[FOM] Question about Freiling's axiom of symmetry

Timothy Y. Chow tchow at alum.mit.edu
Mon Aug 1 23:39:09 EDT 2011

There was some discussion on FOM back in 2004 about Freiling's paper on 
the axiom of symmetry and the continuum hypothesis.


Here is a question I realized that I couldn't answer.  Taking ZF as our 
base theory, what is the relationship among the following statements?

AS:  Freiling's axiom of symmetry.
LM:  All sets are Lebesgue measurable.
CH:  The continuum hypothesis.

In particular, are ZF + AS + LM + CH and ZF + AS + LM + ~CH both 
consistent?  (As usual, assume large cardinals if needed.)

If so, then this would seem to be another argument that AS just supports 
one's intuition that LM is more plausible than AC (the axiom of choice) 
and has little to do with CH.


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