Wikipedia:Reference desk/Mathematics: Difference between revisions - Wikipedia

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*Arithmetic:

**<math>\Pi_1^0</math> - [[Goldbach's conjecture]], [[Fermat's last theorem]]

**<math>\Sigma_2^0</math> - ?[[P=NP]] (conjectured to be false)

**<math>\Pi_2^0</math> - [[Twin prime conjecture]], P ≠ NP conjecture

**<math>\Sigma_2Sigma_3^0</math> - ?

**<math>\Pi_3^0</math> - [[Waring's problem]]

**higher - ?

* Analysis

**<math>\Pi_2Sigma_1^12</math> - [[Continuum hypothesis]] (notassuming sureaxiom of choice, see discussion)

**<math>\Pi_1^2</math> - [[Weak König's lemma]]

**<math>\Pi_2^2</math> - [[Continuum hypothesis]] (alternate formulation)

** ?

* Algebra

**<math>\Delta^2_2</math> Artinian rings are Noetherian

Can anyone add good, mathematically interesting entries to the above? Is there a list like this anywhere? Does the concept make any sense? Thanks.

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:::Compactness ''in general'' is a notion from [[general topology]], not from classical analysis. Certainly if you just want to talk about what subsets of R^''n'' are compact, that's much simpler.

:::The "no intermediate cardinality" version of CH is indeed <math>\Pi^2_2</math>, but the more useful (and in my view more fundamental) <math>2^{\aleph_0}=\aleph_1</math> version is simpler, namely <math>\Sigma^2_1</math>. Given the axiom of choice, the two are equivalent. Without the axiom of choice you could have the "no intermediate cardinality" version without a lot of the consequences that we think of as following from CH. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 17:14, 22 September 2010 (UTC)

Thanks, I've updated the list a little bit. What I'm really wondering though is whether collecting lists like this is interesting and if it has been done elsewhere. [[Special:Contributions/71.141.90.138|71.141.90.138]] ([[User talk:71.141.90.138|talk]]) 21:15, 22 September 2010 (UTC)

== Instantaneous Acceleration/Uniform Circular Motion ==