An uncountable countable set
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From: Tony Orlow on 23 Aug 2006 17:49 Dik T. Winter wrote: > In article <ecdtie$q90$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > ... > > > No. Required because if 0 has also to have an inverse in a ring, trivially > > > there is only one ring. The ring consisting of the single element 0 with > > > the standard operations. Because, suppose we have a ring. It can be > > > proven that in a ring 0*a = 0 for every a in that ring. But if 0 has > > > an inverse, say b, we also have 0*b = 1, and so 0 = 1. > ... > > The ring is broken by the > > reluctance to allow absolute oo to balance 0 in this respect. > > Not reluctance. If we add oo to a ring we have to define the operations > on it (addition and multiplication). Moreover, oo should have an > additive inverse (otherwise it is not a ring). Now suppose we define > oo to be its own additive inverse: oo + oo = 0. Further suppose we > define n + oo as oo for all n. We now have: > 3 = 1 + 0 + 2 = (use oo + oo = 0) > 1 + oo + oo + 2 = (use n + oo = oo + n = oo) > oo + oo = 0. > So we can not define n + oo as oo. So we *must* have n + oo is some > other number, say m. But now we have oo = m - n... > Conclusion, it is not possible to add a single oo without breaking the > ring structure. No reluctance present. OK, Dik, you're probably right, given the exact definition of a ring. I'll have to review that carefully, and see. There is a paradox about oo=-oo, but x^oo <> x^-oo for x<>1. I'll try to take a look at this and think. Thanks. .... Tony
From: Tony Orlow on 23 Aug 2006 18:04 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) because: (2) A y e N, aleph_0>y and (3) aleph_0/2 = aleph_0 = aleph_0^2 < 2^aleph_0 (1) is trivially inductively provable. (2)and (3) are from transfinitology. > > So I take it from your response that you don't have a set of axioms for > your mathematics. So I wonder how you expect people to evaluate whether > something is or is not a theorem of your mathematics. Can you evaluate the relationship between the above three statements? > > MoeBlee > Thanx, Tony
From: Tony Orlow on 23 Aug 2006 18:18 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. Any finite number of bit positions produces a finite set of strings. Any countably infinite set of bit positions produces an uncountable set of strings. And, there's nothing in between. 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: Tony Orlow on 23 Aug 2006 18:28 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> David R Tribble wrote: >>> Tony Orlow wrote: > > <snip> > >>>> Please >>>> show where I am being inconsistent, not with set theory, but within my >>>> own assumptions. Remember, I don't claim to believe in transfinite set >>>> theory, and don't intend to be consistent with it. >>> The problem is that your own assumptions are not consistent with >>> each other - they don't form a coherent theory. Many of these >>> inconsistencies have been pointed out to you before, but you choose >>> to not believe them or simply to ignore them. >>> >> 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. > > No inconsistency so far, I suppose, in your view. See above, and keep in mind the context of the conversation. > > 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. The Finlayson numbers are nilpotent infinitesimal "degenerate" intervals, sequential yet indistinguishable on the finite scale. There is a certain density per unit interval, which is infinite. This is also the number of naturals on the infinite real line, as corroborated using IFR, thanks. > > Does this thing that is the largest natural, that doesn't exist, i.e. > "...1111", does it have a left end? Perhaps it goes on for ever to the > left, and never stops, and the place where it stops is called Infinity. Yeah, like I said, left-endless numbers like that are immeasurable. That's why I invented the T-riffics. > > No, no inconsistency here, eh? You tell me. Nice try, though. > > Brian Chandler > http://imaginatorium.org > :) TO
From: Tony Orlow on 23 Aug 2006 18:35
mueckenh(a)rz.fh-augsburg.de 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. :( > > That is wrong. We can talk about finite sets and about infinite sets > which all have the same magnitude by the measure of intercession > instead of bijection. > >>> What do you call the "size" of a countable set with no end? >>> You don't have a name for it, do you? >> No, I find focused concentration on the Twilight Zone, the "boundary" >> between finite and infinite, to be a rather fruitless exercise. There is >> no such distinct boundary. > > Because there is no boundary at all. All sets are finite, but some are > without end. Those are called potentially infinite. All of them have > the same magnitude if measured by intercession. Intersection? Some sets are actually infinite. The universe is infinite. It's a set of things too. > > Definition: "A and B intercede": An order can be defined such that: if > b_1 and b_2 are elements of set B, then a at least one element a of set > A lies between them in this order, and if a_1 and a_2 are elements of > set A, then at least one element b of set B lies between them in this > order. Like rationals and irrationals, if the two are finitely discernible. > > Example: Rational and irrational numbers intercede in the natural order > by magnitude. Yes, they are both dense on the real line, and actually infinite, if we allow bit positions, or denominators and numerators, to become actually infinite. :) Tony > > Regards, WM > |