Timeline for Are some interesting mathematical statements minimal?
Current License: CC BY-SA 4.0
12 events
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| Apr 23 at 15:48 | comment | added | Joel David Hamkins | Yes, if the minimal model exists, then it will think that ZFC is arithmetically sound. The reason is that if ZFC proves an arithmetic statement $\varphi$ inside the minimal model, then it is really a proof of $\varphi$ in the ambient universe where the minimal model exists, and so $\varphi$ must hold in all ZFC models in that ambient context, and this includes the minimal model. So $\varphi$ is true in that model, as desired. | |
| Apr 23 at 7:41 | comment | added | user21820 | Does it satisfy "ZFC is arithmetically sound"? | |
| Apr 22 at 20:44 | comment | added | Joel David Hamkins | @FrodeAlfsonBjørdal Yes, you can iterate transfinitely, although this becomes a subtle matter for large ordinals, since one requires an ordinal notation system to express the theories that arise, and this is a necessary aspect in order for the assertions to be expressible. One effect is the phenomenon identified by Feferman that you mention. | |
| Apr 22 at 20:43 | comment | added | Joel David Hamkins | @Corbin The minimal model is a model of ZFC, which proves that for every ordinal $\alpha$ both $\aleph_\alpha$ and $\beth_\alpha$ exist. The minimal model is countable, however, so it doesn't have the genuine alephs or beths, but it has ordinals that it thinks fulfill the defining properties of these cardinals. | |
| Apr 22 at 19:43 | comment | added | Frode Alfson Bjørdal | @Joel David Hamkins "The minimal model satisfies Con(ZFC) and Con(ZFC+Con(ZFC)) and so forth, iterated many times, since these are arithmetic consequences of the existence of a transtive model of ZFC." Transfinitely many times, as Feferman had, for PA, to obtain arithmetical completeness? | |
| Apr 22 at 19:02 | comment | added | Corbin | For my own edification: This means that, in the minimal model, all alephs exist but not all beths? Or is "all beths exist" independent? | |
| Apr 22 at 11:09 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 21 at 21:25 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 21 at 21:19 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 21 at 21:15 | vote | accept | Frode Alfson Bjørdal | ||
| Apr 21 at 21:07 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Apr 21 at 21:00 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |