[FOM] Explanation/Continuum Hypothesis

Alasdair Urquhart urquhart at cs.toronto.edu
Tue May 9 12:36:59 EDT 2006

Paul Studtmann wrote:

> Were Goedel's remarks intended to be an explanation of the independence of
> the continuum hypothesis or rather something that one can infer as a result
> of its independence? 

His remarks of 1947 were intended as a reason for why we might expect
the continuum hypothesis to be independent.  That is to say, the axioms
are far from determining a unique notion of set, so it is plausible (Goedel
argued) that they also do not determine a unique cardinal number for
the continuum.  

> It seems to me that an explanation would require an 
> explanation as to why the continuum hypothesis holds for one conception of
> set and not the other.  

I am not sure what you require of an explanation here.  Paul Cohen's proofs
show that for certain forcing extensions, the continuum hypothesis fails.  One 
can also give some kind of intuitive explanation of the idea of generic sets, 
as Goedel did, in conversation with Wang.  But apart from that, I don't see 
what kind of  "explanation" one could give other than presenting the proofs.
In the models constructed by forcing, the failure of CH depends on quite 
special features of the forcing conditions chosen, and cannot be inferred
just from general ideas about generic sets.

Alasdair Urquhart

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