Cantor Confusion
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From: MoeBlee on 1 Dec 2006 14:05 mueckenh(a)rz.fh-augsburg.de wrote: > If you use the term "set" (like for instance for your set A) as defined > in set theory, then all the elements are "there" (where ever that may > be). Therefore you cannot describe potential infinity by means of ZF or > NBG set theory, unless you use completely different definitions of > "set" etc. Potential infinity is as impossible to describe as the set > of all set in ZF. In ZF, we can formulate the description - the set of all sets - easily. What we can't do is prove that there is an object that fulfills that description, as indeed we prove that there is not an object that fulfills that description. MoeBlee
From: Virgil on 1 Dec 2006 14:20 In article <1164967792.130794.251330(a)j72g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > It cannot have a cardinal number at all. We have no cardinals in > > > potential infinity. Cantor knew that. > > > > Piffle. > > > > Extending the concept of bijection from sets to potentially > > infinite sets is trivial. > > May be if you apply your personal definition of potentially infinity, > but not if you apply the generally accepted definition. What "generally accepted" meaning is that? Most mathematicians do not accept that a set can be "potentially" infinite without being actually so. > > The problem is although the term "finite" is used in the > > definition of A, A does not have many of the properties > > that are usually associated with finite sets. In particular > > it is not possible to produce A by induction. > > A potentially infinite set like N can be produced by induction. If it is a set, it is not merely "potentially" anything, and if it is merely potentially something then it is not a set. > An > actually infinite set cannot be produced by induction nor can it be > produced by other means because it is simply nonsense. That may be true in WM's as yet unspecified, and possibly unspecifyable, axiom system, but is false in ZF or NBG or NF. > The only way of > getting it is to postulate its existence by an axiom, although the > degree of nonsense is not reduced by this method. The only way of getting anything in mathematics is by postulating it either directly or by postulating the means of constructing it. > > Try to learn the commonly accepted meaning of "potentially infinite". There is no such thing as a 'commonly accepted meaning of "potentially infinite"'. > I > am not wiling to discuss your personal definitions. We do not wish to discuss your personal definitions either, but you keep thrusting them at us. > > > > Let B and C be two potentially infinite sets such that: > > > > > > For any element a, that can be shown to exist in A, > > it is possible to show that a exists in exactly one > > of B and C. > > > > for any element, b, that can be shown to exist in B, > > for any element, c, that can be shown to exist in C > > and b < c. > > > > for any element b_1 that can be shown to exist in B, there > > is an element b_2 that can be shown to exist in B, and > > b_1 < b_2 > > > > > > The potentially infinite set of pairs (B,C) is the real numbers. > > No. Your definition already fails with the meaning of "that can be > shown to exist". In fact only the existence of a finite number of > numbers can be shown to exist. So your set of real numbers is finite? In WM's system, no numbers at all can be "shown" to exist without assumptions justifying that "showing". Why should we accept WM's assumptions when he rejects ours? Does WM Popishly claim infallability?
From: Virgil on 1 Dec 2006 14:27 In article <45700481.7010300(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 1:38 PM, Bob Kolker wrote: > > Eckard Blumschein wrote: > > > >> > >> Really? Fraenkel, 2nd ed. 1923, approved Cantor having managed not just > >> to battle but also refute an assertion by Gauss. > >> Is there any evidence for this proud claim? No. > >> Whenever Cantor declared mathematicians like Aristotele, Locke, > > > > Locke and Aristotle were NOT mathematicians. Aristotle was at most, a > > logician or an ethicist or a political "scientist" or a literarary critic. > > > > As a scientist he was a failure. Why? He didn't check. > > > > Bob Kolker > > Weren't Aristotele, Galilei, Newton, Leibniz, Locke, Spinoza, Peirce, > Poincar�, Einstein, Weyl so called universal scientists? > Fermat who paved the way for calculus was not a mathematician but a > lawyer. Borel was a politician. One does not have to be a professional mathematician to be a mathematician, and whatever Fermat and Borel were professionally, they were also, at least as supremely competent amateurs, mathematicians. > You are perhaps a mathematician. Do you believe to understand > mathematics deeper than one out of the mentioned? > > Why should mathematics be esoteric? Not all of it is. Various bits of it come at various levels of abstraction, and even children understand the least esoteric bits.
From: Virgil on 1 Dec 2006 14:55 In article <45700723.3060406(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 1:39 PM, Bob Kolker wrote: > > Eckard Blumschein wrote: > >> > >> > >> Large enough is certainly not qualitatively different enough, infinity > >> is the location where two parallel lines are thought to meet each other, > >> and division by zero has been forbidden because it yields anything. > > > > Division by zero in a field yeilds a contradiction. > > Just this contradiction resides already in the notion of (actual) > infinity. On the contrary, it resides in the definition of multiplication in the set of objects one is considering. Given addition in some set of objects, S, the general definition of subtraction in S is that for elements a,b,and c in S, a - b = c if and only if c is the unique x in S for which a = b + x. Similarly given multiplication in a set of objects, S, then for elements a,b,and c in S, a/b = c if and only if c is the unique x in S for which a = b*x. Division by zero in standard sets of numbers is not defined because there is never a unique x in such sets of numbers for which a = 0*x. Either no x works or more than one works. Infinity has nothing to do with it. A finite example: The residues of the integers modulus a prime is always a finite field under the usual addition and multiplication, so there is no "infinity" involved, but division by zero in those fields is still barred for the reason above, a = 0*x can never have a unique solution. > > Isn't it better to understand why it is incorrect than simply to learn > it is forbidden? > > Eckard Blumschein It is better to understand the real reason (see above), but Eckard doesn't seem to understand the real reason. It has nothing to do with "infinity".
From: Virgil on 1 Dec 2006 15:05
In article <1164972793.366818.22230(a)n67g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > > > There is nothing in that that shows that a set is not a fixed entity. > > > > > > The universe of all sets can grow. Define: "The universe of all sets is > > > called the set of all sets", and you see it. > > > > As to what Fraenkel, Bar-Hillel, and Levy wrote, they are underscoring > > the fact that different axioms yield different universes of sets. That > > is what they mean by the universe of sets "growing" (scare quotes in > > original text). > > You should try to distinguish between "to differ" and "to grow". You > may scream as loud as you can. These verbs are different and denote > different processes. > > > > An easy example which > > > should not escape you: The set of states of the EC has been growing and > > > probably will continue to grow. > > > > Which you'll have a hard time proving to be a set in Z set theory. > > Of course in set theory there are variables denoting sets. In any book > on ZF set theory you can find sentences like: "The letters X and Y in > these expressions are variables; they stand for (denote) unspecified, > arbitrary sets." By such tools it is very easy to deal with a set like > the set of states of the EC in ZF. It turns out again and again that > you have very little knowledge about set theory and its philosophy. Each "set" of states of the EC, in the mathematical sense, is only valid during a period in which no states are added or dropped, so there is not just one mathematical set, there are many such mathematical sets. The distinction is between the common usage and the mathematical usage of the word "set". > > I am sorry but as your behaviour parallels your expertise I will no > longer discuss with you. WM is hardly in a position to cavil about either. Do not, as some ungracious pastors do, Show me the steep and thorny way to heaven, Whiles, like a puff'd and reckless libertine, Himself the primrose path of dalliance treads, And recks not his own rede. |