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*To*: fom@math.psu.edu*Subject*: FOM: 53:Free Sets/Reverse Math*From*: Stephen G Simpson <simpson@math.psu.edu>*Date*: Wed, 21 Jul 1999 13:59:43 -0400 (EDT)*In-Reply-To*: <v03110734b3b81b8b3451@[24.31.187.108]>*Organization*: Department of Mathematics, Pennsylvania State University*References*: <v03110734b3b81b8b3451@[24.31.187.108]>*Reply-To*: simpson@math.psu.edu*Sender*: owner-fom@math.psu.edu

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 x1,...,xk. 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

**References**:**Harvey Friedman**- FOM: 53:Free Sets/Reverse Math

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