Timeline for On statements independent of ZFC + V=L
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| S Jul 25 at 8:02 | history | suggested | Lucenaposition | CC BY-SA 4.0 |
added MathJax
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| Jul 25 at 5:25 | review | Suggested edits | |||
| S Jul 25 at 8:02 | |||||
| S Mar 22, 2018 at 16:05 | history | suggested | Alexander Pruss | CC BY-SA 3.0 |
fix typos in text
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| Mar 22, 2018 at 15:45 | review | Suggested edits | |||
| S Mar 22, 2018 at 16:05 | |||||
| Jan 13, 2010 at 3:59 | vote | accept | Ashutosh | ||
| Jan 13, 2010 at 3:59 | comment | added | Ashutosh | I see. Thanks again for your comments. | |
| Jan 12, 2010 at 13:24 | comment | added | Joel David Hamkins | @Ashutosh2: About "interesting". Surely large cardinals should count as interesting, since they have been studied in set theory for over a century (starting well before the Incompleteness Theorem). Part of their interest now, I believe, is the fact that they do transcend ZFC in consistency strength. After all, we know that such transcendent statements must exist by the Incompleteness Theorem, and large cardinals provide a rich hierarchy able to measure such transcendent strength. The point of my answer was that these axioms also can show the rich diversity of models of V=L. | |
| Jan 12, 2010 at 13:13 | comment | added | Joel David Hamkins | @Ashutosh1: No independence-from-ZFC result can be established in ZFC alone, unless ZFC is inconsistent, since the assertion that something is independent of ZFC or of ZFC+anything already implies Con(ZFC). But I think you meant "in ZFC+Con(ZFC)" alone. This makes the question considerably more difficult, for the reasons you mention in your question. I don't know any examples, although I would love to hear about any. | |
| Jan 12, 2010 at 4:03 | comment | added | Ashutosh | The large cardinal assumptions, compatible with $V=L$, are nice but we need to assume the consistency of their existence and the proof of independence is almost the same as for $ZFC$ alone (but that not relevant to the question I asked). One more comment: By using the adjective "interesting", I intend to eliminate Goedel's diagonal sentences and their variants; e.g. stuff like there is a set model of $ZFC$ etc. | |
| Jan 12, 2010 at 4:03 | comment | added | Ashutosh | Thanks! This brief excursion into the theory of large cardinals looks very interesting but allow me to ask a few more question. Can you give an example of an "interesting sentence" $\phi$ whose independence w.r.t. $ZFC + V=L$ can be established in $ZFC$ alone? | |
| Jan 12, 2010 at 3:46 | comment | added | Joel David Hamkins | Yes, I guess one can have in L a transitive model of ZFC+0# exists. Indeed, there is such a model in V iff there is such a model in L, since this is a Sigma^1_2 statement. But of course, in L they will all be fake 0#'s from the perspective of the full L. | |
| Jan 12, 2010 at 3:27 | comment | added | François G. Dorais | +1 Again, a wonderful answer! I really liked your thought provoking examples. They made me realize just how subtle 0# really is... | |
| Jan 12, 2010 at 2:14 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |