Uncountability of the Rationals?
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From: MoeBlee on 11 Jun 2010 19:36 Also, I'd still like to know: (1) What does Orlow mean when he says he doesn't object to ZFC itself but only the way it's been "extended" or "perverted" (my term here, which I'm guessing captures his sense). Orlow says classical logic okay. But with classical logic, from the axioms of Z set theory we prove there exists a limit ordinal. But Orlow rejects that there exists a limit ordinal (right?). So has ever said whether it is the axiom of infinity (alone or with other axioms) that he rejects? Then how is that not to reject ZFC. (By the way, "reject" or whatever word Orlow prefers, since I don't insist that one be in confined to a dichotomy of accepting/rejecting theories, but only that if one does reject a theory, then I don't see how it would rational to say that still one does not reject any aspect of the logic or clause in the axioms.) (2) If Orlow objects to set theoretic cardinality spoken of as 'size', would his objection disappear if instead we used the word 'zize' (especially since 'size' is not ordinarily even an "official" nickname for the operation nicknamed 'card')? MoeBlee
From: Virgil on 11 Jun 2010 20:06 In article <b660a0b3-d8ef-4948-9990-6fda3360854b(a)x27g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 11, 12:00�pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > f(g(x)) cannot equal g(f(x)), can it? �The functions are f:N -> S and > > g:S -> N, so f(g(x)) is in S, while g(f(x)) is in N. > > Yes, it most certainly can. Not unless N = S.
From: Virgil on 11 Jun 2010 20:14 In article <71d1ed73-ecb7-4f7b-9102-c3fb9d59941d(a)k39g2000yqd.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Consider this. Are the following two statements > logically equivalent? > > AneN n<omega > > ~EneN n>=omega > > The first says there is no ntaural that can be the size of the entire > set of naturals. That's true. > > The second says that there is no n in N that is greater than or equal > to this "size". > > Does either imply, actually, that such a size exists? What is your definition of "size"? If one's definition of "size" of a set is is "the smallest ordinal that bijects with the set", then given the axiom of choice, every set has a size. > No. Both assert > that no natural number will suffice. However, this is not to say that > there exists ANY number which does. Logically, you must concede, > neither statement proves that omega exists as a number. I am not > obligated to imagine such a number, or give it any credibility. You must concede N as a set, and one can easily define 'number' so that the number of elements in N exists.
From: Virgil on 11 Jun 2010 20:17 In article <53563acd-1d94-47d2-9970-4aec4d956421(a)k39g2000yqd.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Ordinals have nothing to do with my theory. I've called the von > Neumann ordinals "schlock" for years. You should know that, at the > very least. Considering John Von Neumann, whom I once met, versus Tony Orlow, whom I have no desire to meet, if either ever produced any "schlock", it must have been TO.
From: Virgil on 11 Jun 2010 20:18
In article <631018f5-22b5-4d46-860c-c1c874634cd2(a)r27g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > > (1) What logic at this time do you propose for carrying out your > > mathematics. Classical, multi-valued, para-consistent, some other > > logic that you've devised? > > So far, I am using classical logic. If we could ever get into > probabilistic logic I'd be amazed. If so, TO is using as little of it as possible, and perhaps even less that that. |