Cantor Confusion
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From: Lester Zick on 22 Nov 2006 17:22 On 22 Nov 2006 10:22:47 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> That is not the question any longer. The question is: Can a set >> theorist admit that she is in error? It seems impossible. They all are >> too well trained in defending ZFC. > >In error as to what? Set theorists admit mistakes. It is not uncommon >for books to have errata sheets attached. For what it's worth, Moe, I have yet to see any set theorists admit mistakes in the paradigm. ~v~~
From: William Hughes on 22 Nov 2006 19:50 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > No. You can develop "potential set theory". Just define > > > > an element of a potential set to be an element of any one > > > > of the arbitrary sets that can be produced. Now go through > > > > set theory and add the word "potential" in front of each > > > > occurence of the word set. There is no difference between > > > > saying "a potentially infinite set exists" and saying "an actually > > > > infinite set exists". > > > > > > A potentially infinite set has no cardinal number. It cannot be in > > > bijection with another infinite set because it does not exist > > > completely. > > > > > > > Piffle. You need to study your potential-set theory. > > > > We have defined what it means to be an element of > > a potential set. > > We never have all elements available. > > > > Therefore we can define bijections between potentially > > infinite sets. > > We can never prove hat a bijection fails, like in Cantor's argument. Piffle. Real numbers are represented by potentially infinite sequences. A list of reals is a function that takes an element of the potentially infinite set of natural numbers and returns a potentially infinite sequence. The diagonal number is a potentially infinite sequence. We show the diagonal number is not a member of the list in exactly the same way as before. > > > > Therefore we can define cardinalities of potentially infinite > > sets. > > Yes, we can define that a set is finite or that it is infinite, > denoting the latter case conveniently by oo. That's all. > Defintion. B is a g-subset of a potentially infinite set A if B is a set or a potentially infinite set and any element of B is an element of A. We can now contruct potentially infinite power sets, and the usual theory follows. You are confusing potentially infinite with computable. While it is true that a computable set is either finite or potentially infinite, it is not true that a potentially infinite set is computable. - William Hughes > Regards, WM
From: William Hughes on 22 Nov 2006 20:03 Eckard Blumschein wrote: > On 11/22/2006 1:46 PM, William Hughes wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > >> William Hughes schrieb: > >> > >> > No. You can develop "potential set theory". Just define > >> > an element of a potential set to be an element of any one > >> > of the arbitrary sets that can be produced. Now go through > >> > set theory and add the word "potential" in front of each > >> > occurence of the word set. There is no difference between > >> > saying "a potentially infinite set exists" and saying "an actually > >> > infinite set exists". > >> > >> A potentially infinite set has no cardinal number. It cannot be in > >> bijection with another infinite set because it does not exist > >> completely. > > Since I am convinced that Dedekind and Cantor were wrong, I avoid the > expression cardinal number. Ok call the equivelence classes of the equivalence relation bijection the breadfruits. They still exist. > Why do you not simply write instead: > Something potentially infinite is countable because it is not thought to > exhibit the impossible property to include all elements of something > actually infinite? Because this is nonsense. By defininition a set A is countable if and only if there is a bijection between A and the natural numbers. Because of the way we have defined bijection and potentially infinite set it does not matter at all whether or not A is actually infinite when deciding if such a bijection exists. > > > > > Piffle. > > I recommend to avoid hurting words. > > >You need to study your potential-set theory. > > > > We have defined what it means to be an element of > > a potential set. > > > > Therefore we can define bijections between potentially > > infinite sets. > > If one considers something as actually infinite, then this point of view > even hinders bijection. One cannot eat the cake and have it. > potentially infinite and actually infinite point of view exclude each > other as do discrete numbers and continuity. > "One cannot eat the cake and have it" is not much of an argument. Do you have any other argument why you cannot have bijections between potentially infinite sets. - William Hughes
