FOM: 53:Free Sets/Reverse Math

Stephen G Simpson simpson at
Wed Jul 21 13:59:43 EDT 1999

Harvey Friedman 19 Jul 1999 14:11:48 

 > This concerns the reverse mathematics of the well known free set
 > theorem for omega.

The well known free set theorem?  I hadn't heard of this theorem
before.  Apparently it says:

  For all F:N^k --> N there exists an infinite set A such that for all
  x1,...,xk in A, if F(x1,...,xk) is in A then F(x1,...,xk) is among

Harvey, this seems like a nice consequence of Ramsey's theorem, but
what is its motivation?  Does it have something to do with the
existence of freely generated subalgebras in a given algebra?

I seem to remember some combinatorial set theory about large cardinals
with properties something like this.  Is there a relevant concept
called Jonnson cardinals?  I am away from my library and can't easily
look this up.

Harvey, could you perhaps fill in some of this background?

-- Steve

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