Uncountability of the Rationals?
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From: Jesse F. Hughes on 16 Jun 2010 13:09 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > So it appears to me that I don't know 1/3 is in R at all. This chain of > reasoning does not end in a statement that I know. Can you give me a > finite proof (or, let's be generous, anything that looks like a proof) > that 1/3 is a real number. In fact, I'd be interested also in a proof that 8 is a real number. -- Jesse F. Hughes "I thought it relevant to inform that I notified the FBI a couple of months ago about some of the math issues I've brought up here." -- James S. Harris gives Special Agent Fox a new assignment.
From: Aatu Koskensilta on 16 Jun 2010 13:18 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > In fact, I'd be interested also in a proof that 8 is a real number. You can't come up with a proof on your own? Go crank your non-standard crank, you crank. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 16 Jun 2010 13:33 On Jun 16, 8:15 am, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 15, 8:22 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > If TO accepts all of the axioms of ZFC, but rejects the > > theorem "there exists a cardinal (or ordinal) number," > > then I'll agree to call TO "wrong." Still, I believe that > > if we can show him a theory which does satisfy his > > intuitions, he'll have less of a reason to criticize the > > adherents of ZFC. What do you mean Transfer Principle "adherent of ZFC"? > Okay, perhaps I am "wrong" about this. I am going over the axioms of > ZFC, and I simply don't see any reference to any primitive referring > to ordinality or cardinality Also not among the primitives are mentions of subsets the empty set union intersection pairs ordered pairs natural numbers prime numbers (thank you, Aatu) metric spaces Banach spaces or ANYTHING other than equality (if taken among the primitives) and elementhood and the variables, the sentential connectives, and the quantifiers (and left and right parentheses, if we don't go Polish). MoeBlee
From: Virgil on 16 Jun 2010 14:48 In article <f6d161b0-150a-4c47-bea4-bc898e5a0f87(a)i31g2000yqm.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Actually, I don't see any explanation of how ZFC proves their > existence. I will probably be told to just go read a book... Didn't know you could!
From: Virgil on 16 Jun 2010 14:52
In article <f6d161b0-150a-4c47-bea4-bc898e5a0f87(a)i31g2000yqm.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Sometimes there are logical arguments themselves which don't appeal to > everyone equally. What I have been told is that the size of omega must > be larger than every natural since no natural is large enough to > express it, and so it is some infinite number, aleph_0. However, that > logical argument, as I pointed out, really just proves that aleph_0 > cannot be finite. Along with the argument that any initial segment of N > + of size x contains an xth element whose value is x, which would > imply that aleph_0 or omega is a member of N+, we arrive at a > contradiction implying that aleph_0 cannot actually exist. Which axiom > contradicts this logic? Using the von Neumann naturals, or any other set of naturals which starts with 0, as is becoming fairly standard in mathematics, no natural need be anything like a member of itself, which is a consummation devoutly to be wished. |