Timeline for Compactness-like property for universal generalization?
Current License: CC BY-SA 3.0
9 events
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| Jan 19, 2013 at 22:17 | comment | added | Nick Thomas | Goldstern: Ah, I understand. That's a useful insight; thank you! | |
| Jan 19, 2013 at 21:42 | comment | added | Goldstern | I meant: "there is a model in which there are infinitely many elements satisfying $\phi$". | |
| Jan 19, 2013 at 21:17 | comment | added | Nick Thomas | Goldstern: My trouble is figuring out what other relations between the models might be relevant here. (Obviously I'll post if I figure that out.) Unfortunately, I do not understand the part in quotation marks. :-/ (Care to explain more?) $\phi$ does not mention the well-order. Francois: Thanks for the suggestion! I am going to play with it and see if it gets me anywhere. | |
| Jan 19, 2013 at 15:39 | comment | added | François G. Dorais | A good strategy would be using the Omitting Types Theorem to omit the type $\lnot\phi(x)$. Unfortunately, your hypotheses don't seem to imply that $\lnot\phi(x)$ is not isolated. | |
| Jan 19, 2013 at 15:24 | comment | added | Goldstern | Is there any other relation between the models $M_n$? If not, then your assumption is equivalent to "there is a model $M$ with "$\phi^M$ infinite". (Assuming your formula is first order.) Also: please tell us if $\phi$ mentions the well-order. | |
| Jan 19, 2013 at 8:01 | comment | added | Nick Thomas | Andres: That's an excellent question, and it shows that what I'm asking for can't be done in general. In my specific problem, $\phi(x)$ has a form which excludes that case. But it seems clear that I haven't asked the right question, because I haven't included enough constraints to yield a solvable problem. I will see if I can repair my question; and in the meantime, thanks for your help! | |
| Jan 19, 2013 at 7:40 | comment | added | Andrés E. Caicedo | How are we excluding the situation where $\phi(x)$ says that $x$ has property blah and that there is a $y$ without property blah? | |
| Jan 19, 2013 at 7:36 | history | edited | Nick Thomas | CC BY-SA 3.0 |
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| Jan 19, 2013 at 7:31 | history | asked | Nick Thomas | CC BY-SA 3.0 |