Cantor Confusion
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From: William Hughes on 15 Dec 2006 07:22 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > > > > (It is contained in the union of all lines, but the > > > > > > > > union of all lines is not a line) > > > > > > > > > > > > > > That is a void assertion unless you can prove it by showing that > > > > > > > element by which the union differes from all the lines. > > > > > > > > > > > > Not quite. In order to achieve that the diagoal is not in any linem all > > > > > > that is required is: > > > > > > Given any line there is an element of the diagonal not in THAT line. > > > > > > It is not requires that: > > > > > > There is an element of the diagonal that is not in any line. > > > > > > > > > > > > > > > For linear sets you cannot help yourself by stating that the diagonal > > > > > differs form line A by element b and from line B by element a, but a is > > > > > in A and b is in B. This outcome is wrong. > > > > > > > > > > Therefore your reasoning "there is an element of the diagonal not in > > > > > THAT line. It is not required that: There is an element of the diagonal > > > > > that is not in any line." is inapplicable for linear sets. You see it > > > > > best if you try to give an example using a finite element a or b. > > > > > > > > > > > > In every finite example the line that contains > > > > the diagonal is the last line. > > > > > > Every example with natural numbers (finite lines) is a finite example. > > > > > > > Your claim is that there is a line which contains the diagonal. > > > > > > Because a diagonal longer than any line is not a diagonal. > > > > > > > Call it L_D. Question: "Is L_D the last line?" > > > > > > There is no last line > > > > Then, there is a line that comes after L_D. > > > > Therefore :L_D does not contain every element > > that can be shown to exist in the diagonal. > > All elements that can be shown to exist in the diagonal can be shown to > exist in one single line. > Call it L_D L_D contains a largest element. n. L_D is not the last line, so there is a line with element n+1, Element n+1 can be shown to exist in the diagonal. - William Hughes
From: Bob Kolker on 15 Dec 2006 07:22 Han de Bruijn wrote: > And go to the section "Does mathematics require axioms?". Actually, explicity axioms/postulates are required to make sure proofs are kosher. If one does not explicitly schematize his assumptions, one is likely to sneak in other assumptions along the way. So axiomatizing is a way of keeping proofs on the Straight and Narrow. Consider all the "hidden postulates" that Euclid used in his geomtry. He snuck in ordering along a line segment. He snuck in isometric transformations. Hilbert fixed that very nicely in his -Grundlagen-. Axiomatization helps to liberate the purely logical content of a mathematical system from its intuition based baggage. For example: until real variables was reasonably axiomatized and the convergence os series carefully defined it was not realized that continuity and differentiability are not equivalent. Weirstrass defined an everywhere continous but nowhere differentiable function. Also measure theory required a set theoretical basis. This enabled the integeral to be extended in such a way that integration and limit could commute. This is not generally possible with Reimann integrals. Non formalized set theory enabled real variable analysis to be tightened up. Like it or not, sets are at the basis of most mathematical systems. Just about every mathematical structure starts off with a set of elements and some operations etc etc. Try doing group theory without the concept of set. Bob Kolker
From: William Hughes on 15 Dec 2006 07:28 Han de Bruijn wrote: > William Hughes wrote: > > > Han de Bruijn wrote: > > > >>William Hughes wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Let's repeat the question. Does there exist more than _one_ concept of > >>>>infinity? Isn't unbounded the same as infinite = not finite = unlimited > >>>>= without a limit? Please clarify to us what your "honest" thoughts are. > >>> > >>>You are *way* in deficit on clear answers. Try answering > >>>the following question with yes or no. > >>> > >>> Is there a largest natural number? > >> > >>No. > > > > I there an unbounded set of natural numbers? > > Suppose you mean "Is". What does it mean that a set is unbounded? An unbounded set of natural numbers is a set of natural numbers that does not have a largest element. Please answer yes or no. - William Hughes
From: Bob Kolker on 15 Dec 2006 07:32 Virgil wrote: > > > Actually, we today have a number of improvements on Aristoteles logic, > and Euclid's axiom system had to be revamped by Hilbert to bring it up > to modern standards. But they have both stood the tests of time > remarkably well. Aristotelean logic was sound as far as it went. The main improvements to logic has been broadening and deepening the subject. Most important the embedding of formal logical systems in metatheories has enabled us to see the ultimate limits of logic. For example the Goedel Incompleteness Theorems. This is an approach that Aristotle (at his level of technique and understanding) could not have dreamed of. Be that as it may Aristotle's work on logical inference and semantics still stands up rather well. This is over just about 2400 years. Too bad Aristotle's physics was not of the same quality. If it were, we would be traveling around in Star Ships and not jet airplanes. Bob Kolker
From: William Hughes on 15 Dec 2006 07:32
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > > > > > > > A_1 = {1} > > > > > > > > > > > > A_2 = {1,2} > > > > > > > > > > > > A_3 = {1,2,3} > > > > > > > > > > > > > > > > > > > > > > > > B = {1,2,3} > > > > > > > > > > > > > > > > > > > > > > > > then B is contained in the last A_i. If there is no last A_I, then > > > > > > > > > > > > there is > > > > > > > > > > > > no A_i that contains B > > > > > > > > > > > > > > > > > > > > > > That has nothing to do with "last". > > > > > > > > > > > > > > > > > > > > If A_i contains B, then A_i contains any A_j. > > > > > > > > > > Therefore A_i is "last". > > > > > > > > > > > > > > > > > > > > > > > > > > No comment? > > > > > > > > > > > > > > If B contains any A_i then B is the last. > > > > > > > > > > > > B cannot be the last A_i, as B is not > > > > > > one of the A's (B corresponds to the diagnonal, > > > > > > the A_i correspond to the lines) > > > > > > > > > > In which element does it differ from every A_i? > > > > > > > > In no element. > > > > > > > > We know that B is contained in all of the A_i > > > > put together. > > > > > > How can you put all together, if there is no last one? You will never > > > be sure that you have all. > > > > We know that any element that can be shown to be > > in B can be shown to be in one of the A_j.. > > > > The question is: "Is there an A_D, such that > > any element that can be shown to be > > in B is in A_D?" > > > > The answer is: "There is an A_D, such that > > any element that can be shown to be > > in B is in A_D > > That is correct, because we are dealing with finite linear sets. > > > if and only if there is a last A_i." > > No. Correct is: > If B is an actually infinite set of finite linear segments, then there > must be at least one infinite finite linear segment, which is a > contradiction. Therefore B is not actually infinite but only > potentially infinite, i.e., B contains only a finite (though not > bounded) number of elements. > A_is bounded. B is not bounded. There is an element that can be shown to be in B that is not in A_i. - William Hughes |