[FOM] Why does mathematics reverse?

Sam Sanders sasander at cage.ugent.be
Tue Sep 28 13:46:23 EDT 2010

I invite the members of the FOM to consider the following question:

"Where does the Main Theme of Reverse Mathematics come from?"

In particularl:

"Why is it that theorems of ordinary mathematics correspond to a very small number of logical principles, given
that there are infinitely many such principles?"

Some background on this question:

Reverse Mathematics (RM) is a program in the Foundations of Mathematics initiated by Harvey Friedman in the Seventies.
Stephen Simpson's excellent book 'Subsystems of Second Order Arithmetic' is the standard reference.  The aim of RM is to 
find the minimal axioms that prove a theorem of ordinary mathematics (given a weak base theory, called RCA_0). 

In many  cases, the minimal axioms can be also derived from the theorem, in the base theory RCA_0. (This is the 'reverse' direction, hence RM)
Thus, RCA_0 proves the equivalence of the minimal axioms and the theorem.  With very few exceptions, 
theorems from ordinary mathematics are either provable in RCA_0 or equivalent to one of the other 'Big Five': WKL_0, ACA_0, ATR_0, \Pi_1^1-CA_0.    
This phenomenon is called the Main Theme of RM.

Here, 'ordinary mathematics' should be understood as countable mathematics.  RM typically takes place in subsystems second-order arithmetic.  
As many uncountable objects have a dense countable subset (e.g. the rationals are dense in the reals), the scope of countable mathematics is quite large.   

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