Timeline for On representing Turing Degrees relative to iterates of the jump
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Jul 12, 2017 at 22:05 | comment | added | Noah Schweber | @Dpiz Each countable ordinal can be represented by a binary relation on (a subset of) $\mathbb{N}$; in fact, by many. These are called codes of the ordinal. For any code $c$ of $\alpha$, $0^{(c)}$ has a natural definition; the problem is that $0^{(c)}$ and $0^{(d)}$ may differ wildly, even if $c$ and $d$ are both codes for $\alpha$, if $\alpha$ is not computable. As far as choosing codes goes, this definitely requires more than just DC: the theory ZF + AD + DC proves that we can't pick codes all at once. | |
| Jul 12, 2017 at 21:24 | comment | added | Dylan Pizzo | What do you mean by "fixing a code for $\alpha$", and could you do that with only Dependent Choice? | |
| Jul 12, 2017 at 18:24 | comment | added | Joel David Hamkins | But I see now that you wrote $\not\geq 0^{(\alpha)}$, rather than $\leq 0^{(\alpha)}$, and so you are not actually requiring those degrees to be cofinal. Any strictly increasing $\omega_1$-sequence will indeed define an index function. | |
| Jul 12, 2017 at 18:01 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |