Cantor Confusion
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From: David Marcus on 18 Nov 2006 20:38 Virgil wrote: > In article <MPG.1fc90ff92e8ec01d989922(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > If an actually infinite set of numbers existed, and if neighbouring > > > > > elements had a fixed distance from each other, then the set must > > > > > contain an infinite number. > > > > > > > Is that a "no" or a "yes"? > > > > > > Read again, simplified: If neighbouring elements have a fixed distance, > > > the answer is yes. > > > If neighbouring elements have not a fixed distance like the rational > > > numbers: the answer is no > > > > Let's try a simpler question: > > > > Does an actually infinite set exist? > > WM has already committed himself to a situation in which as set like the > rationals need not contain any "infinite element" but that a proper > subset of it, like the integral rationals, must contain an "infinite > element". > > One wonders how a proper subset can contain an element not in the > containing superset? It does make one wonder. -- David Marcus
From: imaginatorium on 19 Nov 2006 00:33 David Marcus wrote: > imaginatorium(a)despammed.com wrote: > > > > Dik T. Winter wrote: > > > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > ... > > > > > In > > > > > principle no axiom is necessary. But you need a few to have some start > > > > > to work with. > > > > > > > > That's the question. By means of axioms you can produce conditional > > > > truth at most. I am interested in absolute truth. Axioms will not help > > > > us to find it. I don't think we need any axioms. > > > > > > If you want to find absolute truth you should not look at mathematics. > > > > Really? There are two groups of order 4; could any truth be more > > absolute than that? > > I think it depends on what the words mean. If the axioms are correct in > your model, then the theorems are correct in your model. Well, model-schmodel, really. (This stuff is a bit beyond me, actually...) It's not entirely clear what the notion of "absolute truth" refers to. Suppose you think it is a matter of absolute truth that all men are created equal. Then you go to Venus and discover that in their language the word 'All' means flying, 'men' means pigs, 'are' means eat, 'created' means chocolate, and 'equal' means icecream.* Moreover the atmosphere of Venus turns out to be full of flying pigs, but is of such chemical composition that icecream of any flavour self-combusts explosively. Well, has absolute truth varied? I think the reasonable answer is 'No', because a truth is _about_ something, not merely a string of formal symbols. : * Language doesn't work like this - I know, but I haven't time to assemble grammars and whatnot : just to make the same point. Anyway, see the Hilary Putnam stuff about horses and schmorses : (which I have only read secondhand in Dennett). > Why did you pick the statement you did, rather than something like 2 + 2 > = 4? Because as far as I know there is no (normal, sane) interpretation of the _words_ of my statement about groups of order 4 other than the standard one. Whereas, for example, in other contexts 2 + 2 = 1, so while the truth to which "2+2 = 4" refers is absolute, it takes longer to write, because you have to spell out the full context, and in present crank company even saying "integers" may take 2-3 lines. You say this depends on my axioms and my model; but are there such that make my claim about groups of order 4 untrue? Brian Chandler http://imaginatorium.org
From: David Marcus on 19 Nov 2006 01:25 imaginatorium(a)despammed.com wrote: > David Marcus wrote: > > imaginatorium(a)despammed.com wrote: > > > Dik T. Winter wrote: > > > > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > Dik T. Winter schrieb: > > > > ... > > > > > > In > > > > > > principle no axiom is necessary. But you need a few to have some start > > > > > > to work with. > > > > > > > > > > That's the question. By means of axioms you can produce conditional > > > > > truth at most. I am interested in absolute truth. Axioms will not help > > > > > us to find it. I don't think we need any axioms. > > > > > > > > If you want to find absolute truth you should not look at mathematics. > > > > > > Really? There are two groups of order 4; could any truth be more > > > absolute than that? > > > > I think it depends on what the words mean. If the axioms are correct in > > your model, then the theorems are correct in your model. > > Well, model-schmodel, really. (This stuff is a bit beyond me, > actually...) > > It's not entirely clear what the notion of "absolute truth" refers to. > Suppose you think it is a matter of absolute truth that all men are > created equal. Then you go to Venus and discover that in their language > the word 'All' means flying, 'men' means pigs, 'are' means eat, > 'created' means chocolate, and 'equal' means icecream.* Moreover the > atmosphere of Venus turns out to be full of flying pigs, but is of such > chemical composition that icecream of any flavour self-combusts > explosively. Well, has absolute truth varied? I think the reasonable > answer is 'No', because a truth is _about_ something, not merely a > string of formal symbols. > > : * Language doesn't work like this - I know, but I haven't time to > assemble grammars and whatnot > : just to make the same point. Anyway, see the Hilary Putnam stuff > about horses and schmorses > : (which I have only read secondhand in Dennett). > > > Why did you pick the statement you did, rather than something like 2 + 2 > > = 4? > > Because as far as I know there is no (normal, sane) interpretation of > the _words_ of my statement about groups of order 4 other than the > standard one. Whereas, for example, in other contexts 2 + 2 = 1, so > while the truth to which "2+2 = 4" refers is absolute, it takes longer > to write, because you have to spell out the full context, and in > present crank company even saying "integers" may take 2-3 lines. > > You say this depends on my axioms and my model; but are there such that > make my claim about groups of order 4 untrue? I don't know. -- David Marcus
