[FOM] REFORMATTED Correction to my "Cantor'd argument" post REFORMATTED
stevnewb@ix.netcom.com
stevnewb at ix.netcom.com
Mon Feb 10 17:04:11 EST 2003
To: Eudora Nicknames::F.O.M. Subject: REFORMATTED Correction to my "Cantor'd argument" post REFORMATTED
Several people have kindly pointed out to me that the following line from my "Cantor's argument" post is incorrect: "We know already from the Loewenheim-Skolem Theorem that models exist in which the Cantor Theorem is false"
That was VERY sloppily put. I was referring to the non-denumerability results which Cantor **derived** from the theorem in question. Loewenheim-Skolem Theorem tells us that models exist in which the non-denumerability results do not hold, e.g., that for ANY set of cwffs which have an infinite model, there are models of EVERY transfinite cardinality. 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. But since they are contingent, then all cwwfs which materially imply them are also contingent, [or
contradictory!] because
.B => C. <=> .Dom(B) sbst Dom(C). - - - [Contingent Implication]
so if Dom(C) is only a proper subset of the universe, then Dom(B) must also be only a proper subset of the universe, which is impossible unless B is also contingent.
That is the idea which I was attempting to convey, but failed from being in a hurry. [There's a moral in here someplace!]
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.
A contingent cwff can be validly implied only by another contingent cwff, and even then only if [Contingent Implication] holds.
My apologies to anyone who lost time over my sloppy statement.
Steve Newberry
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