Timeline for Set Theory and V=L
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14 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Nov 28, 2012 at 19:29 | comment | added | Joel David Hamkins | Ziggurism, it will depend on the details of your Gödel coding. | |
| Nov 28, 2012 at 18:28 | comment | added | ziggurism | So I gotta ask, what's the least real number? | |
| May 4, 2010 at 12:20 | comment | added | Gerald Edgar | Isn't this in Goedel's original papers? As I recall, he proves four consequences of V=L, but AC and CH are just the first two. | |
| May 4, 2010 at 10:34 | comment | added | Joel David Hamkins | You are right; truth of a single statement (which is all we need here) is arithmetic in the code. The full first order order satisfaction relation is hyperarithmetic in the code, since one says "there is a real coding the satisfaction relation..." or "for all reals coding the satisfaction relation...", since this real is unique, and the property of being the first order satisfaction relation is arithmetic in the code by the inductive definition of satisfaction. | |
| May 4, 2010 at 4:36 | comment | added | user5810 | "So satisfaction of any expressible formula is arithemtic in the code." The problem is that that does not hold uniformly. What I think would be needed is something like an arithmetical formula that takes (n,m,real) and returns whether the resulting formula is true. (exactly that can't actually exists) | |
| May 4, 2010 at 4:04 | comment | added | Joel David Hamkins | For your question about satisfaction, you don't need to translate it separately into an arithmetic statement---the general phenomenon is that when a real codes a countable structure, then quantifying over that structure amounts to quantifying over the natural numbers, via the code. So satisfaction of any expressible formula is arithemtic in the code. Just imagine that you have a real coding a structure, and imagine how you could say using this code that this structure satisfied a first order statements. | |
| May 4, 2010 at 4:02 | comment | added | Joel David Hamkins | The fact that reals are added at a countable stage is part of Goedel's proof that CH holds in L. (Since $L_{\omega_1}$ is the union of all $L_\alpha$ for couintable $\alpha$, it has size $\omega_1$ and contains all reals. This part of the proof uses the Condensation principle. | |
| May 4, 2010 at 3:17 | comment | added | user5810 | "if $x$ is a real number of $L$, then it appears at some countable stage $L_\alpha$ for a countable ordinal $\alpha$" So thaaat's how it works. (Could I find a proof that $\alpha$ must be countable somewhere?) As far as verifying that the structure thinks V=L, how would you define "the nth formula with the mth tuple of parameters constructs set x in the $\beta$th level" arithmetically? Or would you not actually need to do that? | |
| May 4, 2010 at 3:13 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| May 4, 2010 at 2:57 | vote | accept | CommunityBot | ||
| May 4, 2010 at 2:45 | comment | added | Joel David Hamkins | The differences between Baire space or Cantor space or $P(\omega)$ does not matter in the argument, which works equally well for any of them. | |
| May 4, 2010 at 2:23 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| May 4, 2010 at 2:17 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |