Timeline for Question regarding $W$ as not hyperarithmetic
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 19 at 18:58 | comment | added | SSequence | I don't think you should. There are number of useful references in the answer. | |
| Jan 19 at 18:52 | comment | added | Noah Schweber | @SSequence Ah, I see; that wasn't clear to me. I'll delete this answer if you'd like. | |
| Jan 19 at 18:45 | comment | added | SSequence | None of these. I mentioned originally in (Q1) that it is possible to show non-hyperarithmeticity of $W$ using admissibility of $\omega^{CK}_1$. So I was just wondering what showing admissibility (perhaps multiple ways of showing it) of $\omega^{CK}_1$ actually entailed (read: "required in steps of its proof"). This is why the other answer is quite helpful. | |
| Jan 19 at 18:40 | comment | added | Noah Schweber | @SSequence I'm a bit confused: leaving circularity issues aside, which fact on its own are you worried about? The non-hyperarithmeticity of $W$, or the admissibility of $\omega_1^{CK}$? | |
| Jan 19 at 18:37 | comment | added | SSequence | Yes, you are indeed right. While I don't understand your answer fully, it seems that part of what you are saying is that (please correct if I am wrong) once we show that supremum of hyperarithmetic well-orderings on $\mathbb{N}$ is equal to $\omega_{CK}$ then $W$ being non-hyperarithmetic follows easily from that. I can see why that is true (though using $<\omega_{CK}$-time definition of hyperarithmetic tbh). This is what I was trying to say in comments below the answer by Farmer S. Also, thanks for the references. They are certainly useful. | |
| Jan 19 at 18:31 | history | answered | Noah Schweber | CC BY-SA 4.0 |