From: William Hughes on 22 Nov 2006 23:15 Eckard Blumschein wrote: > On 11/21/2006 6:12 PM, William Hughes wrote: > > Eckard Blumschein wrote: > > >> > (i.e. the difference between actual and potential infinity) is > >> > mostly one of terminology. > >> > >> Yes, but I got aware that actual and potential infinity are two > >> different views both directed towards the same object of natural numbers > >> from different levels of abstraction. The potential infinite view > >> belongs to counting and is able to separate from each other all single > >> numbers. The actual infinite one provides the fiction of all natural > >> numbers at a higher level of abstraction. You cannot have both views at > >> a time. > >> > > > > Why not. There is nothing about assuming the set of all natural > > numbers exists that precludes counting or the ability to sepatate > > from each other all single numbers. > > Actual infinity is not as handsome as it seems to be. Non sequitur Do you have any argument to support: assuming that the set of all natural numbers exists precludes counting and/or the ability to separate from each other all single numbers. > > > The elements of the > > potentially infinite set are exactly the same elements with > > exactly the same properties as the elements of the > > actually infinite set. > > I object to exactly this assumption. Actual infinity denotes a diffent > quality, not a larger quality, not a quantity at all. Again a non sequitur. > > > >> And the elusive belief hidden within and conveyed by this name. > >> Aren't I correct? > > > > No. You can develop "potential set theory". > > Quite a while ago, it was a surprize to me: Axiomatic set theory does > not really require the actual infinity. > > > Just define > > an element of a potential set to be an element of any one > > of the arbitrary sets that can be produced. Now go through > > set theory and add the word "potential" in front of each > > occurence of the word set. There is no difference between > > saying "a potentially infinite set exists" and saying "an actually > > infinite set exists". > > Yes. This is indeed a clever method of obscuration. Obfuscation of what? Do you agree with the statement: There is no difference between saying "a potentially infinite set exists" and saying "an actually infinite set exists". or not? If not, do you have anything even remotely relevent to say? > Nonetheless, if one > prefers to distinguish between rational and real numbers, then the reals > are only distinguished by being fictitious and therefore uncountable. > Genuine reals being really real are only required for theoretical > considerations. They do however, have properties quite different from > the properties of genuine numbers. This is my original concern. After I > already settled the somewhat puzzling matter by means of really real > numbers, I took me some time to understand the relationship to the still > thaught naive set theory, axiomatic set theory and what a comprehensive > understanding of mathematics really needs. > This semi-coherent ramble has nothing whatsoever to do with the question of whether there are any real (as opposed to cosmetic) differences between potentially infinite and actually infinite sets. If you do not wish to discuss this question fine, but a simple statement to that effect would have been far preferable to the above. (Even simply not replying would have been preferable.) - William Hughes
From: Virgil on 23 Nov 2006 00:12
In article <1164197588.292514.91250(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164143109.673986.202100(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > What about all contructible numbers or all words? They cannot be > > > > > mapped > > > > > on N though they are a countable set. > > > > > > > > > > > > Absolute piffle. The question is whether the set N can > > > > be mapped to the set N, not whether some other > > > > set can be mapped to the set N. > > > > > > It is the same impossibility, but not so obvious. > > > > What is impossible about: for all n in N, n |--> n ? > > It is impossible to believe that you think all n e N must be > automatically included only by writing n. What is impossible for WM is not necessarily impossible, or even difficult for anyone else. WM seems to have extremely limited capabilities. > > > > > > > > The existence of a cardinal which bijects to a set S is not a necessary > > condition for the identity function on S to exist. Every set, whether > > well orderable or not, has an identity function, at least in such > > relatively sane systems as ZF and NBG. > > Is that an axiom? No. It is a direct consequence of the axiom of separation in ZF. > > > > In order to assign a cardinal, at least one > > > bijection to an ordinal is required. > > > > If one does not assume the axiom of choice, as in ZF, then there ae sets > > without the sort of cardinality that WM requires, but such sets still > > can have identity functions on them, and in ZF, must have identity > > functions on them. > > No. Read the axiom of separation of ZF before making such foolish claims. > > Regards, WM |