An uncountable countable set
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From: Virgil on 26 Aug 2006 01:49 In article <44efa44f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. It may be to large for TO, but it is just the right size for mathematicians. TO has this nasty habit of swallowing camels and straining at gnats, at least with respect to mathematical issues.
From: Virgil on 26 Aug 2006 02:02 In article <44efa5dc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <44eefa93(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >>> 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. Does To claim to be able to do it with any smaller index set? If not then the index set specified as being required is the set that is required. > > > > 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. So TO's argument fails, as expected. > > > > 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. How about to cover 10 but not 11? > > 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? The smallest one is big enough. Anything smaller is too small and anything larger is larger than needed. > > Everybody but TO accepts Card(N) as necessary and (more than) sufficient. > > Excessively sufficient, with no smaller sufficient alternative. Precisely. > > > > 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. Then until you have such a system you know what you will deservedly get. > >>> 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. Not according to those who hold them to be truths. Without some independent standard, there can be no universal truth at all, only personal truths. > > 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. Logic determines only relative truth ( this is true if that is true).
From: Virgil on 26 Aug 2006 02:11 In article <44efa6c5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Either every finite natural has a finite bit string and no infinite set > of positions is required No inifinite set is requires fora any natural but for any finite number of bits, there is a natural that requires more than that many bits, so there cannot be any finite bound on the set of bit needed. In standard set theories, an unbounded set of naturals is infinite by any standard definition. When TO produces a system in which infinite sets are finite, only then will his counterclaims be given any consideration. > Yes, given the current understanding, any unbounded set is infinite. And will remain so until TO's mythical system appears, and possibly even then. > >>> 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.
From: Virgil on 26 Aug 2006 02:19 In article <44efab93(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > 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. > > However, if we have a set which we can prove does NOT have any > cardinality within the system, then there's a hole, yes? If there were a hole, TO is too stupid to find it. And the index set argument doesn't fly. > So, is "every set has a cardinality, either finite or an indexed aleph" > a theorem or not? In what system? > You suggest that the axiom of choice makes it so. I > say that the set of bit positions required by the binary naturals does > not have such a cardinality. As that set is N, TO is wrong, at least in ZF, ZFC and BG. TO is trying to pull off an old swindle of his that falsely declares endless sets finite. The set of bit postitions is endless because for every bit position, say the nth, there is a natural, 2^(n+1) that requires a larger one. Thus there is no required bit position that can be the last one last or the largest one possible, and the set of bit positions is the same as the set of naturals.
From: mueckenh on 26 Aug 2006 05:34
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > In real mathematics (as opposed to matheology) > > we have the valid foundation: What cannot be addressed, that does not > > exist. > > Do you have axioms for this "real mathematics"? Abolish Axioms. Acquire And Ask An Abacus. What it says and what you can derive from that by pure logic: That is mathematics. Regards, WM |