[FOM] independence of RVM/attribution

Harvey Friedman friedman at math.ohio-state.edu
Sun Feb 12 11:46:52 EST 2006

Shipman http://www.cs.nyu.edu/pipermail/fom/2006-February/009748.html
misattributed the unprovability of RVM (existence of a real valued
measurable cardinal) in ZFC to Solovay. In my response to him, I didn't
clarify this properly.

It is a result of Ulam that every RVM is weakly inaccessible, in 1930.

It was known soon after Godel's second incompleteness theorem that the
existence of a weakly inaccessible cardinal is not provable in ZFC (assuming
ZFC is consistent).

Solovay showed that the systems ZFC + RVM and ZFC + MC (measurable cardinal)
are mutually interpretable. See

R.M. Solovay, Real-Valued Measurable Cardinals, in Axiomatic Set Theory, ed.
Dana Scott, Proceedings of Symposia in Pure Mathematics vol. 13, part 1,
American Mathematical Society, 1971.

Harvey Friedman


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