An uncountable countable set
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From: Daryl McCullough on 25 Jun 2006 15:06 >Daryl McCullough schrieb: > >> mueckenh(a)rz.fh-augsburg.de says... >> >> >Such a function exists from |N --> |R for all real numbers which can be >> >individualized, i.e. which are really real numbers. >> >> If such a function exists, then demonstrate it. Give us a function >> f from N to R such that every real number that is "individualized" >> is in the image of f. > >Cantor's general proof breaks down I'm not asking about Cantor's proof. I'm asking you to demonstrate a function f from N to R such that every "individualized" real number is in the image of f. Can you demonstrate such a function? If not, why not? -- Daryl McCullough Ithaca, NY
From: Virgil on 25 Jun 2006 15:29 In article <1151226733.798018.282870(a)u72g2000cwu.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1151153602.449685.154410(a)u72g2000cwu.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > By definition, a set M is countable if and only if there SOME surjection > > > > g:N --> M exists and is uncountable if no such surjection can exist. > > > > > > > > This means that, given a set M, one must allow ANY function from N to M > > > > to be considered. > > > > > > > > So that given that M(f) exists at all, one is not constrained to only > > > > the function f which defines K(f) and M(f), but can consider whatever > > > > functions imaginable from N to M(f). > > > > > > > > Since it is easy to construct surjections from N to the set of all > > > > finite subset os P(N), it is also easy to do it from N to M(F). > > > > > > Such a function exists from |N --> |R for all real numbers which can be > > > individualized, i.e. which are really real numbers. > > > > > > What real numbers are not really real numbers? > > In the Dedekind model, every set of rationals that is bounded below > > determines a real number (or bounded above, for that matter). > > In the Cauchy sequence model, every Cauchy seqence of rationals > > determines a real number. > > Which of the real numbers so determined are not really real numbers? > > Irrationals, for instance, are not numbers at all, because Cauchy's > epsilon cannot be made arbitrarily small. The members of the sequence, 1, 1/2, 1/4, 1/8, ... are all positive but contain members smaller than any given positive , thus rationals can be made arbitrarily small. > But we will keep this > disussion at the reordering of the well-ordered set of rationals. If > you have understood that there are not all rationals, you will > understand easily, that there is no irrational. In my imagination, all of the rationals exist and all of the irrationals as well. I am sorry to hear that "mueckenh"'s imagination is so self-limiting.
From: Virgil on 25 Jun 2006 15:33 In article <1151227006.042370.6320(a)r2g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > There is no mapping including K as the image of at least one source > > > element, even if surjectivity is not at all in question! > > > > That is not the issue in question. > > "Mueckenh" has claimed that he has produced an example of a set that is > > both countable and uncountable (See Subject), but his example is flawed > > because the sets and function he has presented do not exist as he has > > presented them. > > I made this claim well aware that you would substitute f by another > mapping g. This should show you that Cantor's diagonal method can also > be treated in the same way showing that it doesn't show anything about > uncountability. That only shows that "mueckenh"' has no understanding of the Cantor theorem and Cantor's second proof of it at all.
From: Virgil on 25 Jun 2006 15:39 In article <1151227192.733162.241500(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1151154151.237811.168860(a)y41g2000cwy.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > Actually, even if it IS true, it would not allow transmutation of a set > > > > with a first element into one without a first element, at least if the > > > > transpositions are applied sequentially, as they must be. > > > > > > The transpositions are just as sequentially as the construction of the > > > diagonal number. > > > > > > > There would have to be a first transposition in the sequence which > > > > effects the change, and a single transposition is incapable of doing > > > > this. > > > > > > This arguing stems from K?nig (1905). We know that there are only > > > countably many constructible reals. If there is a well-ordering of the > > > reals, then within this order there must be a first non-constructible > > > real. This is a paradox. Hence, there is no well order. > > > > No one here is claiming a well-ordering of the reals. > > You don't accept ZFC??? There is a good deal of difference between claiming that there must be a thing and claiming to have found it. The well ordering of the reals may be as non-constructable as the non-constructable reals but be required to exist as do the non-constructable reals. It is only contructionalists who balk at existence of non-constructable items in mathematics.
From: Daryl McCullough on 25 Jun 2006 15:23
mueckenh(a)rz.fh-augsburg.de says... > > >Daryl McCullough schrieb: >> What is it that you think is paradoxical? > >It is a paradox that mathematicians can believe that 2 + n and 2 * n >share the same level, but 2^n does not, while exponentiation is only an >abbreviation for multiplication which is only an abbreviation for >addition. > >It is a paradox that logicians can believe that the non-existence of a >paradoxical set proves uncountability. > >It is a paradox that we know that only countably man numbers can be >identified, and that numbers do exist only in our minds, hence numbers >which can not be identified in our minds cannot exist. It is a paradox >that we know all this but nevertheless speak of uncountably many >numbers. You don't seem to understand what a "paradox" is. There is nothing paradoxical about any of this. >It is a paradox how this obviously selfcontradictory theory could >occupy mathematics. It isn't self-contradictory. To be self-contradictory means that it is possible, using valid rules of inference, to derive both a statement and it's negation. That isn't the case here. >> Here's an example: f(x) = x/2. That's a function from N to R. >> It generates the decimal expansions >> >> 0.00 >> 0.50 >> 1.00 >> 1.50 >> 2.00 >> ... >> >> The diagonal procedure produces a new real by adding one to >> each diagonal digit: >> >> D(f) = 1.61111111.... >> >> D(f) is definitely *not* in the image of f. > >But it is definitely in the image of a suitable g. Sure. The image of f is a countable set. Add the diagonal, and you get a new countable set. >This can be done >with all diagonal numbers D ever constructed. That's right. And if you put them all into one huge countable set, you still haven't enumerated all the real numbers. >Hence all diagonal numbers ever constructed are countable. That is certainly true. You can never generate all the real numbers by starting with a countable set and applying diagonalization. >> So you think that there is some function f such that *every* >> real is in the image of f? Then show it to me. > >I'll show it to you after you have constructed all numbers you wish to >be contained in my mapping. I just told you: the image should include all real numbers. In particular, the image of the function should have the following property: If <A,B> is any partition of the rationals into two sets such that every element of A is less than every element of B, then there should be a real r such that r is greater than or equal to every element of A, and r is less than or equal to every element of B. >> Do you think that there is such an f? If so, what is it? >> Give me a formula for it. > >There is no such thing like "all reals". There is no such thing as "no such thing". Mathematical objects are *abstractions*. They don't exist in the real world. They are concepts. If we have the concept of "the set of all reals" and that concept is consistent, and we can reason about it, then that's all that matters for mathematics. >> How are you saying anything different from that? But the >> latter statement immediately implies the statement >> "forall f from N to P(N), f is not surjective". > >forall f from N to any set, K(f) is not in the image of f . Yes, that's true. And that implies that "forall f from N to P(N), f is not a surjection". -- Daryl McCullough Ithaca,NY |