Timeline for Non-constructive existence proofs without AC?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 20, 2013 at 19:02 | history | edited | Goldstern | CC BY-SA 3.0 |
you meant L[c], not L
|
| Mar 5, 2013 at 22:14 | comment | added | Joel David Hamkins | Ben, could you elaborate on which kind of systems you might have in mind? All that was really required for the example is the existence of a single model of the theory having instances of a property but no definable instances of the property. For any such theory, the same trick will work. | |
| Mar 5, 2013 at 22:12 | comment | added | Joel David Hamkins | Rodrigo, indeed, my example about generic Cohen reals has your form. I have edited the final example also into this same form, since it shows that every subtheory of ZFC also admits the properties. | |
| Mar 5, 2013 at 22:10 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 201 characters in body
|
| Mar 5, 2013 at 20:29 | comment | added | user21349 | I wonder whether this is a universal idea or just something specific to ZF(C). Are there, e.g., any examples of interesting, widely studied systems in which the nonconstructive axioms are an infinite axiom schema? In a case like that, the argument given above would fail. | |
| Mar 5, 2013 at 14:45 | comment | added | Rémi Peyre | Terrific! I did not expect that the proof would be so simple... I wish I asked it on MO earlier ;-) Thanks again! | |
| Mar 5, 2013 at 14:42 | vote | accept | Rémi Peyre | ||
| Mar 5, 2013 at 12:29 | comment | added | aws | Something interesting to note about your final paragraph is that it only works in classical logic. There are examples in intuitionistic set theory where it does not work. | |
| Mar 5, 2013 at 12:10 | comment | added | Rodrigo Freire | The problem with this (bad) understanding of the nonconstructive character of AC arises within logic itself. Suppose that $P$ has the property for ZFC. Let $P^′$ be $\exists x P\rightarrow P$. In this case, $\exists x P^′$ is a theorem from logic which has no definable witness in ZFC. This stuff is analyzed in my paper "On Existence in Set Theory", NDJFL, v. 53, n.4, 2012. | |
| Mar 5, 2013 at 11:46 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 574 characters in body; added 2 characters in body
|
| Mar 5, 2013 at 11:33 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 584 characters in body; added 3 characters in body
|
| Mar 5, 2013 at 11:26 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |