Timeline for Is every ordinal potentially definable?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2015 at 13:47 | comment | added | Theodore Slaman | I think it is worth investigating Joel's idea in a different generic extension of L. Harrington has a paper on Long Projective Wellordings. He can force over L to increase the size of the continuum (C) and ensure that there is a lightface projective well ordering of the reals and also ensure that every set of reals of size less than C is boldface projective, $\Pi^1_2$ I think. If we use Harrington's method to make C $\aleph_{\alpha+1}$ then we should be able projectively define the ordering on the real numbers that occupy the positions of cardinals. | |
| Oct 12, 2015 at 11:39 | comment | added | Joel David Hamkins | Concerning homogeneity: suppose $\alpha$ is uncountable and there is a real $x$ coding $\alpha$ that is definable in my model $V[G]$. Since the forcing is homogeneous, it follows that $\text{HOD}^{V[G]}\subset V$, and so $x\in V$, contradicting $\alpha$ uncountable. And I agree with your point about parameters (which I had also realized shortly after posting my comment). | |
| Oct 12, 2015 at 8:39 | comment | added | Noah Schweber | Re: lightfaceness, I think if we allow a parameter in the generic extension, then every ordinal becomes potentially $\Sigma^0_0$ - take $V[G]$ which collapses $\vert\alpha\vert$ to $\omega$, and use a real coding a copy of $\alpha$ as a parameter. | |
| Oct 12, 2015 at 1:40 | comment | added | Noah Schweber | Could you elaborate on " . . . by homogeneity considerations"? It's not obvious to me. | |
| Oct 11, 2015 at 23:26 | comment | added | Noah Schweber | I was hoping for a lightface definition, yes. | |
| Oct 11, 2015 at 23:23 | comment | added | Joel David Hamkins | My answer shows that if you start in $L$, then for any ordinal $\alpha$ there is a forcing extension $V[G]$ in which the collection of reals coding relations on $\omega$ with order type $\alpha$ is projectively definable. But you want there to be a particular such real that is (lightface) projectively definable? For large $\alpha$ that won't be true in my model for any real, by homogeneity considerations. If I can use $G$ as a parameter, though, then I can use the $L[G]$ order. | |
| Oct 11, 2015 at 23:19 | comment | added | Noah Schweber | Awesome! But yeah, the problem is normally I would use large cardinals to go from a projective definition to a projective copy . . . :P | |
| Oct 11, 2015 at 23:17 | comment | added | Joel David Hamkins | My answer is about defining the order-type, which is somewhat different than what you asked, since to have a projective copy we'd need to pick out a particular copy. I'll think a bit more about the connection. | |
| Oct 11, 2015 at 23:15 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |