An uncountable countable set
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From: Virgil on 23 Aug 2006 22:18 In article <ecijd7$7mn$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > MoeBlee wrote: > > Albrecht wrote: > >> There is no relevance in which system the axiom is found. > >> E.g. the Axiom A: "Axiom A is wrong", is self contradicting, regardless > >> of which other axioms are used, I think. The same holds for the axiom > >> of infinity. > > > > You miss the point. Since you've not shown any contradiction in set > > theory, whatever contradiction you claim to have found must be a > > contradiction between set theory and something else outside of set > > theory. But if you can't articulate that something else as a > > mathematical formula, then no one much cares that set theory conflicts > > with your not mathematically articulated principles. > > Hi MoeBlee - How are you? > > Set theory contradicts with: > > (1) E y e N, A x>y, x< 2*x < x^2 < 2^x (y=2) That isn't even a well formed statement so it is not even false, but merely meaningless.
From: Virgil on 23 Aug 2006 22:27 In article <ecik82$975$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> I have never said I think there is a largest natural. I have said that > >> some of your assumptions lead to that conclusion. > > > > A while ago you said you don't claim set theory is inconsistent, only > > that it is not consistent with your own notions. But what you just said > > is tantamount to claiming set theory to be inconsistent. > > True. That was quite a while ago. The more I reflect on that wonderful > newbie's contribution, the question as to how many bits are required to > list all the naturals in binary, the more I see it as that gaping hole > in the hull of this wreck. If set theory claims to have a cardinality > which fits every set, then this set stands out as the counterexample. That is no more a problem with set theory that the problem of finding the exact number of binary bit positions necessary to count up exactly to 10 but no farther. Most "numbers" do not have an exact number of binary bit positions that allow you to count up to that number but no farther. The only issue is whether one can count up to at least that number, and infinitely many bit positions are enough for the naturals. This number doesn't exist. It > can't be classified. So, yes, I think you have an actual flaw within. > Sorry for the bad news. Release the orange, before the zookeeper comes > with real shackles. :) > > > > > >> Huh? You mean, if I reject your axiom system, and concoct another, the > >> fact that yours came first means that contradictions between the two are > >> the fault of mine? Hmmmm..... Nah. If my ideas all fit together and > >> provide at least as robust a system as the status quo, then seniority > >> really doesn't count. > > > > Seniority should have some weight, but should not be the final arbiter. > > If your system is markedly better, then it deserves adoption. But > > you've never come close to presenting a system, so the question is > > nugatory. > > Did you just give me a nuggie? Okay, that's war! > > I have presented a system, with a unit infinity of reals per unit, and > the same infinity of naturals per infinite line. I don't contemplate the > navel like the standard system, that is, the "least" infinity. There is > no such thing. Divide any finite interval containing an infinity of > reals with a distinguishable midpoint, and you have two smaller infinite > sets of reals. > > > > >> Define "truth". > > > > Formal definition is given in a formal meta-theory. Greatly simplified, > > a sentence S is true in a model M iff the evaluation function per M > > (definition of this function given courtesy of the defintion by > > recursion theorem) with the sentence as argument yields the set of all > > functions on the variables into the domain of the model. > > Maybe that's a little simplified. Not sure what you're saying. Sorry. > > > > > MoeBlee > >
From: Virgil on 23 Aug 2006 22:37 In article <ecikqd$a1o$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > imaginatorium(a)despammed.com wrote: > >> It's easy to say that without mentioning any specific inconsistencies. > > > > Well, here are a few statements from you, culled from nearby posts (at > > least in the google "chronological order")... > > > > (Quoted in post by Moeblee above) > > Tony Orlow wrote: > >> I have never said I think there is a largest natural. I have said that > >> some of your assumptions lead to that conclusion. > > > > So you think there is no largest natural? > > Right. There is no end to counting. > > > > > (Quoted in post by you, three down the list) > > David R Tribble wrote: > >> Tony Orlow wrote: > >>> But Monsieur, what about the injection from P(N) into N, via the bit > >>> strings which denote set membership, each of which also corresponds to a > >>> binary natural? Tsk, tsk. Mustn't forget that one! Remember, the only > >>> set which doesn't map is the entire set, and that maps to the largest > >>> natural, that is, ...1111 with all bits in finite positions. > > > > But you think that "...1111" _is_ the largest natural. > > I said, given that all bit positions are finite, this constitutes a > finite value (previously inductively proven), and is the largest finite > (given that no other string can be greater in any bit position), as well > as the mapping to the entire set. Since neither finite naturals nor > countably infinite subsets thereof have an end, and this string doesn't > either, it represents both equally well. It's not a proper value, > however, within the T-riffic system. In order to calculate relative size > of infinite sets, one needs a variable range. Thus TO claims that there is no largest and also that there is a largest. Any system like that is to be avoided at all costs. > > I wonder if you see any inconsistency in the Finlayson thingies: where > > between every pair of rationals is another rational, yet each rational > > has an adjacent rational to the right of it? > > I'm not sure Ross has ever said any such thing. Others are sure. >The Finlayson numbers > are nilpotent infinitesimal "degenerate" intervals, sequential yet > indistinguishable on the finite scale. This garbage about scales again. If a < b on any scale then a < b on every scale. A MATHEMATICAL inequality does not become an equality when you put away your magnifying glass. Nor does a MATHEMATICAL equality become an inequality when you look at it under higher magnification. Does TO also claim that while x is a member of A at one magnification, it need not be so at another, or that A being or not being a subset of B depends on the magnification used by the observer? Silliness compounded!
From: Virgil on 23 Aug 2006 22:39 In article <ecil8v$aqh$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > Tony Orlow schrieb: > > > > > >>> Ordinals and cardinals are necessities if we want to talk about > >>> set "order" and "size" in any kind of logical, well-defined way. > > I didn't write that. :( Pity! You should have.
From: Virgil on 23 Aug 2006 22:47
In article <ecile2$aqh$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > Since Dedekind, unbounded means infinite, but it's just not so. ROll > back the film a little. Find any dictionary which allows unbounded to mean finite, even if it predates Dedekind, if you can. The conjoining of unbounded and infinite goes back at least to Zeno of Elea, who predates Dedekind by a couple of millennia at least. Webster?s Concise Electronic Dictionary 4 sense(s) for ?finite? 1. fi?nite (adjective) having definite limits fi?nite (noun) [fi?nites] fi?nite?ly (adverb) fi?nite?ness (noun) Proximity/Merriam-Webster U.S. English Thesaurus 1 meaning(s) for ?finite? 1. (adj) having definite or definable limits or boundaries (synonym) bound, bounded, limited (related) confined, restricted, definable, defined, definite, determinate, fixed, terminable, exact, precise, specific (contrast) boundless, unbounded, unlimited, absolute, complete, total (antonym) infinite Proximity/Franklin U.S. English Thesaurus 1 meaning(s) for ?finite? 1. (adj) having only finitely many elements (synonym) numbered (antonym) infinite ====================================== Websters Concise Electronic Dictionary 6 sense(s) for infinite 1. infinite (adjective) having no limit or extending indefinitely vast infinite (noun) infinitely (adverb) infiniteness (noun) infinitude (noun) [infinitudes] Proximity/Merriam-Webster U.S. English Thesaurus 2 meaning(s) for infinite 1. (adj) being without known limits (synonym) eternal, illimitable, perdurable, sempiternal, supertemporal (related) everlasting, perpetual (contrast) bounded, circumscribed, limited, restricted (antonym) finite 2. (adj) having no limits (synonym) limitless, boundless, endless, immeasurable, indefinite, measureless, unbounded, unlimited, unmeasured (related) bottomless, countless, incalculable, incomprehensible, inexhaustible, innumerable, undrainable, unfathomable, vast, wasteless (contrast) bound, bounded, finite, fixed, limited, measurable, comprehensible, fathomable, confined, restricted (antonym) limited Proximity/Franklin U.S. English Thesaurus 2 meaning(s) for infinite 1. (adj) greater than any finite value (synonym) boundless, transfinite, uncountable, limitless, unbounded, unlimited (antonym) finite 2. (adj) lasting for all time (synonym) ageless, timeless, dateless, eternal, everlasting, permanent, amaranthine, perennial, all-time, for keeps (related) lasting (antonym) fleeting |