Cantor's Diagonal?
in [Math]
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From: Nam Nguyen on 11 Feb 2010 21:41 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> I'm glad that I'm neither crank nor standard. You must be a standard >> then? > > Why do you think I must be a standard? > Well, I don't think you're a crank and, though I could be wrong, I very much doubt if you are a "rebel" in the foundation of logic matters. Then, seems like you must be in that standard school of reasoning. (And many of your posts do seem to suggest so. No?)
From: Ken Quirici on 11 Feb 2010 22:54 On Feb 2, 9:59 pm, zuhair <zaljo...(a)gmail.com> wrote: > Hi all, > > I have some difficulty digesting the diagonal argument of Cantor's. > > The argument is that the set of all infinite binary sequences cannot > have a bijection to the set of all natural numbers, thereby proving > that the former set is uncountable? > > However the argument looks to me to be so designed as to reach to that > goal? > > One can look at matters from an alternative way such as to elude > Cantor's conclusion! > > Examine the following: > > Lets take the infinite binary sequences of the letters O and H > > so for example we have the sequence > > X = OHOHOH........ > > in which O is coupled to the even naturals and H coupled to the odd > naturals. > > so the sequence above is > > X= {<O,i>,<H,j>| i is an even natural, j is an odd natural} > > so X is just an example of a infinite binary sequence. > > However lets try to see weather we can have a bijection between > the set of all infinite binary sequences and the set w+1 > which is {0,1,2,....,w} > > so we'll have the following table: > > 0 , 1 , 2 , 3 , ... > 0 H , O , O , H ,.... > 1 O , H , H , O ,.... > 2 H , H , H , H ,.... > 3 O ,O , H , H ,.... > . > . > . > . > > w O, O , O, O ,... > > Now according to the above arrangement one CANNOT define a diagonal ! > since the w_th sequence do not have a w_th entry > to put H or O in it. > > So if we can have a diagonal then this would be of the set of all > infinite binary sequences except the w_th one, i.e. of the following > > 0 , 1 , 2 , 3 , ... > 0 H , O , O , H ,.... > 1 O , H , H , O ,.... > 2 H , H , H , H ,.... > 3 O ,O , H , H ,.... > . > . > . > . > > Suppose that the diagonal of those was D=HHHHH.... > i.e. D={<H,n>| n is a natural number} > > Now the counter-diagonal would be D' = OOOO... > i.e. D' = {<O,n>| n is a natural number} > > However there is nothing to prevent the w_th infinite binary sequence > from being D' ? > > So neither we can have a diagonal of all the infinite binary > sequences, nor the diagonal of a subset of these sequences would yield > a successful diagonal argument such as to conclude that the set of all > infinite binary sequences is uncountable? > > Thereby Cantor's argument fail in this situation! > > What I am trying to say is that this Diagonal argument seems to be > purposefully designed to reach the goal of concluding that > the set of all infinite binary sequences is uncountable, by merely > selecting a particular bijection with the set {0,1,2,3,....} > in a particular arrangement, such as to make a diagonal possible, such > as to conclude the uncountability of these infinite binary sequences, > While if we make simple re-arrangement like the one posed above then > this argument vanish! > > There must be something wrong with the way I had put things here, but > I would rather want to read the full proof of this diagonal argument > in Zermelo's set theory. > > Zuhair The proposition we are attempting to prove is that the reals are uncountable. If we show the reals in the interval [0,1) are uncountable, we're done. The reals in the interval [0,1} are all decimal representations of the form: 0.d11 d12 d13 d14 .... where each dij is a decimal digit. If the reals in the interval [0,1) are in fact countable we can list them - here is some arbitrary listing of them: r1 = 0.d11 d12 d13 d14 ... r2 = 0.d21 d22 d23 d24 ... r3 = 0.d31 d32 d33 d34 ... r4 = 0.d41 d42 d43 d44.... (I'm getting tired of writing) We can now construct a new decimal representation which differs from each rn in the nth place. Is the method of construction in dispute? If not, then we're done, aren't we, since we've found a valid decimal representation which differs from every decimal representation in an arbitrary list of the reals in [0,1). Therefore there IS no arbitrary list of the reals in [0,1), and since they can't be listed, they're not countable. Does that work?
From: Aatu Koskensilta on 12 Feb 2010 09:11 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Then, seems like you must be in that standard school of reasoning. Let's by all means stipulate I'm the most ardent conformist, a spineless lapdog of the orthodoxy, incessantly biting at the ankles of poor logical rebels -- but what is this "standard school of reasoning"? > (And many of your posts do seem to suggest so. No?) What my posts seem to suggest is for others to decide. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 12 Feb 2010 11:29 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Then, seems like you must be in that standard school of reasoning. > > Let's by all means stipulate I'm the most ardent conformist, a spineless > lapdog of the orthodoxy, incessantly biting at the ankles of poor > logical rebels -- but what is this "standard school of reasoning"? The school of reasoning in which those who subscribe to it would be, in your phrasing, "ardent conformist" and "lapdog orthodox" not just to the technical methods of reasoning (which is a good trait) but also to certain accumulated collective thoughts, traditions, philosophies, which are detrimental to the knowledge about mathematical abstraction (and which altogether is a bad trait, imho).
From: Nam Nguyen on 12 Feb 2010 12:24
Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Then, seems like you must be in that standard school of reasoning. >> >> Let's by all means stipulate I'm the most ardent conformist, a spineless >> lapdog of the orthodoxy, incessantly biting at the ankles of poor >> logical rebels -- but what is this "standard school of reasoning"? > > The school of reasoning in which those who subscribe to it would be, > in your phrasing, "ardent conformist" and "lapdog orthodox" not just > to the technical methods of reasoning (which is a good trait) but > also to certain accumulated collective thoughts, traditions, philosophies, > which are detrimental to the knowledge about mathematical abstraction > (and which altogether is a bad trait, imho). An example of such collective tradition would be the belief there isn't such a thing as a formula that it's impossible to assign a truth value in the naturals (i.e. arithmetically undecidable formula). Another example, is the undermining the rigidity of the rules of inference, as syntactical game of symbol manipulation. To this school, subjective interpretations, assumptions in models would override what we can *or can't* know about the undecidability of a T through rules of inference. To paraphrase, in meta level school would reason by intuition's "Basic Instincts" instead of logical strength and constraint of symbolic manipulations. |