Cantor Confusion
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From: Virgil on 14 Oct 2006 18:39 In article <1160858282.925275.281630(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > You think we cannot name the M, so it cannot exist? You know we cannot > > > construct a well-order of the reals. But nevertheless it does exist, > > > according to Zermelo's proof. And here, there *is* such an M, for all > > > finite positions, according to a very simple proof, independent of > > > whether or not we can name it. > > > > > > No. We do not have one M. We have a lot of > > different M(N)'s, one for each digit position. > > And each one covers all it predecessors. > > > We can > > show that if there is a last N, call it K, then all > > the M(N) can be replaced by a single M(K). > > Exactly. > > > But there > > is a last N iff the number of digit positions in 0.111.... > > is finite. > > Exactly. Either the number of digit positions is finite or there are > some positions undefined or an infinite set does not actually exist, > meaning that the number of digits is finite though unbounded. These are > the alternatives. Or an infinite set does actually exist.
From: Virgil on 14 Oct 2006 18:46 In article <1160858469.901946.319320(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > >This statement does not refer in any respect to > > > the number of digit positions. All finite digit positions are covered > > > by one finite number M. > > > > What about position M + 1? > > > All finite digit positions N =< M+1 are covered by the finite unary > number M+1. You will fail in your attempt to find a finite number up to > which all numbers cannot be covered by one single unary number. That > should show you that your opposite assertion is wrong. My only "opposite assertion" is that for every finite natural number there is a larger finite natural number, without end. > > > > > > > > No, the fact that every digit position of 0.111... is finite does not > > > > mean that M does not depend on N, and that is what we need. > > > > > > Why should we need that? > > > > Because for every M there is an N = M+1 which M does not cover. > > > But it is finite, no matter how you name it. But never the last possible. > > > > > > The problem is that you believe in an actually infinite series of > > > finite numbers. This belief leads to intermingling digit position and > > > number of digits just according to the needs and, after all, it leads > > > to such self-contradictive statements as we have seen here with "the > > > vase". > > > > There is nothing self contradictory in "the vase" but there are > > contradictions between the axiom systems on which "the vase" is based > > and the assumptions of those who oppose those axiom systems. > > > > According to the ZFC system: The vase is empty at noon, because all > natural numbers left it before noon. > By means of the ZFC system we can formulate sequences and their limits > in mathematical language. From this it follows that lim {n-->oo} n > 1. > And from this it follows that the vase is not empty at noon. By what axiom do you conclude that the limit as t increases towards noon of any function and the value of that function at noon must be the same? This assumption requires something like continuity at noon, which clearly cannot be the case.
From: Virgil on 14 Oct 2006 18:50 In article <1160858900.410979.56520(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > This is the Fall > and original sin of [Cantor's] set theory ..." > > > No one has actually > > shown a contradiction yet. > > To be repeated 10 times per day, in order to fix the belief of those > adherents which have critical minds. (It is the rumor that such ones do > in fact exist, but nobody knows where.) When, if ever, someone actually finds that rumored contradiction, you will be free to say you told us so, but until then we can continue to say that the best efforts of minds more brilliant than any posting here have failed to find any such contradictions in Cantor of SF or NBG.
From: David Marcus on 14 Oct 2006 18:53 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > Han de Bruijn wrote: > > > Dik T. Winter wrote: > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > ... > > > > > It is not > > > > > contradictory to say that in a finite set of numbers there need not be > > > > > a largest. > > > > > > > > It contradicts the definition of "finite set". But I know that you are > > > > not interested in definitions. > > > > > > Set Theory is simply not very useful. The main problem being that finite > > > sets in your axiom system are STATIC. They can not grow. Which is quite > > > contrary to common sense. (I wouldn't imagine the situation that a table > > > in a database would have to be redefined, every time when a new row has > > > to be inserted, updated or deleted ...) > > > > Is your claim only that set theory is not useful or is contrary to > > common sense? Or, are you claiming something more, e.g., that set theory > > is mathematically inconsistent? > > It is not useful and contrary to common sense but above all it is > mathematically inconsistent. If I remember correctly, you offered to > formulate the vase problem in your language. Perhaps you can see the > contradiction there. Actually, I offered to give my translation of the vase problem into Mathematics. Be happy to. First, in English: Problem: At one minute to noon, balls 1 thru 10 are added to a vase and ball 1 is removed. At half a minute to noon, balls 11 thru 20 are added and ball 2 is removed. Etc. How many balls are in the vase at noon? Now, the translation into Mathematics: Problem: Let 1 signify that a ball is in the vase. Let 0 signify that a ball is not in the vase. For n = 1,2,..., define A_n = 12 - 1 / 2^(floor((n-1)/10)), R_n = 12 - 1 / 2^(n-1). Now, for n = 1,2,..., define a function: B_n(t) = 1 if A_n < t < R_n, 0 if t < A_n or t > R_n, undefined if t = A_n or t = R_n. Let V(t) = sum{n=1}^infty B_n(t). The number of balls in the vase at time t is V(t). What is V(12)? The sentences that use the words "ball" or "vase" aren't really part of the mathematical statement. They are translator's comments. Sorry, I don't see a contradiction, yet. Please point out the contradiction. -- David Marcus
From: David Marcus on 14 Oct 2006 18:59
Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > Han de Bruijn wrote: > > > I'm not interested in the question whether set theory is mathematically > > > inconsistent. What bothers me is whether it is _physically_ inconsistent > > > and I think - worse: I know - that it is. > > > > What does "physically inconsistent" mean? Wouldn't your comments be > > better posted to sci.physics? Most people in sci.math are (or at least > > think they are) discussing mathematics. > > ONE world or NO world. Sorry. No idea what you mean. Do you have anything to say about mathematics or would you prefer we all just ignore you? -- David Marcus |