[FOM] REFORMATTED Correction to my "Cantor'd argument" post REFORMATTED
neilt at mercutio.cohums.ohio-state.edu
Mon Feb 10 23:14:17 EST 2003
On Mon, 10 Feb 2003 stevnewb at ix.netcom.com wrote:
> I should NOT have said "models exist in which the Cantor Theorem is
> false", but rather that the Cantor theorem **materially implies** the theorems
> which state that Pwr(omega) and the Reals in (0, 1) are non-denumerably
> infinite. These latter theorems are contingent because there are
> countable domains in which their negations are true.
No, there are not. The theorems in question are a priori and necessary.
You are confusing the non-standardness of a model of a theory with the
falsity of a theorem of that theory. Every theorem of the theory is true
in every model of the theory, even if the model is question is a
non-standard model of that theory.
> That is the idea which I was attempting to convey, but failed from being
> in a hurry. [There's a moral in here someplace!]
The moral is: respect mathematical proof as a source of a priori knowledge
of necessary truths!
> I should probably repeat the definition of contingency:
> A cwff is ABSOLUTE iff it is either a contradiction or a tautology [=the
> negation of a contradiction]. A cwff is CONTINGENT iff it is not absolute.
So what about the cwff "~0=1"? Is this a contingent arithmetical truth?
> A contingent cwff can be validly implied only by another contingent
So are you a relevantist logician, then? Do you maintain that the
non-contingent cwff A&~A does NOT logically imply the contingent cwff B ?
I happen to believe that the answer is affirmative, but many on this list
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