Cantor Confusion
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From: G. Frege on 9 Feb 2007 09:04 On Thu, 8 Feb 2007 13:20:04 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >>>> >>>> Every element is a set. There are only sets in ZFC. >>>> Two slightly different claims (prima facie). Let's define: set(x) =df Ey(y e x) v x = 0. Then we have the following two theorems in ZFC: (1) Ax(Ey(x e y) -> set(x)). "Every element is a set." (WM) and (2) Ax set(x). "There are only sets in ZFC." (WM) No? Corresponding to this theorems is the usual claim that everything is a set in (for?) ZFC. No? F. -- E-mail: info<at>simple-line<dot>de
From: William Hughes on 9 Feb 2007 10:22 On Feb 9, 4:15 am, mueck...(a)rz.fh-augsburg.de wrote: > On 8 Feb., 13:37, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > In other words the statement > > > A: every set of finite even numbers contains numbers which > > are larger than the cardinal number of the set > > > is false. > > No, it is true. How can you conclude it was wrong? You said that E was a set of finite even numbers. Thus you cannot use "every set of finite even numbers" and have a true statement. I said: > > > > The statement can be proved correct for an actually existing set E > > > with a cardinal number |E| which is a number that can be compared by > > > size with natural numbers. > > The statement > > > B: every set of finite even numbers with a finite cardinal > > contains numbers which are larger than the cardinal > > number of the set > > > The statement B is true, and of little interest. > > You are misinterpreting again. Correct it: > > > B: every set of finite even numbers > > contains numbers which are larger than the cardinal > > number of the set. > > This implies that the cardinal nunmber of the set is finite. Only if the statement is true. If the statement is false for a set K we do not know whether -K does not have a cardinal number or -K has a cardinal number but K does not have an element larger than its cardinal number. > > > As has been pointed out before, if you wish to claim > > that only sets with finite cardinality exist, then > > you are free to do so. > > I do not wish it, but the finity of numbers enforces it. > > > > > However, saying that E does not have a cardinal > > does not change the properties of E. E still > > has a sparrow (recall a sparrow is an equivalence > > class under the equivalence relation equitransform > > which generalizes the concept of bijection to include > > potentially infinite sets). This sparrow can be > > compared with other sparrow's including the sparrows > > of finite sets, which are just the finite cardinals. > > There is nothing contradictory about defining > > a sparrow. > > Unless And at this point you acknowlege that is it possible to define the sparrow of E. So the sparrow of E exists. The question is "can we call it a number that is bigger than any natural number?". > you say that it is a number larger than any natural number. > If we decide to call the sparrow of E a number, then it is not a natural number and this statement is not a contradiction. No statement you make about things that are true of every set with finite cardinality, or things that are true for every natural number, can be used to show something about the sparrow of E. The set E is not a set with finite cardinality and the sparrow of E is not a natural number. The fact that E is composed of sets with finite cardinality does not mean that E is a set with finite cardinality. The fact that the cardinality of E can be seen as the limit of natural number does not mean that the cardinality of E must have the same properties as the natural numbers. A limit of a sequence does not have to have the same properties as the elements of a sequence. > > > > Extending the concept of cardinality to include > > potentially infinite sets does not lead to > > a contradiction. > > Unless you say that it is a number larger than any natural number. > No. If you extend the concept of cardinality to potentially infinite sets, then the cardinality of a potentially infinite set is not a natural number, so it is not a contradiction to call the cardinality of a potentially infinite set a number and to say that this number is larger than any natural number. - William Hughes
