infinity ...
in [Math]
|
Prev: math
Next: The proof of mass vector.
From: David R Tribble on 21 Oct 2005 16:47 Tony Orlow wrote: >> But S is infinite. For any element of P(*N) you name, I can name an element >> in *N that corresponds to it. Try me. > David R Tribble said: >> How about the subset B = {0, ...999}? > Tony Orlow wrote: > Let's stick to binary, use N digits per naural, and say B = {0,111...111}. > Then, the natural number corresponding to this subset is 100...001, with 2^N > digits. 111...111 has N digits, but 100...001 has 2^N digits. So do the naturals have N digits or 2^N digits? If N < 2^N, you've managed to contradict yourself. Or maybe N = 2^N, so there is no contradiction?
From: Virgil on 21 Oct 2005 16:47 In article <MPG.1dc2e55059dc7f7798a525(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > You mean A. Yes, that's what it means to have a > > surjection f between two sets A and P(A) (a bijection > > is a kind of surjection). It means that for every w in P(A) > > there exists a y in A such that w = f(y). If y is not in > > A, then the fact that w = f(y) is irrelevant to whether > > there is a bijection between A and P(A). > But, is S the COMPLETE set, and if so, how many bits does it have? Every set, if it is to be a set at all, is complete. Incomplete objects, whatever they may be, are not sets. Sets have members, not bits. > > > > >If you are assuming you have the complete set of naturals, > > >that you have identified the last The what? There TO goes again on his perpetual quest for a non-existent "last natural", which in his saner moments he concedes does not exist. To seems to be ragarding sets as processes, rather thatn as static objects. In standard matematics, sets are static, however dynamic they may appear to be in the wonderland of TOmatica. > > The point is that for any rule for generating > > subsets, there exists a subset that will *never* be generated > > by that rule. So no rule can generate all possible subsets > > of the naturals. (Cantor's theorem is not limited to sets > > and functions that are defined by any particular rule, > > but for concreteness, let's restrict ourselves to those.) > The process defined by the rule will never end, but as much as the > Peano axioms define the infinite set of naturals, the bijection > generates the infinite set of sets, the power set. Such a non-existent bijection does not generate anything except more chaos in TOmatica. > But, if there are N finite naturals There are not. If N is the set of all finite naturals then it is not the number of that set as well, at least outside TOmatica. > , there are still 2^N finite subsets. If N is the set of all finite naturals, and there are Card(N) members to a set then it has Card(N) finite subsets but Card(P(N)) ~= 2^Card(N) subsets altogether. > > > > Even though you can never complete the process of generating > > an infinite set, you *can* complete the process of giving > > a rule for one. For example, the rule "x is in the set if and > > only if x is even" defines the infinite set of even natural > > numbers. The rule "x --> 2*x" is a rule for mapping the set > > N of natural numbers to the set E of even numbers. > Right, and the rule I offered gives a mapping to a unique subset for each > natural in N But it is not onto. At lest no version I have seen is. > Please tell what rule I broke when forming my bijection? Since TO did not even give a rule for determining the image of one of your *n's from *N, TO has not even formed a function, much less a bijection. If TO has some rule, so far known only to himself, call it f:*N -> P(*N), let him find for that rule some *n in *N such that f(*n) = {x in *N: x not in f(x)}.
