An uncountable countable set
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From: Virgil on 25 Aug 2006 14:51 In article <44eef04d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <44ece1fb(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >>> Tony Orlow wrote: > >>>> 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 > >>> I don't know what you intend '<' to stand for. For the domination > >>> relation? The less than relation on ordinals? > >> The standard "less than" operator, commonly used for finite reals. > > > > But since one is only talking about naturals, which are separate from > > the reals in ZF and NBG, that makes the TO (1) statement nonsense. > > If the naturals are not a subset of the reals in ZFC and NBG, then those > theories are even more screwed up than they already seemed. While there are sets of objects within the rationals and reals which look very similar to the urset of naturals, they are not quite the same. They are merely copies of the naturals within the larger systems. > > >>> I don't know what is meant by '(y=2)' in the larger formula. > >> I mean that for x>2, 2*x < x^2 < 2^2. For x=2 they're equal. Then say so. > > > > Then (1) is a very peculiar way of trying to convey that claim. > > What's peculiar about it? The fact that To has snipped it to start with. > > > > >> Is aleph_0>2? > > > > AS TO's comparison operator is according to TO defined only for reals, > > the question is nonsense. > > > > Then aleph_0 is not a count of any sort, or it would exist on the > hyperreal line. It's not a number of any real sort. Partly right, it is not a real number, and does not appear on the real line. But it is allowed in determining cardinality. So if TO says cardinalities are not "numbers" to him it is not a number, even though to others it still can be a "number". > > > > > > >>> What is transfinitology? > >> You know, Cantorian religion. That Hollywood stuff. ;) > > > > As separate from TO religion and all of that Broadway stuff? > > Broadway was my stomping ground as a kid. That explains TO's flash without substance. > > >>> What is the definition (and in what theory is > >>> this definition?) of '/' where w (omega) is in the numerator? > >> Well, I was calling it Bigulosity. The definition of x/y is "how many > >> intervals of length y fit into an interval of length x?". > > > > As we have no definition of length valid for intervals whose endpoints > > are not finite reals, it means nothing. > > > > That's because you have no infinite values which behave sufficiently > like numbers to support such a definition. Our 'infinite values' behave like cardinal number or ordinal numbers, as appropriate, with no problem, they just do not behave like real numbers. > > > > > >> This is > >> constructible using straightedge and compass. > > > > > > DO let us see your straightedge and compass construction of > > Aleph_0^2/Aleph_0, TO. I need a new wall hanging. > > > The selection of any unit is done by simply choosing a point separate > from the origin. When it comes to division using infinite values, one > translates the geometric definition into symbolic form and applies > induction formulaically. Let's see TO perform this gobbledegook operation, justifying each step according to the rules of some recognizable system of axioms and definitions.
From: MoeBlee on 25 Aug 2006 14:54 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> If the naturals are not a subset of the reals in ZFC and NBG, then those > >> theories are even more screwed up than they already seemed. > > > > With either of the usual constructions of the reals (Dedekind cuts or > > equivalence classes of Cauchy sequences), both the system of naturals > > and the system of rationals are isomorphically embedded in the system > > of reals. Mathematicians usually speak of the system of natural numbers > > and the system of rational numbers as subsystems of the the system of > > reals. Strictly speaking, that is incorrect, but it is harmless given > > the isomorphism. > > Harmless, even though incorrect? What makes it incorrect, if not > inconsistency? Doesn't inconsistency cause a problem? Now you're being captious. The informality is harmless, as I said, because we are within ISOMORPHISM. It has nothing to do with the consistency of the theory. It's only a matter of INFORMAL convenience to not have to say each time "the system that is isomorphically embedded" but instead speak directly of the system as if it were a subsystem, since, to WITHIN ISOMORPHISM, it is a subsystem. This kind of informality is common throughout mathematics and is harmless. > I used y in the equation (or statement of inequality) and then specified > in parentheses at the end, with a space, which condition for y made that > statement true. I didn't think that was tooooo confusing. It was unclear enough that I had to check with you what you meant. It's not a big deal, but you could acheive clarity by being more explicit. > > 'number of any real sort' is not a defined predicate of set theory. w > > is not in the standard ordering of the reals. That does not make w an > > undefined object. > > My experience is that asking amthemticians for a definition of "number" > results in.....nothing. Because it's more a philosophical issue or an issue of terminology outside the system. The purpose of set theory is not to address the question of what is and is not a number. Rather, among the purposes of set theory is to axiomatize and construct the various number systems that are of interest. > >> you don't see scientists accepting transfinitology either. > > > > Of course, you have a survey of scientists to support your claim. > > Uh, yes, right here. Why don't you survey thise that object, regarding > what they do? That's quite an unscientific survey method of yours. > > You have no coherent system of definitions at all. It's not the job of > > set theory to define objects that obey the whims of your informal > > notions. > > It's the job of mathematicians to work with numbers. Numbers are among the primary concern of mathematics. Set theory axiomatizes and constructs number systems. > >> The selection of any unit is done by simply choosing a point separate > >> from the origin. When it comes to division using infinite values, one > >> translates the geometric definition into symbolic form and applies > >> induction formulaically. > > > > "Translates the geometric definition into symolic form and applies > > induction formulaically" is no less doubletalk than "Coordinates the > > numeric form into geometric postulates and applies the recursive > > definition metrically." > > > > Is that a question? Is that a rhetorical question? MoeBlee
From: Virgil on 25 Aug 2006 14:57 In article <44eef224(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <ecikqd$a1o$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> 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! > > No, it all rests on the notions of identity and equality. TO would have one believe that equality is conditional on scale rather than being, as it must be in mathematics, absolute, and independent of scale. Does 1 + 1 = 2 become false under a sufficiently strong magnifier, TO? Does 2 < 3 become false if one moves far enough away?
