Timeline for Example of a $\Pi^2_2$ sentence?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 19, 2021 at 16:08 | comment | added | Paul Blain Levy | Thanks, Elliot, for this great example (now that I understand at least the first part). | |
| Aug 19, 2021 at 16:06 | comment | added | Paul Blain Levy | So Elliot didn't fail the count-to-three test. | |
| Aug 19, 2021 at 15:55 | vote | accept | Paul Blain Levy | ||
| Aug 19, 2021 at 15:55 | comment | added | Paul Blain Levy | Thanks @NoahSchweber, your comment makes it clear to me. | |
| Aug 19, 2021 at 14:59 | comment | added | Noah Schweber | @PaulBlainLevy An element $\sigma\in\{0,1\}^{<\omega_1}$ can be coded by a single real: as a pair $\langle A,B\rangle$, where $A$ is a well-ordering of (an initial segment of) $\omega$ with ordertype $\vert\sigma\vert$ and $B$ a subset of the domain of $A$ (thought of as the set of bits on which $\sigma$ takes on value $1$). Of course there is no "canonical" way to assign a code to an element of $\{0,1\}^{<\omega_1}$ in this way, but that's not a problem here. | |
| Aug 19, 2021 at 12:38 | comment | added | Paul Blain Levy | Thanks, Elliot, but I can only see that Suslin's hypothesis is $\Pi^3_2$, by the following argument. Let's take SH in the form: "The tree $\{0,1\}^{<𝜔_1}$ has no subtree in which every chain and every antichain is countable." A countable ordinal can be represented by a set of natural numbers. So an element of $\{0,1\}^{<𝜔_1}$ can be represented by a set of sets of natural numbers. So a subset of $\{0,1\}^{<𝜔_1}$ can be represented by a set of sets of sets of natural numbers. This gives $\Pi^3_2$. How can we get $\Pi^2_2$? | |
| Aug 19, 2021 at 7:14 | history | answered | Elliot Glazer | CC BY-SA 4.0 |