[FOM] CH and mathematics

Alex Blum blumal at mail.biu.ac.il
Sun Jan 20 05:50:26 EST 2008

Is Feferman's question  in effect an expression of doubt about the 
meaningfulness of CH? Would you say that the same would have been true 
of Fermat's last theorem would have been proved independent of the 
axioms of arithmetic?
    You speak of mathematicians intuitions and refrain from speaking of 
mathematical truth. But intuitions about what? Can't all mathematicians 
be mistaken?  I now quote from H.Jeome Keisler, in the same book:"What 
would happen if a contradiction were found in a very weak system on 
which we rely, such as primitive recursive arithmetic?"p180
    I would rather think that the status of CH as a set theoretic 
proposition has not changed over time. We now know that it is 
independent of the other axioms but its truth is no more a problem than 
is that of any of the ZF axioms or for that matter any other 
mathematical proposition; or is it?

Alex Blum

Arnon Avron wrote:

>In two recent postings asked:
>>In his recent entry in Philosophy of Mathematics:5 Questions, Solomon
>>Feferman asks: "Is the Continuum Hypothesis a definite mathematical 
>>If is not a mathematical problem, what kind of a 
>>problem is it?
>If you assume apriorily that CH is a * definite problem*, then 
>of course it is a definite mathematical problem. 
> For some reason you omitted the crucial word "definite" when you
>formulated *your* problem. But this is the main issue.
>  Set-theoretical platonists would say that CH is a definite
>problem. But non-platonists like me, who have always had a 
>problem with the concept of "arbitrary set of natural numbers", 
>not understanding in what sense such sets "exist", and who 
>doubt that the term "P(N)" has a definite (=absolute) meaning, 
>necessarily doubt that CH is a definite proposition, with a definite
>truth-value. The fact that CH cannot be decided in the strongest
>systems that the overwelming majority of  the mathemaricians 
>can claim to have some intuitions about,` casts even stronger
>doubts that CH can be said to have a definite truth value.
>Arnon Avron 

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