From: Daryl McCullough on 21 Oct 2005 16:32 Tony Orlow says... >Do you think it is lazy to try to construct a better foundation for >mathematics? You haven't done that. You've just posted ill-considered nonsense. You haven't done any actual mathematics. >Well, it is a lot easier than regurgitating what I learned in >books. Nobody has been regurgitating what they read in books. The people who have been arguing with you have been trying to make sense of *your* statements. We've tried to understand what you were saying about "bigulosity", we've tried to understand what you've been saying about "infinite naturals". People have been working *much* harder at making sense of your nonsense than you ever worked at making sense of actual mathematics. >It's so much easier to paint than go to the museum. Only >lazy people make anything new. The real work is in defending >the status quo against the crackpots. As I've pointed out to you before, nobody has any trouble accepting different points of view about mathematics. I personally am somewhat familiar with recursion theory, ZF set theory, Quine's New Foundations set theory, Aczel's non-well-founded set theory, constructive type theory, the lambda calculus, category theory, nonstandard analysis, combinator theory, Conway's surreal numbers. These are all *drastically* different ways of looking at the foundations of mathematics. I don't reject any of them just because they are different from the status quo. I'm always on the look out for new theories, new ways to look at things. But the difference between all these subjects and Tony Orlow mathematics is that they are all *rigorous*. The people that developed them put *thought* into them. They actually tried to get clear what their basic assumptions were, and they tried to rigorously work out the consequences of those assumptions. You flatter yourself to think that we are rejecting what you say because it goes against the status quo. People aren't rejecting your ideas because they are challenging, or because they are new, or because they attack the status quo. People are rejecting your ideas because they are idiotic. They are ill-thought-out. They are nonsensical. They are a complete mess. -- Daryl McCullough Ithaca, NY
From: David R Tribble on 21 Oct 2005 17:05 Tony Orlow wrote: >> But S is infinite. For any element of P(*N) you name, I can name an element >> in *N that corresponds to it. Try me. > David R Tribble said: >> If you use your binary encoding method to map members of *N to the >> subsets of *N (i.e., the members of P(*N)), you will find that all of >> the finite naturals in *N map to some of the finite subsets of *N, >> and that all of the infinite naturals in *N map to some of the >> infinite subsets of *N. All of the subsets mapped contain only >> finite naturals, though. (You should check this, so you can see this >> for yourself.) > Tony Orlow wrote: > No, that is according to standard theory. A bijection can be made between the > finite naturals and sets of finite naturals, which are always finite, since > the set is unbounded and seemingly endless. The bijection between *N and > P(*N) also continues forever, this time actually. Since you can always keep > borrowing naturals from further down the set, you never run out of naturals > to map to your subsets. Of course, the power set IS bigger, bijection > genuflection notwithstanding. But no matter how much you "keep borrowing naturals from further down the set", you never "get to" any naturals in *N that map to subsets containing infinite naturals (which we'll call the "I-subsets"). No matter how "far" you go with the mapping process, you always end up mapping only subsets (finite and infinite) containing only finite naturals (which we'll call the "F-subsets"). You never get "far enough down" to map any of the I-subsets. No matter what "infinite digital length" L you choose for your infinite naturals, you "use up" all of them of that length and have to choose a bigger L, then another bigger L, and so on. Which proves that you'll never have an infinite L that's big enough to hold all the numbers you need for mapping all the F-subsets so that you can start mapping the I-subsets. Your "borrowing from further down" is your own proof that there cannot be any such "largest" L. (It does not exist, like the largest member of N, which does not exist.) But I don't expect you to understand any of this.
From: Virgil on 21 Oct 2005 17:11
In article <MPG.1dc2eab2e50dd9fc98a526(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > William Hughes said: > > > > Tony Orlow wrote: > > > > <snip> > > > > > I gave the mapping rule: natural <-> binary string <-> subset, > > > ordered naturally from 0. But this presumes conditions contrary to fact. For example, that there is a "natural" ordering of these infinite strings of binary digits. What is TO's supposedly "natural" ordering? If it matches the usual ordering for finite strings whe a atring contains only finitely many 1's, then it can be shown to be ambiguous for lots of pairs of strings with infintiely many of both 0's and 1's. > Okay, so you want to claim that the infinite strings cannot be mapped > to the infinite binary strings representing n in *N. Sure. Let's see > which version of the largest finite you bring out. Existence of "Largest finites" is TO's mantra, not ours. They can only exist in TOmatica. > I will repeat what I have said regarding the diagonal proof regarding > the reals. It has nothing to do with the reals, and proves nothing > about the ability to count a set. In that proof, as in this one, you > are constructing a full list of digital representations, traversing > it diagonally, and inverting those bits to form an antidiagonal which > is supposedly not in the list. At each point, as you add another > string to the list, you can perform this operation to get a string > not on the list. But, this proof is so flawed so as to be more of a > joke than a logical argument. But those flaws exist only in TOmatica, not outside. So only in TOmatica does TO's complaint hold. |