Cantor Confusion
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From: Virgil on 10 Dec 2006 14:48 In article <457c2070(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > David Marcus schreef: > > > >> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> Ah, now don't act as if you didn't have those heated debates with some > >>> manifest opponents of set theory, i.e. Wolfgang Mueckenheim. > >> If you've been reading the threads, then you should know that WM has so > >> far failed to present any mathematics at all. All he does is present > >> incorrect arguments that he insists follow from the standard axioms and > >> then proclaim: "Behold, standard mathematics is inconsistent." Big deal. > > > > Nonsense. The arguments presented by WM are quite reasonable. > > > > I agree. Now why does that fail to convince?
From: stephen on 10 Dec 2006 14:49 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > stephen(a)nomail.com wrote: >> >> But writing down all the details using nothing but sets would >> be cumbersome. Just saying that numbers can be represented as sets >> is not the same as actually demonstrating how a calculation would >> be represented using sets and nothing but sets. As I said, it >> is like writing a Turing machine that computes a Fourier transform. >> Anyone who understands the theory of computation knows that it >> can be done, but nobody is likely to do it, especially just to >> satisfy who has no understanding of the theory of computation. > True, but all mathematics is built up from more primitive concepts. We > can prove that something can be done without actually doing it. "We" can, but many of the people in this thread seem incapable of it. Stephen
From: David Marcus on 10 Dec 2006 14:52 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > > It is just supposed to work. No one is saying zero really is the empty > > set (whatever "really is" means). > > Then no one is saying the von Neumann successor ordinals "really are" > the naturals? Good. Correct. No one is saying that. It is simply a model that has the same mathematical properties, i.e., it satisfies the Peano axioms. > Not when the alternative is a construction that establishes the > equivalent of a new primitive through a dubious connection between > succession and containment. What is more "parsimonious" in inventing > some weird model of the naturals and declaring that it exists, rather > than having succ() be a primitive relation? I don't see the advantage. Basically, the smaller the language and the fewer the axioms, the better. If you are doing logic, you may not need your succ function, so it will just clutter up your proofs. If you are doing arithmetic, you need it, but then you can define it. > If the sequence consists of segments of the form (0,x) or (x,0), there > is no segment which is diagonal in direction. If every segment in the > sequence is of the form (x,x), there is no segment which is not. This > information is not evident when the pairs describing the curve are > locations, because locations don't have direction. OK, but the lines connecting successive points do have direction. > So, my question remains. If this is a valid formulation of the two > objects, and an explanation for Chas' counterexample to infinite-case > induction, where does this fit with set theory? Has nothing to do with set theory. If your "infinite-case induction" doesn't do what you want, then you need to construct something that does. > I don't think the von > Neumann ordinals as a model of a sequence can suffice, since they only > allow finite values until a leap is made to the limit ordinals, and > continuity is violated. This is the problem I have with the vNO's, and a > large part of my problem with transfinitology. The notion that a > sequence must be "countable" simply is not correct in the bigger picture. If the usual notion of a sequence doesn't do what you want, then come up with one that does. > > Then come up with an approach that does what you want. By "approach", I > > mean definitions (of objects, convergence, etc.) that let you prove the > > theorems you want. That's what everyone does. For example, Einstein came > > up with Brownian motion, but it wasn't clear how to mathematically model > > it. Wiener figured out how. > > Well, I'm suggesting a definition of the curve as a sequence of pairs > which denote xy offsets, rather than a set of pairs of xy coordinates. > Is that not a concrete enough description of an "approach" to spark a > new neuron in your head? It should be. If you think there is something > wrong with it, please elucidate. Fine. State your definition of a curve, state a theorem, and state the proof of your theorem. If you do that, then you will be doing mathematics. That's what Wiener did: he came up with a model, then proved it had the properties that were needed for Einstein's Brownian motion. > >> I'm sure that cleared things up for you, eh? > > > > Pretty much. > > Did it? I rather thought you'd accuse me of being totally nonsensical, > though I know I'm not. Nice surprise (unless you're being as sarcastic > as I was). Nope. Wasn't being sarcastic. > PS - No good counterexamples to infinite-case induction? Too bad. :( I didn't look. Is "infinite-case induction" a theorem? -- David Marcus
From: Virgil on 10 Dec 2006 14:52 In article <1165762822.356809.126820(a)16g2000cwy.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > step...(a)nomail.com schreef: > > > Han.deBruijn(a)dto.tudelft.nl wrote: > > > > > Let us say that set theory is half the truth. Within set theory, any > > > function is a special relation between commodities and products, i.e. > > > domain and range. But the production process itself involves _labour_, > > > hence time, and this aspect is not covered by the static set theoretic > > > framework, where functions are reduced to "mappings" between sets. > > > > You can include it. You are just talking about a cost being > > associated with a function. That is trivial to do. > > Exactly! In the eye of the capitalist beholder labour is identical with > "cost". And that cost is an easy thing to incorporate. But a fact is > that labour is not cost and not static and it involves time. And time > is not a set. Is HdB so ignorant that he does not recognize velocity and acceleration as being functions of time. If they are, why not 'labor', or anything else that is not static, as a function of time, with domain an appropriate set of times?
From: David Marcus on 10 Dec 2006 14:56
Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > > > Ah, now don't act as if you didn't have those heated debates with some > > > manifest opponents of set theory, i.e. Wolfgang Mueckenheim. > > > > If you've been reading the threads, then you should know that WM has so > > far failed to present any mathematics at all. All he does is present > > incorrect arguments that he insists follow from the standard axioms and > > then proclaim: "Behold, standard mathematics is inconsistent." Big deal. > > Nonsense. The arguments presented by WM are quite reasonable. State one that is reasonable. > > > I'm talking about rejecting any monolithic foundation for mathematics, > > > any "foundation" that narrows down my freedom of thinking. I hate any > > > form of NewSpeak, whether it is called Set Theory, Category Theory or > > > Object Oriented Programming. I've seen too many of these. > > > > Fine. Give some evidence that it "narrows your thinking". I.e., present > > some mathematics that can't be done using ZFC as a foundation. > > I'm pretty sure you can twist and bend any thought in such a way that > it fits into your set theoretical paradigm. And as soon it doesn't fit, > you'd simply say that it is not mathematics. Wolfgang Mueckenheim has > come up with several of such examples. You'd not be better off with me. > See e.g. the "Probability XOR Calculus" thread, which was initiated by > this author: > > http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962 Yes, I've read the thread. The replies to you were quite reasonable. Basically, you wanted a theorem, but the theorem as you stated it wasn't true. The solution is to come up with a concept and a revised theorem that does what you want. It isn't our fault that you didn't do this. -- David Marcus |