Talk:Reverse mathematics

message: See the discussion page there.  (Just mentioning it here for reference.) --Gro-Tsen 00:35, 7 February 2006 (UTC)
thread_title:  Criticizing the move to [[second-order arithmetic]] 

message: See my comment in the discussion under second-order arithmetic
--CMummert 02:10, 20 March 2006 (UTC)
thread_title:  Changes on 2006-3-20 

message: I moved this from the main page.  I don't think it should be deleted, so I am putting it here until I can work it back into the article.
 For example, while usual (forward) mathematics is done in 
 the language of set theory and in the system of  
 Zermelo-Fraenkel set theory with the axiom of choice 
 (which, unless the contrary is explicitly stated, is assumed 
 to be the foundation system taken for granted by working
 mathematicians), this system is really much stronger than is necessary.  
CMummert 15:39, 28 June 2006 (UTC)
thread_title:  Saving a phrase 

message: I am a mathematician, but not at all expert in logic. After seeing "...than recursive comprehension can..." under the title "Additional systems" I was puzzled. A minute after I'he guessed that "recursive comprehension" and "RCA0" are probably the same. But still, should we replace the title "The base system" with "The base system: recursive comprehension" (or something like that)? Boris Tsirelson (talk) 18:59, 20 September 2008 (UTC)
replies: text: Yes, RCA stands for "recursive comprehension axiom", and the 0 indicates that the subsystem only includes induction axioms for Σ^0_1 formulas. Similarly, ACA stands for "arithmetical comprehension axiom". I added the explanation of RCA to the article. Please feel free to edit the articles yourself, as well. — Carl (CBM · talk) 18:55, 21 September 2008 (UTC)
replies: text: Thank you! It is definitely better than what I would do myself. I guessed that the last A means "arithmetic", and had no guess about the 0.Boris Tsirelson (talk) 19:34, 21 September 2008 (UTC)
thread_title:  Does RCA<sub>0</sub> stand for Recursive Comprehension? 

message: I added the Jordan curve theorem to the list of WKL0-equivalent statements, based on a mention and citation that I found in the Jordan curve theorem article.  I hope this was correct. 69.228.171.150 (talk) 16:05, 22 October 2009 (UTC)
replies: text: Yes, it's correct to add it here. Thanks for noticing it. — Carl (CBM · talk) 16:10, 22 October 2009 (UTC)
replies: text: I guess my concern was over whether the statement needed to be qualified in some way.  I'll leave that part to you, if there is any issue. 69.228.171.150 (talk) 16:15, 22 October 2009 (UTC)
thread_title:  Jordan curve theorem 

message: I changed the redirect for Weak König's lemma to point to a section of this article instead of to second-order arithmetic.  It may have been left from before the article forked.  Was that right?  If yes, then there are a bunch of other such redirects that I can also change. 69.228.171.150 (talk) 17:04, 22 October 2009 (UTC)
replies: text: I think you are right; the other article doesn't even mention the lemma. — Carl (CBM · talk) 18:59, 22 October 2009 (UTC)
thread_title:  redirect 

message: Adding a note about the subscript 0 is a good idea. The only thing that's tricky is the usual definitions make it hard to give a one-line summary of what's going on.  I think that the regular editors here, like R.e.b., are aware of this, but for other people, here's a summary of the situation. 
RCAo has the \Sigma^0_1 induction scheme, which happens to have the particular "second order induction axiom"
:[0 \in A \land (\forall n)(n \in A \to n+1 \in A)] \to (\forall n)(n \in A).
For systems ACAo and stronger, that one axiom would be enough, since those systems have sufficient comprehension to construct the sets for which induction proofs are used. For RCAo, the whole scheme of \Sigma^0_1 induction is needed, because only \Delta^0_1 comprehension is included, and we regularly prove things about properly \Sigma^0_1 sets by induction. — Carl (CBM · talk) 11:18, 20 October 2010 (UTC)
thread_title:  induction and subscript 

message: Is that quote from Rényi verified?  He died in 1970, before reverse mathematics was invented according to the article.  I have also heard a category-theoretic version of the same saying: a comathematician is a machine for turning cotheorems into ffee.  A cotheorem is of course something one deduces from a rollary.
71.141.88.54 (talk) 10:50, 14 February 2011 (UTC)
replies: text: I'm sure it was a joke. It was added by a non-logged-in editor from Cambridge . — Carl (CBM · talk) 12:46, 14 February 2011 (UTC)
thread_title:  Theorems into coffee 

message: Should Ulm's theorem be included under ATR_0? The formulation in the article Ulm's theorem looks like the statement in Simpson's theorem V.7.1, which is provable in RCA_0. After proving theorem V.7.1, Simpson says that the logical strength to reach ATR_0 is in asserting that the Ulm extensions exist, and instead theorem V.7.3 is the equivalence with ATR_0. Usually a theorem stated like "for any gizmos G, H, their widgets are equivalent" will include the assertion "the widgets of G and H exist" in its statement, as it is formulated as \forall(G,H\in\mathrm{Gizmos})\exists W,Z(W,Z\text{ are widgets of }G,H\land W\equiv Z), however Ulm's theorem appears to be "for all gizmos G, H, if their widgets are equivalent, then ...", with existence of widgets before the conditional. C7XWiki (talk) 18:16, 5 June 2024 (UTC)
thread_title:  Ulm's theorem