An uncountable countable set
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From: Tony Orlow on 5 Sep 2006 21:59 Virgil wrote: > In article <44fdc5c9(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> I am NOT accepting the axioms. The axioms are artificial statements >>>> which can be made to work together, but which do not follow from >>>> elementary logic the way they should. Every axiom should be justifiable >>>> based on fundamental concepts. I don't see that here. >>> You keep avoiding what I've told you several times. To get an adequate >>> amount of mathematics, you have to adopt axioms that are not derivable >>> from pure logic alone. So whatever axioms you produce will be what you >>> call "artificial statements" just as you call the axioms of set theory >>> "artificial statements." >> Perhaps I ignore that statement because you have not given any >> justification for it. I am not convinced that it's necessary to invent >> rules which do not follow from more elementary fundamentals. > > Those more elementary fundamentals will themselves be expressed as rules > which then would require existence of yet more elementary rules, and on > ad infinitum. > > You cannot get anywhere without having place to start from. Start at the origin, 0. > >> Inadequate why? N=S^L is inductively provable > From what? One needs the entire arithmetic of the natural numbers before > it even make sense, and then only makes sense for N,S and L all being > natural numbers. Yes, if you include the infinite naturals. The formula applies equally well for infinite N, S or L. > >> and inductive proof is derived directly from logic by forming an >> infinite loop. > > An inductive proof involving a statement about natural numbers requires > the natural numbers at the very minimum. If N, S and L cannot be > numbers, what does "N=S^L" mean? > That axiom is indeed built on the naturals as a consequence of increment/count/successor, on addition as repeated increment, multiplication as repeated addition, and exponentiation as repeated multiplication. It also requires the notions of symbol, alphabet, string and language. Then, you're all set. :) > > > >> IFR is justified primarily geometrically > Which, even if true, would presume a priori the entire structure of the > real number system, and/or the geometry of the real line. Worse than that, it requires the Cartesian plane. But that's not hard to define, is it? > > So that neither "N=S^L" of "IFR" can be an axiom independent of all > those many other axioms necessary for either to have any meaning at all. > if they are to hold at all in any system, they must be theorems, not > axioms. > Unless they are derived from other logical statements, they're not theorems. If the justification for the axiom is not other axioms, then it's an axiom. > > >> The axioms of internal and external infinity are >> constructive axioms like Peano's, and therefore somewhat "artificial" > > While I am not sure I recall them, I suspect that, like TO's "N=S^L" > of "IFR", those "axioms" must be theorems, not axioms in any system in > which they are to hold true. No, they were my extension to Peano's. External: y -> x<y<z. Internal: x<z -> x<y<z. You can insert the existential quantifiers if it makes you more comfortable.
From: MoeBlee on 5 Sep 2006 22:02 Tony Orlow wrote: > > I didn't say that you said that all objects in a class are equal. In > > fact that is why it is imporatant to keep clear the difference between > > equality and equivalence. > > Ugh. You keep missing the point. It must be deliberate. Yes, I do it because it makes my dog happy. > >> but given some > >> criterion for distinguishing objects, one can form CLASSES where a given > >> property is the same for all members of any given class, ignoring all > >> other properties. > > > > That's pretty much what we do in set theory. Except we don't "ignore" > > other properties; rather, we just have only a finite number of > > primitives. And we don't need to refer to classes; rather sets x and y > > are the same set. > > Yep. You're doing it on purpose. There is no other explanation for you > ignoring that I have already said that finite theories with a finite > number of primitives can only address a finite number of properties. I don't know what you mean by a 'finite theory', since every theory, being a set of sentences closed under entailment,is infinite. But YOU missed MY point again. A finite number of primitives does not limit us to finitely many properties, since properties are compound. > > I understand that it is your postulate. But it is not the same > > statement as the symmetry of identity. And apparently that IS difficult > > for you to understand. > > Ugh again. Is is = = ???? Is IS =. = = is. Got it? No. I don't know what you mean by those strings. But I can tell you that you'd profit by reading something on the subject of identity theory. > x IS the set S of property values defining it. Now a different formulation from you. Okay, your postulate is that x is the set of properties that define x. Now you just need a logistic system, axioms, primitives, and defintions to go along with that. > x=S <-> S=x. > > Comprende? Dios mio! What in the world