From: Virgil on 25 Aug 2006 15:04 In article <44eef7f0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Please say what sentence and its negation you believe are both theorems > >>> of set theory. > >> I'm not sure how this would be proven in set theory (I don't think it > >> is), > > > > So you don't know that set theory is inconsistent (by 'inconsistent' I > > mean the usual definition). > > > >> but it appears to be a belief, anyway, that all sets can be > >> classified through cardinality. > > > > With suitable axioms, we can define a cardinality operation 'card' so > > that we get a theorem: card(x)=card(y) <-> x equinumerous with y. > > Given a set x, can we always determine card(x)? Depends. In ZF m not necessarily, but in ZFC or NBG, at least theoretically yes. > > No, but you are obligated to define a cardinality for this set which is > consistent, if you claim the theory is consistent. You can't. For each index value there is a natural whose binary string requires that index value. Thus anything less than N is too small. Thus N is required, with cardinality Card(N).
From: Virgil on 25 Aug 2006 15:27
In article <44eefa93(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <44ed9670(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> MoeBlee wrote: > > > >>> Please say what sentence and its negation you believe are both theorems > >>> of set theory. > >> I'm not sure how this would be proven in set theory (I don't think it > >> is), but it appears to be a belief, anyway, that all sets can be > >> classified through cardinality. > > > > In ZF it is NOT the case that every set must have a cardinality > > comparible with that of other sets, but it is a thereom in both ZFC and > > NBG. > > > > Good, so there must be a cardinality for the set of bit positions in the > naturals. Yup. Card(N). > > > > > > >> However, the set of bit positions > >> required to list the naturals in binary defies classification in this > >> system. > > > > On the contrary, that set of bit positions has cardinality equal to that > > of N. The fact that the same set of bit positions is capable of more > > does not mean that there need be any smaller set which is sufficient. > > That depends on how many more it produces. Then TO should be able to give the precise number of binary bits required to represent every natural up through , say, ten, but not to represent any larger naturals. If he can not then his whole argument fails. >An extra bit doubles the > number of strings, so if you have gotten to the least bit string > necessary for n naturals, you will have at most n-1 unused strings. Do > you have at most aleph_0-1 unused strings, when using aleph_0 bit > positions for the naturals? No, you have an uncountable number of unused > strings, according to your theory. Unless TO can give the number of bits to count up to ten but no farther, the issue of unused strings is a straw man. > > > > > There are exact analogies in finite cases. E.g., the number of bits > > required to ennumerate all the naturals up to, say, seventy is > > sufficient to ennumerate considerably more than seventy. > > But less than twice seventy. Can you get within a factor of 2 for your > set of naturals? Nope. So what? Unless TO can do every natural with a finite set of bits, an infinite set is required. Not that the same thing happens for the rationals and decimal representations. Infinitely many decimal places are required. > i > > > > In fact the only cases in which the same set of binary digits will NOT > > enumerate more is when you want them to enumerate every natural up to > > 2^n-1 for n=some natural n. > > Uh huh. So? > > > > > As these cases become ever more rare as n increases, they are in effect > > infinitely rare for infinite n, so the result that TO object to is quite > > normal . and any other would be infinitely unusual. > > I am not sking for an exact number of bits, but just an acceptable > cardinality for this set. There is none. Everybody but TO accepts Card(N) as necessary and (more than) sufficient. > > >>>> I have presented a system > >>> No you haven't. You've posted disconnected pieces of undefined > >>> terminology. > >> No, I've shown how IFR works with the notion of Big'un. > > > > You have made all sorts of unsupported claims, but none of them work in > > any extant system. > > > > If you say so. If TO claims to have a workable system, and wishes those claims to be treated with anything but distain,then he must present that system in its entirety with some better evidence than he has so far produced that it actually does work. > > > > >>>>>> 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. > >>> My statement is too compressed and not pinpoint accurate given the > >>> limitations of a one paragraph answer. See Enderton's mathematical > >>> logic textbook. The full formalization is culminated in the exercise in > >>> which you are asked to make sets of functions from the variables into > >>> the universe the values of a certain recursively defined function. > >>> > >>> MoeBlee > >>> > >> Okay. That's not a very fundamental definition, as far as I can tell. > > > > > > Let's see if TO can do any better, then. > > > > So, TO, what is YOUR definition of TRUTH? > > Truth is the value put on a statement, from 0 to 1. Truth is consistency > with reality. There are a few ways to talk about truth. If truth is merely a value one puts on a statement, everyone's "truths" will be different, which is not very useful. As no one has been able to produce any workable tests for "consistency with reality" that apply to mathematics, truth in mathematics appears to be entirely independent of reality. So so far, TO has no workable notion of "truth". A workable mathematical definition depends on the logical notion of a logical tautology. Any logical tautology derived only from a gIven set of axioms is TRUE in that axiom system. This is form of relative truth, and mathematics can be certain of no other. |