Cantor Confusion
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From: Jacko on 23 Nov 2006 15:13 the_wign(a)yahoo.com wrote: > Cantor's proof is one of the most popular topics on this NG. It > seems that people are confused or uncomfortable with it, so > I've tried to summarize it to the simplest terms: > > 1. Assume there is a list containing all the reals. > 2. Show that a real can be defined/constructed from that list. > 3. Show why the real from step 2 is not on the list. > 4. Conclude that the premise is wrong because of the contradiction. so reflect the integers about the right most decimal point and you have a countable infinity of fractionals, each fractional can be a fractional added to an integer so set size is Z*Z -> countable. this is a direct proof. of the countability of the frationals, or infinite rationals as one may choose to see them ;)
From: Virgil on 23 Nov 2006 16:16 In article <1164284605.171771.206840(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164192462.824282.134620(a)m7g2000cwm.googlegroups.com>, > > 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. > > > > Not true in any extant axiom system. In every axiom system that has been > > presented so far, there are finite sets and possibly infinite sets, but > > nothing in between. Something which is potentially, but not actually, > > infinite cannot be a set at all in any set theory yet seen here. > > And just that is the reason why any set theory is waste. Not to mathematicians.
From: Virgil on 23 Nov 2006 16:23 In article <1164285024.129501.161870(a)l39g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164195797.399821.219210(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > One counter example contradicts ZFC: > > > > > There is not one single element of the diagonal which is not contained > > > > > in a line. This line contains this and all preceding elements. > > > > > > > > While true, it is irrelevant to the issue of whether there is one line > > > > containing every element of the diagonal, which there is not. > > > > > > > > WM conflates "every element of the diagonal is in SOME line", which is > > > > true, with "every element in the diagonal is in THE SAME line", which > > > > is false. > > > > > > That means we need at least two lines for the elements o the diagonal? > > > > Not to anyone who understands logic. > > What is the opposite of "at least two"? What we need is infinitely many lines, "at least two" is necessary but not sufficient. > > > > What it does mean is that we need infinitely many finite lines to get > > all of the infinitely many members of the diagonal. > > So infinite is less than two? What we need is infinitely many lines, "at least two" is necessary but not sufficient. > > > > At least to people of any sense. > > Look, what is the value of the statement of a fool that his companion > is not a fool? I think, there is no value at all. Then WM makes a fool of himself calling others foolish. > > > > > Please give an example which requires that at least two different lines > > > are needed to contain two elements of the diagonal. > > > > Please give any line for which every element of the diagonal is in THAT > > line. > > Project the diagonal in horizonat direction. Then you have it. WM seems disoriented. In WM's own model, that forms a column, not a line.
From: Virgil on 23 Nov 2006 16:26 In article <1164285212.255489.92230(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164195919.471077.233250(a)f16g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <456304B0.70705(a)et.uni-magdeburg.de>, > > > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > > > > Being admittedly not very familiar with set theory, I nonetheless > > > > > wonder > > > > > if sets are considered like something going on forever. > > > > > > > > Sequences do, sets do not. > > > > > > > Sequences are sets. > > > > > > Regards, WM > > > > Sequences are functions, a very special sort of set, those with domain > > N. > > > > Since not all sets are sequences, not all sets have the properties that > > sequences must have in order to be sequences. > > Have you lost the rest of your mind? I said "all sequences are sets", > but not "all sets are sequences." I have not lost anything, though WM seems to have. I did not contradict what you said, I merely expanded upon it.
From: Virgil on 23 Nov 2006 16:33
In article <1164285630.630550.112930(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164196369.064190.253870(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <1164126395.211430.7520(a)h54g2000cwb.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > Tell me which point is not accepted: > > > > > 1) Every line which contains an index of the diagonal, contains all > > > > > preceding indexes of the diagonal too. > > > > > 2) Every index of the diagonal is in a line. > > > > > 3) In order to show that there is no line containing all indexes of > > > > > the > > > > > diagonal, there must be found at least one index, which is in the > > > > > diagonal but not in any line. > > > > > > > > This one is flat out false, unless one assumes, a priori, a last line > > > > and a last member of the diagonal. > > > > > > We assume not a last line, but we assume that eery line has finitely > > > many indexes. And this is true. > > > > What you allege in (3) does not follow from this. > > It is true that given any line there will be a diagonal element not in > > that line. > > It is false that given a diagonal element there is no ine which contains > > it. > > > > WM again dyslexes his quantifiers. > > > > > > > > It is certainly false in ZF or NBG, where such an assumption is also > > > > false. > > > > > > For finite indexes it is correct. > > > > Infinite things are not constrained to behave in all respects like > > finite ones. That is because "infinite" means "not finite". > > But *the lines are all finite*. The *set* of lines in WM's so-called EIT is not finite. > That's just why I chose the EIT as > example. > > > > > > > > It is quite enough to show that for every index in the diagonal, > > > > except 1, there is some line not containing that index. Then no line > > > > can contain every index. > > > > > > Name two finite indexes which cannot be in one line. > > > > That is not at all relevant to what I said. > > It is relevant since every line is finite. WM again has cart-before-horse-itis. What is relevant is not whether one can always find a line holding a given element but whether one can always find a particular element in a given line. Quantifier dyslexia hits WM again. > So your magic belief is easily > destroyed. Not by those who cannot keep their quantifiers straight. |