An uncountable countable set
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From: Tony Orlow on 25 Aug 2006 21:31 Virgil wrote: > 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). And yet, N is infinitely too large.
From: MoeBlee on 25 Aug 2006 21:36 Tony Orlow wrote: > I beg your pardon? What is the difference between Dedekind infinite and > infinite? I have been saying there is another sense in which the > Dedekind definition is not always appropriate, and have been told that > it's the only standing definition, and you're telling me he just > explained the difference between the only mathematical definition of > infinite, and infinite? You must have meant "finite", in which case, > thank you, I am well aware of the Dedekind distinction. YOU ARE NOT LISTENING. You haven't been listening since I first read a post by you. I and others have explained to you MANY TIMES the difference between Dedekinf infinite and infinite. For about the hundredth time: Definition: x is finite <-> there exists a natural number equinumerous with x Definition: x is Dedekind finite <-> there does not exist a proper subset of x that is equinumerous with x Definition: x is infinite <-> x is not finite Definition: x is Dedekind infinite <-> x is not Dedekind finite Theorem of Z: x is Dedekind infinte -> x is infinite Theorem of Z with the axiom of choice: x is infinite -> x is Dedekind infinite Not only are you ignorant of basic set theory, in which you would learn that 'infinite' and 'Dedekind' infinite are different, but you've ignored several explanations of that in the threads. You are so full of yourself and full of self-grandiosity that it takes at least a few dozens iterations of a definition for you to STILL not recognize it. MoeBlee
From: Tony Orlow on 25 Aug 2006 21:37 Virgil wrote: > 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. > > No, one needs to be able to specify to the nearest bit. > >> 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. To cover 10 but not 20: 4. >>> 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? So, with each additional bit the number of strings is doubled. How many times do you remove bits and halve the uncountable number of strings before you cull it down to some countable level? > > 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. But, which infinity? >> 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. Excessively sufficient, with no smaller sufficient alternative. >>>>>> 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. I know. >>>>>>>> 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. In that sense, many truth values are false. > > 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. It may appear to be so to those that prefer it so. > > So so far, TO has no workable notion of "truth". > > A workable mathematical definition depends on the logical notion of a > logical tautology. That's a statement wich always has a value of 1, or at least >.5. > > 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. Logic determines truth. Induction is more than just a form of proof.
From: Tony Orlow on 25 Aug 2006 21:41 Virgil wrote: > In article <44ef2769(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <44ed99b7(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> TO has claimed it, but not proved it. >>> TO claims a lot but never proves any of it. >>> Since the set of digit positions in any positional notation for the >>> members of N is indexed by N, the cardinality of the set of bit >>> positions equals the cardinality of its index set. >> Since the bit strings are the power set of the set of bit positions > > False. At least unless one means the SET of INFINITE bit strings has the > same cardinality as the power set of the set of bit positions. > > But since no natural requires an infinite bit string, that is irrelevant. If no natural requires an infinite bit string, even the very largest, and all bit positions are included in it, then no infinite set of bit positions is required. > >> the set of naturals is power set to the set of bit positions. > > False! TO deliberately conflates the set of infinite bit strings with > the set of finite bit strings. > > So anything based on this false equivalence, is irrelevant. > Either every finite natural has a finite bit string and no infinite set of positions is required, or countably infinite strings are naturals, and still there are no infinite bit positions. > >>>> Yes, given the current understanding, any unbounded set is infinite. >>> Find any dictionary anywhere that says otherwise, that allows TO's >>> self-contradictory unbounded but finite. >> Dictionaries reflect the widest and most accepted usage of words. > > They also, if they are any good, include most, if not all, special > meanings. So if it cannot be found in any of them, even the OED, it is > because it ain't so. Oh. >>>>> A set is infinite if it is not finite. A set is Dedekind infinite if >>>>> it can be mapped to a proper subset of itself, and it is Dedekind >>>>> finite if it is not Dedekind infinite. When we assume the axiom of >>>>> choice, the two notions are identical. Without that axiom there >>>>> can be infinite sets that are Dedekind finite. Do you want to know >>>>> more about set theory? >>>>> >>>>> Now, using, this terminology (pretty standard), what is a "finite but >>>>> unbounded" set? >>>> Using that system, the phrase is senseless, but that system is not the >>>> real universe. It's a concoction. >>> TO's concoctions are even less sensible and less part of any "real" >>> universe. >>> >>> In every standard dictionary, finite means being bounded or having ends, >>> and endless means infinite. >>> >>> TO thinks he can get away with saying endless means having ends, but TO >>> is, as usual, wrong. >> When you have the set of reals in [0,1] there are distinct endpoints to >> the set. The question when dealing with an infinite set is whether >> enumeration of the elements therein can ever terminate as a process. >> Where there is a distinct range and the enumeration never completes, the >> set is infinite. Unfortunately, because the enumeration of the naturals >> never ends, that set is considered actually infinite, when it's really >> only potentially so. The endlessness is due, not to actual infinitude, >> but to the non-boundary between finite and infinite. > > In other words, TO is claiming that everyone in the world except TO is > wrong about non-finite sets being non-finite. > > NOT bloody likely!!! Well, me and WM here. :D
From: MoeBlee on 25 Aug 2006 21:47
Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Given a set x, can we always determine card(x)? > > > > I'm guessing, but I'm pretty sure that there is not an algorithm that > > will tell you the ordinal index of the aleph that is the cardinality of > > any given set. > > Good guess, except I have no idea what your driving at. Of course you don't, because you are ignorant of even of the basics of your own question. With the axiom of choice (which is one method), every set has a cardinality. That cardinality is a cardinal number. And that cardinal number is an aleph indexed by an ordinal. So I answered your quesion: No, I don't think there is a procedure to determine which cardinal is the cardinality of a given set. But in my first answer I gave an even sharper formulation by pining the question down to the ordinal index. Of course, for some sets, we can prove that a certain cardinal is the cardinality that set; but I don't think there can be a general procedure to produce an answer for any set that might be submitted to test. > > First you need to define the set in set theory and prove that it exists > > in set theory. Then, set theory would not be inconsistent for there not > > being an algorithm to determine the ordinal index of the aleph that is > > the cardinality of any given set. > > So, you're saying that identifying a set with no cardinality doesn't > make set theory inconsistent? I think I saw a different opinion yesterday. You INSIST on twisting just about every answer given you. If it is a theorem that every set has a cardinality, then it would be inconsistent if it were also a theorem that there exists a set without a cardinality. But the fact that we don't have a procedure to always announce what SPECIFIC cardinality a set has does not contradict any theorem of set theory. > > Again, you're just yapping without regard for the specific definitions > > of such things as 'consistent'. > > Consistent: Adj. Without contradiction. Yes, and your remarks were without sense of that definition, as SPECIFICALLY you've been told about a million times that a theory is inconsistent iff there is a contradiction in the theory, which means that there is sentence and its negation that are both members of the theory. MoeBlee |