Timeline for What are some reasonable-sounding statements that are independent of ZFC?
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13 events
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| Jul 23 at 13:37 | comment | added | Alex Kruckman | I've checked my knickers, and I don't see any twists. I just think it's important to correct common misconceptions :0) Statements in the language of ZFC are usually not number-theoretic - they are not equivalent to statements about the natural numbers in the language of arithmetic. But statements about ZFC, e.g. about what it can and cannot prove, are number-theoretic. | |
| Jul 23 at 13:23 | comment | added | Alex Kruckman | @GerryMyerson after Gödel coding, CH is a particular number, not a number-theoretic statement | |
| Jul 23 at 13:08 | comment | added | Gerry Myerson | @Alex, Asaf and I were just having a little fun, so don't get your knickers in a twist. (But, as long as we're here, why isn't CH number theoretic, after Gödel coding?) | |
| Jul 23 at 12:48 | comment | added | Alex Kruckman | @GerryMyerson It's not true that every mathematical statement is a number theoretic statement (with any reasonable definition of "number theoretic statement"). For example, $\mathrm{CH}$ is not number theoretic, while $\mathsf{ZFC}\vdash \mathrm{CH}$ is number theoretic (after Gödel coding). And there's a big difference between these two statements, since the first is independent while the second is provably false. | |
| Feb 15, 2019 at 21:40 | comment | added | Asaf Karagila♦ | @Gerry: I draw the line at $0<1$. But that's just me... | |
| Feb 15, 2019 at 21:19 | comment | added | Gerry Myerson | @Asaf, of course, every mathematical statement is a number theoretic statement. But we have to draw lines somewhere. | |
| Feb 15, 2019 at 15:21 | comment | added | Asaf Karagila♦ | @Gerry: To be fair, Con(ZFC) is a number theoretic statement... :-) | |
| Feb 15, 2019 at 12:18 | comment | added | Gerry Myerson | The question asks for "statements that aren't directly set-theoretic". I don't think any statement containing "ZFC" qualifies. | |
| Feb 15, 2019 at 9:21 | comment | added | Asaf Karagila♦ | And according to even stronger theories, it's provable... | |
| Feb 15, 2019 at 9:19 | comment | added | Christopher King | @AsafKaragila Well, according to stronger set theories, the negation is also unprovable. That makes it independent, right? | |
| Feb 15, 2019 at 9:16 | comment | added | Asaf Karagila♦ | It's not independent. It's unprovable. | |
| S Feb 15, 2019 at 8:08 | history | answered | Christopher King | CC BY-SA 4.0 | |
| S Feb 15, 2019 at 8:08 | history | made wiki | Post Made Community Wiki by Christopher King |