[FOM] CH and mathematics

Arnon Avron aa at tau.ac.il
Wed Jan 23 15:03:38 EST 2008

On Mon, Jan 21, 2008 at 04:08:37PM -0500, joeshipman at aol.com wrote:
> Some set theorists (for example, Patrick Dehornoy here:
> http://www.math.unicaen.fr/~dehornoy/Surveys/DgtUS.pdf.
> ) believe that recent work of Woodin makes CH a much more definite 
> question.
> Woodin has argued, roughly speaking, that the properties of the sets of 
> hereditary cardinality aleph_1 are invariant under forcing only if CH 
> is false; therefore only a set theory where CH is false can settle many 
> open questions about sets that are quite small and "early" in the 
> set-theoretic hierarchy. If we believe those questions 
> themselves have 
> definite answers, then we should accept that CH is false.

First, I do not believe those questions have definite
answers either. Second, even if we do, the argument at most
implies that only if CH is false we may be able to 
*know* their definite truth value. But it is quite
possible that there are definite questions whose definite 
truth-value we shall never be able to actualy know 
(maybe the twin-primes problem or GC, or the consistency
of NF are such problems. Who knows). 

 It seems to me that you confuse "having a definite truth
value" with "it should be possible to determine this
truth-value". Perhaps for constructivists the two claims
are identical, but not for ordinary people.  

> I don't necessarily buy this line of reasoning myself, but a lot of 
> people regard it as genuine progress, so you should address it if you 
> want to argue for CH's indefiniteness or our inability to settle it.

I do not buy this line at all, so I feel no need to address it. 

Let me add that in general I cannot understand an argument 
that something is true because things will look  bad if it is not. 
What a kind of a *mathematical* argument is this?? (and 
if productivity is the issue rather than meaninmgfulness and 
truth, than why not accept as true V=L, which is a very 
fruitful axiom?).

Arnon Avron

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