Timeline for What does it mean to suspect that two conjectures are logically equivalent?
Current License: CC BY-SA 3.0
12 events
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| Mar 24, 2021 at 15:04 | comment | added | user147820 | thank you very much, the references are much appreciated. | |
| Mar 24, 2021 at 15:02 | comment | added | Noah Schweber | @AtticusStonestrom I learned reverse math by starting with the paper Ideals in computable rings and alongside that reading the first bit of Subsystems of second-order arithmetic. The latter is more useful as a reference than a self-study text, but the first chapter is an exception to this, and is a good brief introduction. There's also a more recent book Slicing the truth, but I haven't actually worked with it in detail. | |
| Mar 24, 2021 at 14:59 | comment | added | user147820 | woah!!! (+1) I find the commutative ring example here really mindblowing, somehow it "feels" much weaker to me than the other statements. do you have any recommendations for a (brief) introduction to reverse mathematics? | |
| May 4, 2018 at 23:18 | comment | added | Brendan W. Sullivan | Relevant to the concept of reverse mathematics, I highly recommend James Propp's article from the American Mathematical Monthly entitled "Real Analysis in Reverse": arxiv.org/abs/1204.4483 | |
| Feb 6, 2018 at 23:59 | comment | added | jeq | You say "Historically, of course, the most well-known example is ...". Arguably, a better-known example is the 1785 equivalence of Playfair's Axiom with Euclid's Fifth Postulate, see Wikipedia. | |
| Oct 26, 2017 at 23:01 | comment | added | Matthew Kvalheim | Very interesting, +1! One small quibble: do you mean "completeness of the real numbers", as opposed to compactness? | |
| Oct 24, 2017 at 12:20 | vote | accept | Dustin G. Mixon | ||
| Oct 24, 2017 at 4:23 | comment | added | Noah Schweber | @PaceNielsen RCA$_0$ is really weak - it corresponds to computable mathematics, and computability is a seriously strict condition! RCA$_0$ doesn't know that $xR$ exists: $xR$ is defined in a non-computable way (to tell that $y\not\in xR$, we have to search through all $z\in R$). In fact, "$xR$ always exists" is equivalent to ACA$_0$! See this paper. | |
| Oct 24, 2017 at 3:49 | comment | added | Pace Nielsen | Noah, are you sure that "every non-zero non-field has a nontrivial proper ideal" is equivalent (over RCA$_0$) to the weak Konig lemma? This seems to just be a consequence of the definitions: If $R$ is non-zero, and not a field, we can fix $x\in R$ such that $x\neq 0$ and $x$ has no inverse. The ideal $xR$ is proper and nontrivial. | |
| Oct 24, 2017 at 2:59 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| Oct 23, 2017 at 19:44 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| Oct 23, 2017 at 19:37 | history | answered | Noah Schweber | CC BY-SA 3.0 |