[FOM] Question about Freiling's axiom of symmetry

[Timothy Y. Chow]

Thanks to Ali Enayat for his response to my question.

I should have said that by "CH" I was thinking of what Ali called "WCH" 
(every uncountable subset of R can be put into 1-1 correspondence with R) 
because in the absence of AC, I'm wary of stating CH in terms of aleph_1.

The consistency of ZF + AS + LM + WCH already indicates to me that AS 
doesn't really provide evidence against (W)CH.  All it does is confirm 
that AC and probability don't mix well, which we already knew.  More 
specifically, I would argue as follows: "Let's exploit our probabilistic 
intuition to come up with axioms.  Then AS is plausible, and so is LM.  
Can we now refute WCH?  No.  Freiling's refutation of CH relies crucially 
on the well-orderability of R, and we know that the well-orderability of R 
by itself already contradicts our probabilistic intuition.  So CH is a red 
herring."

Establishing the status of ZF + LM + ~WCH would help clarify the picture 
further, especially if it turns out to be inconsistent (though I would be 
very surprised to learn this), though I don't think it's crucial for the 
point I'm hoping to make.

Tim