An uncountable countable set
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From: Virgil on 15 Sep 2006 19:22 In article <450b0042(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >>> Yes, I am including infinite values on the number line, since it's > >>> "infinitely long". > > > > David R Tribble wrote: > >>> Yet another thing you have to define or prove. Where do these > >>> "infinite values" appear on your real number line? > > > > Tony Orlow wrote: > >> Further from 0 than any finite number. > > > > Then how are they "on" the same "line"? > > > > By trichotomy. For all x ad y on the line, either x>y, x=y or x<y. > That's what makes a line in concept. But until TO proves trichotomy for all his imaginings, they are not on any line. And he has not done so. Note that using either the Dedekind cut or Cauchy sequence definition of reals, one can prove trichotomy for them > > > > > David R Tribble wrote: > >>> Are these infinite values connected (in the mathematical sense) to > >>> the finite values on the line? If not, how is it a "line"? > > > > Tony Orlow wrote: > >> Yes, they are on the line, with all finite values between them and 0. > > > > That's not what "connected" means. If they are not connected, > > they are simply two separate sets, which means it is more correct > > to visualize them as two unconnected (but "ordered") "lines". > > > > That is like saying the starting point of a line segment is disconnected > from the rest of the line. Until TO proves his trichotomy, and connectivity, he has neither. Note that one can prove the density and "continuity" of the reals (contunuity being the LUB and GLB properties. Accordingly TO must show that any non-empty set of his extended reals which is bounded above must have a least upper bound. The set of standard reals is bounded above if there are any infinite reals, so must have a LUB, so what is it TO? > > > > Tony Orlow wrote: > >>> You are mistakenly applying > >>> the standard cardinalistic fact that the number or reals in [0,1] is the > >>> same as the number of all reals. That is false in my system. What "system" is that?
From: Virgil on 15 Sep 2006 19:27 In article <450b037c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > >>>>> Tony Orlow wrote: > >>>>>> Mike Kelly wrote: > >>>>>>> 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: > >>>>>>>>>>> > >>>>>>>>>>>> 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. > >>>>>>>>> In response to a silly definition. > >>>>>>>>>>>> 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) > >>>>>>>>> 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, 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. :) > >>>>>>> Does it not bother you that nobody else agrees with, or even > >>>>>>> understands, your proof? > >>>>>>> > >>>>>> I find it disappointing, but not surprising, that you don't understand > >>>>>> such a simple proof, since it's contradictory to your education. I do > >>>>>> find it annoying that you feel the right to disagree with it without > >>>>>> understanding it. If you feel there is a problem with the proof, > >>>>>> please > >>>>>> state the logical error I made. If the string is all finite bits, and > >>>>>> none of them ever can possibly achieve an infinite value, then how can > >>>>>> the string have an infinite value? There's nowhere in the string where > >>>>>> that can occur. It's that simple. Grok it. > >>>>> 1) A finite string of 1s represents a (finite) natural number. > >>>>> 2) An infinite string of 1s represents a (finite) natural number. > >>>>> > >>>>> 1) doesn't imply 2). > >>>>> > >>>> If the string is unbounded but finite, then 2) follows. > >>> What's a finite but unbounded infinite string? > > I missed that you slipped "infinite" in here. Unbounded but finite may > be considered potentially, but not actually, infinite. > > >>> > >> One with all finite bit positions but no greatest. > > > > So... why does 2) follow from 1? > > Because then the string is considered finite. It's not finite AND infinite. > > > > > 1) A finite string of 1s represents a finite natural number. For > > example, 101 represents > > > > 1* (2^0) + 0* (2^1) + 1*(2^2) > > = 1 + 0 + 4 > > = 5 > > > > this representation bijects finite bit strings of 1s and 0s and natural > > number. > > > > 2) doesn't work however... Take ...11111. Then the natural this > > represents would be > > > > 1 + 2 + 4 + 8 + 16 + 32 + .... > > > > but there is no such natural. This sum is divergent. > > > > The sum diverges to an infinite value as n APPROACHES oo. Divergent series do not diverge 'to' any number at all. that is what "divergent" means. If they did, then what number does 1 + (-2) + 4 + (-8) + ... add up to?
From: Virgil on 15 Sep 2006 19:37 In article <450b05e4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45085df2(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > > > >>>>> It doesn't. I never said it does so please STOP asking that question. I > >>>>> say that 111..... doesn't represent a (natural) value at all. > >>>> If it is a whole number (no fractional component), is finite, and has > >>>> successor and precedessor, then ....1111 is a finite natural. > >>> It isn't. Now what? > >> It's not what? A whole number? Show me the fractional component. Finite? > >> SHow me the bit position where an infinite sum is possible. With > >> successor and predecessor? There is no successor to this largest > >> natural, unless the number line is taken to be circular. > > > > TO is claiming numerals (digit strings) are numbers, which shows that he > > has no notion of what numbers are. > > > > And in order for a number to be a natural number, it must be a member of > > the smallest inductive set, which means that it must be representable by > > a digit string with both a first and last digit. > > Proof: Let S be the set of naturals(i.e., the smallest inductive set) > > representable by digit strings with a first and a last digit (which can > > be the same digit). > > (1) the first natural is a member of S > > (2) if x is a member of S the so is the successor of x. > > Therefore EVERY natural is representable by a digit string with both a > > first and a last digit. > > Please give a concise definition of "number", so that I may mend my > ways, O Virgil. You've never been able to before. There seems to general > disagreement anyway. I have given a precise definition of natural number, as being a member of the minimal inductive set required in both ZF and NBG axiom systems. As my comments were only about what can be called natural numbers, and I have given a precise definition of what I mean by that, TO has no case. > > To equate the naturals with "the smallest inductive set" is artificial. Why? When one is starting up a set theory from nothing but axioms, it seems reasonable to use a set which one can prove exists as a standard. If TO wants to start up his system in other ways, let him present that system in its entirety. Until then we go with what we have, ZF or NBG. > The naturals are reals on the line identified by repeated offsets of the > unit from the origin. That presumes both the reals and some complete system in which they allegedly exist, but for which we have no axiomatic justification.
From: Virgil on 15 Sep 2006 19:40 In article <cf41b$450a799d$82a1e228$10666(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > But, fortunately, reality is more simple than this. Every equality is > an equivalence relation. And every equivalence relation is an equality. > So the bijection between tally marks or collected pebbles with counted > objects means, indeed, that tally marks and moving pebbles ARE numbers. > > Han de Bruijn Not in mathematics.
From: Virgil on 15 Sep 2006 19:40
In article <7f951$450a7b04$82a1e228$10666(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Representation is number. There is no difference. Numerals have no > > "soul". > > Whew! I've never heard someone expressing this fact so lucidly! > > Han de Bruijn or so falsely. |