is wrong with you? Of course I understand symmetry of equality. > > Sorry for not reading your mind when you mentioned, in your own scare > > quotes, "subjective". As to your point, I don't want to comment at this > > time since I see some philosophical complexities here I couldn't > > address properly within only a paragraph. Anyway, my original point is > > that a theory such as set theory avoids having the identity of objects > > depend upon DISCOVERY (which is epistemological) of properties. > > > > Of course not. It IGNORES any other properties besides those explicitly > defined within the theory. There is no room for other possibilities > within the theory. That's what I'm saying. No theory accounts for all properties. And of course any theory does not allow what is not possible in the theory. > >>> See, that is what is subjective (or epistemological). We don't define > >>> equality by "way to distinguish" but rather by FORMULAS. > >> Formulas are a fine way to distinguish objects. For instance, I > >> distinguish a vastly greater number of different infinities than > >> cardinality simply by ordering formulas on a unit infinity. Good suggestion. > > > > As usual, you seem not to recognize my point. > > I believe I do, but if you would like to clarify your point, here's your > chance. Requote what I said to which you were in response and I'll consider your offer. > >>> No, we may do better than that in theories in which there are only > >>> finitely many primitive predicate symbols, such as set theory. I told > >>> you all about that already. > >> If there are only finitely many primitive predicate symbols, then there > >> are only finitely many properties being addressed by the theory. > > > > No, because properties are not just primitive but are also compound. > > Unless you compound them infinitely, It's not a process such as that. > you still have a finite set of > compound properties. Perhaps yuo could give an example you think > contradicts that statement. The set of definining formulas of Z set theory is countably infinite. There's your example. > When I specifically distinguished it in my first mention (after the > initial one-liner comment) from deductive logic and inductive proof > (which is deductive), and then get further confusion, then I start to > wonder a little. No, at THAT time, in that other thread, I said that if you are talking about inductive logic as opposed to induction in mathematics, then my remarks would have to be adjusted for that. I posted that immediately upon your saying that you had inductive logic in mind. > > Now, if I were you, I'd say you don't know the difference merely from > > the fact that you misspoke. But I'm not you, so I don't resort to that > > kind of cheap tactic. > > It's not that you misspoke. That's Virgil's gig, anyway. It's that, > after specific clarification you still seemed not to understand what I > was saying, and still don't. Any clue as to why I would bring up > inductive logic, as opposed to inductive proof? Since the incident I just mentioned, please show exactly where you mentioned "inductive logic" and I posted as if otherwise. > >> Number the objects in the universe starting from > >> 0. Every unique set is therefore a unique bit string representing which > >> elements are members. > > > > If that's your axiom, fine. But it contradicts set theory. So, again, > > we need to be clear what theory we're talking about. But you don't have > > a theory, so it's nugatory anyway. You just keep announcing disparate > > axioms without stating a logistic system, primitives, or definitions. > > You say it contradicts set theory, but I imagine that depends on how you > define the universe. No it doesn't. I just contradicts set theory. > If it's "the set of all sets", sure, that leads to > contradictions. The contradiction has nothing to do with that. > If it's "the set of all objects, properties, and > relations between them", perhaps that a different story? Is there a > universe in ZFC. Ross says not. It makes one wonder how a statement > about the universe could contradict ZFC then. I'm not paid to sort out such utter confusions as in the above paragraph. > > You may define words however you like. However, you should at least be > > aware when your definitions and meanings depart from those of most of > > the other people in a converstation. > > Which is most of the time, but
From: Tony Orlow on 5 Sep 2006 22:06 MoeBlee wrote: > Tony Orlow wrote: >> A better way to view the universe of sets is as a relation between the >> objects within it and 1-place predicates concerning objects, where each >> object has a truth value associated with each predicate, between 0 and 1 >> inclusive. Thus, these predicates define subsets of the universe, eh? > > I bet you could find that there's is work that's been done on model > theories with a continuum of values. > > I might be barking up the wrong tree, but I think there might be > something in the work of C.C. Chang (not included in the book) that > might at least suggest some points of departure, if not directly, then > at least through the bibligographic trails. > > MoeBlee > Thanks for the reference. I'll see what I can find.
