there are just too damn many of them. :-) | |
| Oct 17, 2013 at 22:54 | comment | added | user36136 | Dear Asaf, Thanks for your useful answer. Also I have a personal question: What is your most favorite fact (theorem, phenomena, project) in set theory without $AC$? | |
| Oct 17, 2013 at 22:49 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |