Timeline for Example of a $\Pi^2_2$ sentence?
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 19, 2021 at 14:22 | comment | added | Andreas Blass | @Carl-FredrikNybergBrodda Apparently some dialects of French are more reasonable. I once heard a waiter at the Marseille airport counting bills as "soixante, septante, huitante, ...." | |
| Aug 19, 2021 at 12:05 | comment | added | Carl-Fredrik Nyberg Brodda | @NoahSchweber That still makes more sense than counting in French... :-) (...sixty-eighteen, sixty-nineteen, four-twenties...) | |
| Aug 18, 2021 at 23:00 | comment | added | Peter LeFanu Lumsdaine | @NoahSchweber: I remember one term in undergrad where I was taking courses in set theory and in extremal combinatorics; in the same week, they both had lectures impressing us with “large number” hierarchies — ordinals on the one hand, towers of iterated exponentials on the other (arising as numerical bounds in things like the Szemerédi regularity lemma). The enormous combinatorial bounds felt to me then, and still feel, far larger than small things like countable ordinals… | |
| Aug 18, 2021 at 18:24 | comment | added | paul garrett | @NoahSchweber, :) | |
| Aug 18, 2021 at 18:22 | comment | added | Noah Schweber | @paulgarrett Personally I count "$0$, some, lots, oh dear, what?, twenty, $\omega$." (... $\omega_1^{CK},\omega_1,\omega_2,\mathfrak{c},\kappa$, ...) | |
| Aug 18, 2021 at 17:42 | comment | added | paul garrett | The lesson is that people studying infinite stuff have trouble with small integers? :) | |
| Aug 18, 2021 at 17:01 | comment | added | Paul Blain Levy | Your "too complicated" example is actually too simple, since it is both $\Sigma^2_2$ and $\Pi^2_2$, just like your first example and my first suggested answer. See my second suggested answer and @FarmerS's comment. | |
| Aug 18, 2021 at 16:28 | history | edited | Noah Schweber | CC BY-SA 4.0 |
deleted 106 characters in body
|
| Aug 18, 2021 at 11:46 | comment | added | Paul Blain Levy | I appreciate the fact that you described subscript 2 as "high complexity". | |
| Aug 18, 2021 at 11:27 | comment | added | Paul Blain Levy | For the translation of "the continuum is greater than or equal to some uncountable limit cardinal", I think you want to say "There exists an $\omega$-sequence of infinite sets of reals of strictly increasing complexity." | |
| Aug 18, 2021 at 11:25 | comment | added | Farmer S | Oops, that should have been "for each $z$ with $x<^¢z<^¢y$, $W$ codes a bijection between the predecessors of $x$ and those of $z$". It should also say that if $S$ has a largest element $x$ (wrt the wellorder) then for each $z>^¢x$, $W$ codes a bijection between the predecessors of $x$ and those of $z$. | |
| Aug 18, 2021 at 11:18 | comment | added | Paul Blain Levy | I too failed the counting-to-three test. | |
| Aug 18, 2021 at 6:52 | comment | added | Farmer S | Hmm, I think "$c<\aleph_c$" is also $\Sigma^2_2$. Say "There is a wellorder $<^¢$ of the reals and sets $S,W$ of reals such that $S$ has cardinality $<c$, and is closed under limits in the sense of $<^¢$, and for all reals $x\in S$, the set of $<^¢$-predecessors of $x$ is a cardinal, and for all reals $x,y\in S$ with $x<^¢y$, if there is no $z\in S$ with $x<^¢z<^¢y$ then $W$ codes a bijection between the set of $<^¢$-predecessors of $x$ and those of $y$". | |
| Aug 18, 2021 at 6:15 | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 338 characters in body
|
| Aug 18, 2021 at 4:00 | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 50 characters in body
|
| Aug 18, 2021 at 3:59 | comment | added | Noah Schweber | @AndreasBlass Oh dear goodness, I failed to count to three. Fixing ... | |
| Aug 18, 2021 at 2:59 | comment | added | Andreas Blass | I should probably be sleeping instead of writing this (and maybe I am sleeping), but your formulation of "$\mathfrak c$ is a limit cardinal" looks $\Pi^2_3$ to me, the relevant quantifiers being "for every X", "there is $Y$", and "there is no surjection". | |
| Aug 18, 2021 at 2:50 | comment | added | Paul Blain Levy | Noah, that would be more than I asked for, but would still be a good thing to have! | |
| Aug 18, 2021 at 0:06 | comment | added | Noah Schweber | @PaulBlainLevy Thanks, but I don't think you should accept this just yet - I don't actually have a proof that there isn't a simpler equivalent. | |
| Aug 18, 2021 at 0:05 | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 333 characters in body
|
| Aug 18, 2021 at 0:01 | comment | added | Paul Blain Levy | Thanks Noah, that's great. | |
| Aug 18, 2021 at 0:00 | vote | accept | Paul Blain Levy | ||
| Aug 18, 2021 at 11:06 | |||||
| Aug 17, 2021 at 23:23 | history | answered | Noah Schweber | CC BY-SA 4.0 |