From: MoeBlee on 9 Feb 2007 15:26 On Feb 8, 11:59 pm, mueck...(a)rz.fh-augsburg.de wrote: > > "Finished" is informal. > > It expresses exactly the state of art: "There is" > > > Meanwhile, ExAy(yex <-> y is a natural number) > > (where 'is a natural number' is here a rendering of a certain defined > > predicate symbol) is a theorem of many recursive axiomatizations. > > Spare this. I know it. But I understand that finished infinity cannot > be swallowed other than formal, i.e., without thinking and with eyes > wide shut. So, at least you do recognize the theoremhood, in Z set theory, of the existence of the set of natural numbers. (On the other hand, as best I can tell, you seem to contend that Z set theory is inconsistent, so for you theoremhood in Z set theory is trivial even as to any question of derivability.) However, that the existence of the set of natural numbers is a formal result does not entail that it does not have import otherwise, especially not that it can only be grasped "without thinking and with eyes wide shut". > > Meanwhile, you still have not stated from what system of logic and > > what mathematical axioms your own mathematical conclusions are > > supposed to be derived from. > > MatheRealism. What are the axioms and rules of MatheRealism? Are they recursive? > > Or perhaps (?) you said that you reject > > formal axiomatization? Okay, then I don't see what objective basis you > > offer for determining which mathematical propositions have been > > established and which have not. > > Reality. What reality? Empirically testable physical reality? A mathematical statement is not of that kind. Even if you say, "Let's see what happens when we place an apple next to another apple" and we count apples, that is an instance of confirming that 1+1=2 is correct as to such situations and is not itself a mathematical proof of 1+1=2. Moreover, such notions of physical reality very quickly fade in the rear view mirror as we drive towards even a moderate degree of mathematical abstraction. > > "The finity of the infinite" is whose expression? > > Cantor's, Hilbert's, Fraenkel's etc. (Informal, of course.) Where did they that expression? Morevover, as I've mentioned before, the informal writings or even the rigourous mathematics of any particular mathematician are not controlling over a formal theory such as fully formalized Z set theory. > > If 'finity' (as > > opposed to 'is finite') and 'the infinite' (as opposed to 'is > > infinite') are supposed to be about formal set theory, then what are > > the set theoretic definitions and what specfic axiom do you have in > > mind and how do you think it states such a requirement? The axiom of > > infinity? It states that there exists a set that has 0 as a member and > > is closed under successorship. Nothing about 'the finity of the > > infinite'. > > "There exists" means "it is ready, complete(d), finished". Actually, you said 'finished' means 'there exists'. So, you take them as equivalent, I guess. Meanwhile, 'Ex' does not require 'ready', 'complete', nor 'finished' for its mathematical explication. > But better > ask Fraenkel et al. or Hilbert or other mathematicians who know about > the things they talk about. (Levy for instance is still alive and can > be contacted.) We only need to use clear mathematical terminology in our own discussion. We don't need to consult authorities just to do that. And in that regard, 'Ex' does not require 'ready', 'complete', nor 'finished' for its mathematical explication. MoeBlee
From: David Marcus on 9 Feb 2007 15:38 MoeBlee wrote: > On Feb 8, 11:59 pm, mueck...(a)rz.fh-augsburg.de wrote: > > But better > > ask Fraenkel et al. or Hilbert or other mathematicians who know about > > the things they talk about. (Levy for instance is still alive and can > > be contacted.) > > We only need to use clear mathematical terminology in our own > discussion. We don't need to consult authorities just to do that. I've always found it odd that WM and EB quote authority so much considering how far from any authority their beliefs actually are. Of course, most of their authorities are dead, so perhaps they believe that only they can understand what these people actually meant. It must feel odd to live in a world where everyone who would agree with you is already dead. -- David Marcus
From: Ross A. Finlayson on 9 Feb 2007 15:49
Franziska Neugebauer wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > I am about to show that infinity does not exist. > > Good luck! > > > So I will not create sci.math.infinity. > > Surely not. > > F. N. > -- > xyz Ha ha ha! Oh, Mueckenheim, there is infinity, you can't deny it. At least in my opinion it's not reasonably deniable. I consider that so as I prove it to myself and others. Infinity is there, try to integrate without it: oo. There's no reason for confusion about Cantorian transfinite arithmetic, nor yours. Thank you for your consideration, Ross F. |