From: imaginatorium on 5 Sep 2006 22:09 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> And yet, you are saying it's not technically correct. So, 1 is a natural > >> but not a rational or real? If rigorous formulations come to that > >> conclusion, then rigor does not ensure correctness. > > > > You're hopeless. You didn't understand a thing I said, which may be my > > fault for not providing adequate explanation in the context of your > > ignorance of the subject; but it is no my fault that you won't even > > look at a book on mathematics, such as even a introductory text in real > > analysis. > > > > MoeBlee > > > > Your whole point here is ludicrous. You are arguing that the naturals > are not a subset of the rationals, which are not a subset of the reals. > While each superset may require a more complex construction than the > subset, all elements of the subset are covered by the superconstruction. > As points on the real line, they are all real numbers. What you're > saying amounts to what I said above, which is indeed absurd. So, no, I > don't understand a thing you're saying, when you say naturals aren't > reals. Sorry. Why don't you try reading an elementary book on set theory? You might just then understand enough to ask Moeblee an intelligent clarifying question. Remind us what your ultimate objective is here: just to maunder around arguing with people on Usenet until the day your physical powers fail you, or something else? If something else, what exactly? Do you hope to persuade anyone mathematically informed of anything, or are you just going to look for a New Age publisher? (The former amounts to the maundering option; for the latter, you may as well get started finding the publisher, and putting words together.) Brian Chandler http://imaginatorium.org
From: Dik T. Winter on 5 Sep 2006 22:05
In article <1157465168.349273.136370(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1157367467.816725.158560(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > Cantor's list has as many lines as my tree. The diagonal has as many > > > digits as each path of my tree. The only difference is that the paths > > > split while the diagonal does not. > > > > And the difference is that given the list we can immediately state what > > the n-th digit of the diagonal is for arbitrary n. You can not state what > > the n-th path is for arbitrary n. > > You intermingle numbers (paths) with digits. I can state the n-th digit > of any path you want. I do not. The digits are countable because I can immediate state the n-th digit when n is given. The paths are not countable because I can not state the n-th path when n is given. The edges within a given path are countable because you can state which is the n-th edge when n is given. > > > The same is true for my tree. > > > > What is the tenth path in your tree? > > You intermingle numbers (paths) with digits. I can state the n-th digit > of any path you want. (I do not ask what is the tenth entry of Cantor's > list.) No, of course you do not ask that. You should ask that to the one who provided the list. Which is *not* Cantor, because he does start with an arbitrary list, not a specific list. And shows that given an *aribitrary* list, the anti-diagonal produced is not in the list. As this proof holds without any reference to the particulars of the list, it holds for *all* lists. So, if someone gives me a list and tells me that it contains all infinite sequences of two symbols, I can show him such a sequence not in the list. > > What is the tenth path of that tree? > > What is the tenth entry of the list? That is completely irrelevant. See above. But also, the list is a *given* thing with some assumptions. And a list is *by definition* a function from N to a (sub)set something. So in the case of Cantor, the list is the function f: N -> S where S is the set the set of infinite sequences of two symbols. You ask what the tenth element is, and I answer: f(10). I need not go further because the list is a given entity. On the other hand, you *claim* that your set of paths is countable, so you should be able to state what the tenth path is. But you refuse to do so. > > > Give my a tree of infinite paths consisting of 0's and 1's, and I show > > > that there are not less edges than paths. > > > > Indeed. If all edges terminate, also all paths terminate, and both are > > countable, and 1/3 is not in the tree. If edges do *not* terminale, > > What are you talking about? Every edge terminates because it is the > connectio between subsquent nodes of a path. And so all paths terminate, and 1/3 is not in your tree. > > > You know that sets of order 2^omega are countable? > > > > No. Can you prove it? It is precisely Cantor's diagonal proof that > > shows that it is not countable. > > You intermingle 2^aleph_0 and 2^omega. No, I did not know the strange exponentiation used in ordinals. Apparently it can be defined as the set of functions from a set with ordinal number omega can be mapped to a set with ordinal number 2, with the proviso that only finitely elements are mapped to the second element. > A well known example is the set > of the unit fractions 1/n used for the proof of the divergence of the > harmonic series: > > 1 + 1/2 + (1/3 + 1/4) + (1/5 +...+ 1/8) + ... > > It has omega terms in parentheses and 2^omega fractions which biject to > the natural numbers. How do you come at 2^omega fractions? (Yes, I sort of understand.) Exponentiation can also defined as: (1) a^0 = 1 (2) a^(b+1) = (a^b).a (note right handed multiplication) (3) if d is a limit ordinal, a^d = lim{b -> d} a^b. So this implies that 2^omega = lim{n -> omega} 2^n, which is omega. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |