[FOM] CH and mathematics
Vladimir Sazonov
Vladimir.Sazonov at liverpool.ac.uk
Mon Jan 21 18:04:33 EST 2008
joeshipman at aol.com wrote:
> >From: Arnon Avron <aa at tau.ac.il>
> >
> >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.
>
> I don't see why our inability to know something should cast any doubt
> on its definiteness. That's epistemological arrogance.
>
> We'll very likely never know whether the googolplexth decimal digit of
> pi is even or odd; does that cast doubt on its definiteness?
Why not? Anyway, "the" intended world of natural numbers is imaginary
one. In that imaginary world this digit d should have some value, just
according to our intuition which agrees with the formal rules governing
this world (and our intuition). But I absolutely do not see which ever
way this could lead to definiteness of the value of this digit. In
principle I can imagine that (like for non-standard models or like for
Godel completeness theorem) some variation of this imaginary world of
numbers might "exists" where the digit in question can be different.
May be both PA + d = 0 and PA + d = 1 are (practically) consistent, and
there is no good choice (in any sense we could imagine) for either
alternative.
If this question would have some relation to the real world (like when
asking "what is the fifth decimal digit of pi?"), then we could refer
to something real and objective. We can DEFINITELY derive the value of
this digit. (This was actually done!)
Thus, I even do not understand at all what is the objective meaning of
this "definiteness" (of course, assuming that we, human beings, will
never derive/compute/deduce the value of this digit).
Please, will you explain, WHAT DOES THIS DEFINITENESS REALLY MEAN?
Vladimir Sazonov
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