Timeline for Saturated extensions of ZFC models
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2011 at 7:54 | vote | accept | Sumac | ||
| Mar 11, 2011 at 2:03 | answer | added | Justin Moore | timeline score: 4 | |
| Mar 10, 2011 at 18:19 | comment | added | Sumac | @Justin : I am not sure what you mean. My reasoning in my question was wrong, because I forgot that notions like "finite" are not absolute since the saturated model $N$ won't be transitive. The saturated model $N$ of course still must satisfy Foundation. To tell you the truth I can't really visualize clearly how $N$ can satisfy Foundation and at the same time have elements like $v'$, but I guess the whole construction of finite ordinals in $N$ must be strange enough to permit such thing. | |
| Mar 10, 2011 at 17:59 | history | edited | Sumac | CC BY-SA 2.5 |
deleted 4 characters in body
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| Mar 10, 2011 at 1:49 | comment | added | Justin Moore | @Andreas and Sumac: isn't the problem really well foundedness, not transitivity? | |
| Mar 9, 2011 at 23:00 | comment | added | Andrés E. Caicedo | @Sumac: Yes, that's correct; in this case we have absoluteness of "finite" and "natural number". But the extension $M[G]$ is not countably saturated, and (unless $M=M[G]$) it is not an elementary extension of $M$. | |
| Mar 9, 2011 at 22:12 | comment | added | Sumac | @Andreas Blass, Andres Caicedo : Thank you very much for your fast response. Indeed I forgot to notice that since $N$ won't be transitive (even if $M$ is) then most of the concepts like "finite" etc won't be absolute. Just to be sure I understand though... If we had extended a countable transitive model of ZFC $M$ to some $M[G]$ through forcing, since $M[G]$ is also c.t.m. of ZFC, we wouldn't be able to add any natural numbers in $M[G]$. | |
| Mar 9, 2011 at 21:14 | comment | added | Andreas Blass | @Sumac: Your belief that "natural numbers and $\omega$ are absolute" would be correct if you were dealing with transitive models or, more generally, if $M$ were transitive from the point of view of $N$. But in the situation you consider, there is no transitivity available. | |
| Mar 9, 2011 at 20:53 | comment | added | Andrés E. Caicedo | If $a$ is a natural number of $M$, it is also a natural number of $N$, but not vice versa, so being a natural number is not absolute, only upwards absolute. What you are adding is a new natural number. Of course, $M$ doesn't see it. Similarly, "finite" is not absolute. There are finite objects in $N$ that $M$ has no way of seeing. If $a$ is in $M$ and $N$ thinks $a$ is finite, then of course it is finite in $M$, but that is weaker than saying that "finite" is absolute. | |
| Mar 9, 2011 at 20:37 | history | asked | Sumac | CC BY-SA 2.5 |