From: mueckenh on 19 Nov 2006 04:24 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > I claim case iia: There is a potentially infinite sequence N = > > > > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > > > > treat all of its elements. In particular we can never complete this > > > > set. We can never put it into a list > > > > > > OK. Knock yourself out. > > > > I read this several times from you. What does it mean? > > > > > Note however that this case > > > is not consistent with assuming the axiom of infinity. > > > The axiom of infinity says that the set N exists. > > > > But it does not say that it has an ordinal number and a cardinal > > number. My case iia is in complete agreement with the axiom of > > infinity. > > > > > The axiom of infinity says that the set N exists. Your iia says > > "we cannot recognize or treat all of its elements" This is no contradiction to the axiom. Compare the proof that the real numbers can be well-ordered. We cannot construct or define or recognize a well ordering. > > The axiom of infinity does not say anything about ordinal or > cardinal numbers. However, given that the set N exists and > the defnition of ordinal and cardinal numbers, it is easy to > see that if N exists it must have both an ordinal and a cardinal > number. > No. You assume the possibility of a bijection of the set with itself. That is not proven from the mere existence of the set if we cannot recognize or treat all of its elements. And even if we could, the axiom of infinity does not prove that infinite ordinal and cardinal numbers are numbers, i.e., that they stand in trichotomy with each other. > Ordinal. > > Every natural number is an ordinal. > Every initial sequence of ordinals is an ordinal. > (an initial sequence is a set of ordinals, such that > if a is in the set every ordinal less than a is in the set). > The set of all natural numbers is an initial sequence of > ordinals. Therefore the set of natural numbers is an > ordinal. > > Cardinal. > Every natural number is a cardinal number too. > Cardinal "numbers" are equivalence classes of > sets under the equivalence relation bijection. Since any > set has a bijection to itself, Why? There are even models without a bijection to the natural numbers. > every set must be in an > equivalence class. So the set of natural numbers > has a cardinal number.. And even if it had. By means of the equilateral infinite triangle I proved that this cannot be the case. Therefore we have a contradiction. Regards, WM
From: Virgil on 19 Nov 2006 04:54
In article <1163928247.023859.86200(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > The axiom of infinity says that the set N exists. Your iia says > > > > "we cannot recognize or treat all of its elements" > > This is no contradiction to the axiom. Compare the proof that the real > numbers can be well-ordered. There is no such proof in ZF, it requires ZFC to prove that the reals can be well ordered, and even there , the axiom of choice does not provide any more that an existence proof. > We cannot construct or define or > recognize a well ordering. We can easily define one: any ordering in which every non-empty subset has a first element. We have no reason to suppose the we could not recognize one if it were presented for inspection. It is true that no one has actually constructed any explicit well ordering. > > > > The axiom of infinity does not say anything about ordinal or > > cardinal numbers. However, given that the set N exists and > > the defnition of ordinal and cardinal numbers, it is easy to > > see that if N exists it must have both an ordinal and a cardinal > > number. > > > No. You assume the possibility of a bijection of the set with itself. The identity function on any set bijects it with itself. And such functions are guaranteed, via the axiom of replacement. > That is not proven from the mere existence of the set if we cannot > recognize or treat all of its elements. It is proven in ZF, even without C. > And even if we could, the > axiom of infinity does not prove that infinite ordinal and cardinal > numbers are numbers, i.e., that they stand in trichotomy with each > other. But it is possible to prove trichotomy for ordinals in ZF, and for cardinals in ZFC. > > > every set must be in an > > equivalence class. So the set of natural numbers > > has a cardinal number.. > > And even if it had. By means of the equilateral infinite triangle I > proved that this cannot be the case. Therefore we have a contradiction. Since WM assumes properties for his "triangle" which contradict themselves, as well as contradicting ZF, ZFC and NBG, among others, he is welcome to all contradictions of his own making, but there are no other contradictions than those due to his self-contradictory assumptions. Among other idiocies, WM keeps claiming that in his triangle there is a finite line as "long" as the infinite diagonal. |