
Re: Peerreviewed arguments against Cantor Diagonalization
Posted:
Oct 29, 2012 11:44 PM


On Monday, October 29, 2012 9:38:59 PM UTC5, Peter Webb wrote: > Arturo Magidin wrote: > > > > > On Monday, October 29, 2012 9:37:20 AM UTC5, JRStern wrote: > > > > On Sun, 28 Oct 2012 19:20:51 0700 (PDT), Arturo Magidin > > > > > > > > <magidin@member.ams.org> wrote: > > > > > > > > > > > > > > > > >> How can this be unclear? > > > > > > > > > > > > > > > > > > Because I did not simply state the conclusion. > > > > > > > > > I gave you the "diagonal argument". > > > > > > > > > > > > > > > > I did not recognize the argument you gave as a/the "diagonal" > > > > > > > > argument. > > > > > > That suggests you are not very familiar with the argument. > > > > > > > I saw a separate argument for the same conclusion. > > > > > > It was not. > > > > > > > The diagonal argument as I've seen it starts with enumerations of > > > > > > > > numbers, selects out digits, and constructs from those digits, and > > > > > > > > then makes an assertion about the result. > > > > > > Like I said: if you want to discuss specific steps in a specific > > > argument, then you must provide that argument explicitly and in > > > detail. Otherwise, we may be talking about different things. > > > > > > I will allow that there are many expositions of the argument that are > > > not very precise, or very clear, or that are overly complicated (it > > > is very common to present the argument as an argument by > > > contradiction, although it really is not such). But that only makes > > > it all the more necessary that you be very explicit and very precise > > > about exactly what argument you are talking about. > > > > > > > > If you are questioning whether it is "coherent", then you should > > > > > > > > > point to whatever point you find incoherent, rather than simply > > > > > > > > > quote and then give a sentence fragment. > > > > > > > > > > > > > > > > This is (perhaps too obviously) not my area of expertise, and rather > > > > > > > > than try to present my own proof or even formal objection at this > > > > > > > > point  I came looking for further background. > > > > > > A common presentation of the corollary (which in reality can be > > > obtained from the presentation I gave), is the following: > > > > > > THEOREM. If N is the natural numbers, (0,1) are the real numbers > > > between 0 and 1, not inclusively, and f:N>(0,1) is any function, > > > then f is not onto. That is, there exists a real number r(f) (which > > > depends on f) such that f(n)=/=r(f) for every n in N. > > > > > > Proof. Suppose that f:N>(0,1) if any function from the natural > > > numbers to the numbers in (0,1) (this is equivalent to a function > > > from N to the reals, using the bijection between (0,1) and the reals > > > afforded by the arctan function and a simple linear map, like I did). > > > > > > I claim that f is not onto. > > > > > > To that end, for each n in N, consider the decimal expansion of f(n) > > > (if f(n) has two expansions, select the one with only finitely many > > > nonzero entries). That is, > > > > > > f(n) = Sum_{i=1}^infty (a_{n,i}/10^n) 0<= a_{n,i} <= 9 > > > > > > (this is equivalent to saying f(n) = 0.a_{1,1}a_{1,2}a_{1,3}.... is > > > the decimal expansion) > > > > > > We construct a sequence b_n of digits as follows: > > > > > > b_n = 5 if a_{nn}=/=5 > > > b_n = 6 if a_{nn} = 5. > > > > > > The existence of (b_n) is guaranteed by the Principle of Induction, > > > which is a theorem of ZF (follows from the construction of N, which > > > is welldefined by the Axiom of Infinity). > > > > > > Let r(f) = Sum_{i=1}^{infty} b_n/10^n. > > > > > > This is a real number, since the sequence is Cauchy. It is less than > > > 1, because it is term by term strictly smaller than Sum (9/10^n) = 1; > > > it is greater than 1 because it is term by term strictly larger than > > > Sum (1/10^n) = 0.11111.... Thus, r(f) is a real number in (0,1). > > > > > > I claim that r(f)=/=f(n) for all n. First, note that r(f) has a > > > unique decimal expansion; thus, if f(n) has two decimal expansions, > > > then it cannot equal r(f). Hence, in order to ensure that r(f)=/=f(n) > > > it suffices to show that there is a k such that b_k=/=a_{n,k} > > > (because either f(n) has a unique decimal expansion, in which case > > > the fact that b_k=/=a_{n,k} suffices; or else f(n) has two decimal > > > expansions, in which case it cannot equal r(f)). > > > > > > By construction, b_k=/=a_{k,k}; hence r(f)=/=f(n). Thus, r(f) is not > > > in the range of f, so f is not onto. QED > > > > > > As to its validity: we are assuming that f is given; the fact that > > > f(n) has a decimal expansion follows by construction of the real > > > numbers as equivalence classes of Cauchy sequences. Our ability to > > > define b_n is guaranteed by the fact that we know f and we know the > > > expansion of f(n) (because we know f); and the fact that we have > > > defined a full sequence follows by induction. That r(f) is a real > > > number follows because the sequence of partial sums is Cauchy, hence > > > it represents a unique real number. > > > > > > > > > > You have made the proof slightly more rigorous by pointing out the > > antidiagonal is Real using Cachy sequences. > > > > Two problems with this: > > > > 1. The cranks don't dispute the antidiagonal is a Real, so you are not > > addressing their issue (whatever it is, but its not that the > > antidiagonal is not Real).
This is only a "problem" if you think that my presentation was an attempt at convincing or arguing with a crank. It was not. I was addressing the Original Poster of this thread, which seems to suffer not from crankiness or "antiCantorianism", but rather from simple and pure lack of knowledge about the subject (and a lack of desire to acquire that knowledge, prefering instead to ask and talk in vague terms). It was an attempt at presenting a specific instance of the argument so that he could provide specific questions about specific steps... something, alas, he seem sunwilling to do.
> 2. Your average Cantor crank wouldn't recognise a decimal expansion is > > a Cauchy sequence, or that Cauchy sequences specify Real numbers.
Again, only a problem if you think I believe I have provided an "answer" to cranks. I have neither the intention nor the desire to do so.
> You may have been better off proving P(S) > S (where you have > > access to the axioms of ZF)
Amusingly enough, I did do so earlier.
> and then showing c > N is a special > > case where S = N. As of course others have done in this thread.
Where "me" happens to be one of the value of "others [...] in this thread".
 Arturo Magidin

