[FOM] independence of RVM/attribution

[joeshipman@aol.com]

I didn't exactly misattribute the unprovability of RVM to Solovay. I 
was aware it was previously known to be unprovable, what I meant to say 
was that Solovay showed how powerful RVM could be. (What I actually 
said was that Solovay showed that RVM was "more powerful", when what I 
meant was "MUCH more powerful").

Friedman is correct that the unprovability of RVM was known in the 
30's, but it's a bit misleading say it was known "soon after Godel's 
Second Incompleteness Theorem". Godel's Second Incompleteness Theorem 
immediately gives the unprovability of the existence (or even 
consistency) of STRONG inaccessibles, but to get the unprovability of 
the existence of WEAK inaccessibles you need Godel's later work on 
Constructible sets and the consistency of GCH.

-- JS

-----Original Message-----
From: Harvey Friedman <friedman at math.ohio-state.edu>
To: fom <fom at cs.nyu.edu>
Sent: Sun, 12 Feb 2006 11:46:52 -0500
Subject: [FOM] independence of RVM/attribution

  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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