An uncountable countable set
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From: Mike Kelly on 7 Sep 2006 09:58 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <44fd9eba(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Dik T. Winter wrote: > >>>>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > >>>>> writes: > >>>>> Your axiom uses things that are not defined. What is the *meaning* of > >>>>> "x<z"? > >>>> Geometrically it means that x is left of z on the number line. > >>> And for someone standing on the other side of the number line would x be > >>> on the right of z? > >>> > >>> And does the line stay horizontal as one moves around earth? Which way > >>> is larger if the line ever goes vertical. And how does the "larger" work > >>> at antipodes? > >>> > >> Silly questions. > >> > >>> > >>>> It means > >>>> A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it > >>>> needs to, wouldn't you say? > >>> Not hardly. > >>> A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) > >>> is a bit better but still insufficient. > >> True, I should have specified y<>x and y<>z. I guess it's usually done > >> using <= for this reason, eh? > >> > >>>>> > > That is not a definition, because it makes no sense. "The set of > >>>>> > > naturals > >>>>> > > is as large as every natural"? > >>>>> > > >>>>> > It is not larger than all naturals > >>>>> > >>>>> That is something completely different again. > >>>> It's not LARGER than every finite. > >>> Which natural(s) is it "not larger" than", in the sense of not being a > >>> proper superset of that natural or having that natural as a member? > >> ....11111 binary (all bit positions finite) > > > > That isn't a natural number, Tony. > > > > Are you sure? Pay close attention. > > For any finite bit position n, it and all predecessors can only sum to a > finite bit string value of 2^(n+1)-1. OK. Any string 111....1111 with a finite number of 1s represents a natural in binary. > Since there are only finite bit positions in the string, it can never achieve any infinite value at any position in the unending string of bits. OK. Any string 111....1111 with a finite number of 1s represents a natural in binary. >Therefore the value must be finite. Why? You're supposed to be *demonstrating* that the string represents a value, but you're *assuming* it instead. You've shown that any finite bit string of all 1s represents a finite natural number. And concluded from this that an infinite string of 1s represents a finite natural number. Why? Total non sequitur. > Furthermore, since any such number does have a predecessor and > successor, in this case ....1110 and ...0000, respectively, it fits in > the successorship model. The only concept this breaks is that 0 is now a > successor as well, creating an infinite ring of successorship. Other > than that, it works as a natural, and in fact, this is the way signed > integers work in your very computer. > > So, while ...111 may not be considered a standard natural, I see no > reason why it should not be considered, say, an extended natural. And you think this is a better model of "natural numbers" than standard ones? You think it accords better with the intuitive picture people have of what "counting numbers" are? -- mike.
From: Dik T. Winter on 7 Sep 2006 09:57 In article <4500215f(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Mike Kelly wrote: .... > >> ....11111 binary (all bit positions finite) > > > > That isn't a natural number, Tony. .... > Furthermore, since any such number does have a predecessor and > successor, in this case ....1110 and ...0000, respectively, it fits in > the successorship model. The only concept this breaks is that 0 is now a > successor as well, creating an infinite ring of successorship. Other > than that, it works as a natural, and in fact, this is the way signed > integers work in your very computer. > > So, while ...111 may not be considered a standard natural, I see no > reason why it should not be considered, say, an extended natural. Why not use the proper name such numbers already have? 2-adics. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 7 Sep 2006 10:00 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> Your whole point here is ludicrous. You are arguing that the naturals >> are not a subset of the rationals, which are not a subset of the reals. >> While each superset may require a more complex construction than the >> subset, all elements of the subset are covered by the superconstruction. >> As points on the real line, they are all real numbers. What you're >> saying amounts to what I said above, which is indeed absurd. So, no, I >> don't understand a thing you're saying, when you say naturals aren't >> reals. Sorry. > >> Tony > > int main() > { > int x=1; > float y=1; > } > > Does x equal y? > > Stephen Yes, they even have the same bit string representing them (ignoring length). Try bigger numbers next time. Tony
From: imaginatorium on 7 Sep 2006 10:08 Tony Orlow wrote: > Virgil wrote: > > In article <44fe2642(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <44fd9eba(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: <yawn> > >> ....11111 binary (all bit positions finite) > > > > Unless that string has only finitely many bit positions as well as only > > finite bit positions, it is not a natural at all, as it is then neither > > the first natural nor the successor of any natural, and every natural > > has to be one or the other. > > It is the successor to ....11110. Duh. I've already proven that this is > a finite value, No, you have Tproved that it's a Tfinite value, where Tfinite is some private notion of yours you have never been able to define. (Tdefine, Tperhaps, but only circularly in terms of Tinfinite...) given that all bit positions are finite, and that > therefore no place in that string can achieve an infinite value, and > that any such number has predecessor and successor. The cute thing is > that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) Hmm. So -1 is "essentially" a lot larger than 1, for example, whereas add 5 to both sides and you get the other thing. Well, that's faintly amusing ice pose. Come on Tony, either write something new, or just get on with your Tbook... Brian Chandler http://imaginatorium.org
From: Mike Kelly on 7 Sep 2006 10:11
Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Dik T. Winter wrote: > >>> In article <44ef3b88(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > >>> > Dik T. Winter wrote: > >>> ... > >>> > > But what is the case is that if you accept the axioms, you also have > >>> > > to accept what follows from the axioms. > >>> > > >>> > Yes, I understand that, and much to the consternation of some, I don't. > >>> > >>> Yes, much consternation, I can understand that. So apparently you are > >>> accepting the axioms, but not what follows from the axioms. What kind > >>> of logic are you using? > >> I am NOT accepting the axioms. The axioms are artificial statements > >> which can be made to work together, but which do not follow from > >> elementary logic the way they should. Every axiom should be justifiable > >> based on fundamental concepts. I don't see that here. > > > > Which axiom of ZFC set theory do you find objectionable? > > I object to the claim that the Axiom of infinity defines an infinite > set, and that the Dedekind definition of an infinite set is appropriate > for sets defined using finiteness as a criterion for set membership. I > object to the von Neumann model as being truly representative of the > naturals, and the concept of limit ordinals, transfinite cardinalities, I can't tell from this which axiom you are objecting to. I asked which axiom(s) you find objectionable. Please answer the question directly with the name(s) of the axiom(s) you find objectionable, and why. > > Can you give an example of an axiom "justified by fundamental > > concepts"? > > > > N=S^L is easily proven inductively using successive concatenations of > symbols to a string. The Inverse Function Rule is justified by the fact > that, if we have a function that gives us the change in value over a > given number of successive elements, then the inverse function gives us > the number of elements over a given value range. Simply reflect the > graph over the line y=x, and you now have a graph of the number of > elements in any given range. The principle of induction in the infinite > case is justified by the fact that there is no reason to exclude it, > except in the case of inequalities where the difference has a limit of 0 > as n->oo. That kind of stuff. And you really think these are derivable from pure logic/"fundamental" concepts? Hillarious. -- mike. |