Talk:Cantor's diagonal argument/Arguments: Difference between revisions

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This provides a very simple proof for Godel’s incompleteness theorem (the relevance of the encompassed natural number system is clear). Please read my discussion text in the Wikipedia article “Godel’s Incompleteness Theorems”. [[e-mail] --- 10 February 2006]

== A layperson's objection to Cantor's proof ==

I have a question and possible objection to Cantor's diagonal proof. I am not a mathematician, so please don't jump on me if there are technical errors in how I express this.

To the best of my understanding, the basic idea of the diagonal proof is this. Make a list that you imagine to contain all the real numbers between 0 and 1. Then it is always possible to construct a number not on that list by defining the constructed number to differ from the first number in the frst decimal place, from the second number in the second decimal place, and so on. It may simplify thinking about the argument to specify that all the numbers are expressed in binary. That will insure that for any given list, one and only one number can be constructed using Cantor's method.

Now let's make our list and construct our "Cantor number".

Now here's the slick trick. Create a second auxilary list, call it "Extras" (or whatever). Place the newly constructed number on that list. Now if we combine the original list with the "extra" list, we have a list that contains precisely the number that was suposedly "not on the (original) list".

But there's an obvious objection. We now have a new list that is subject anew to the diagonalization process. We can still create a number not on the new list.

No problem. Just put that number as the second entry in the "extra" list, recombine the extra and original lists, and you have a new list. Sure you can now construct another number not on the list. And I can put it on the list. For as long as you can construct numbers not on the newest list, I can put those numbers on a newer list.

Maybe I missed something obvious, but this process looks a lot like enumeration to me. I would suggest that far from proving that the reals cannot be enumerated, Cantor in fact provided precisely the method for enumerating them.

I'm sure that the overwhelming majority of mathematicians here who accept Cantor's proof will see flaws in this argument, but I would be very interested to know what they might be. I have thought about it a lot, and I can't see any obvious objection.

Although, there is one possible question that occurs to me, but I don't know enough to answer it. It can be objected that we are not actually enumerating "all" the reals in the specified interval, because even if we construct diagonal numbers and amend our list "forever", there would be reals that would never show up on the list. If so, this would disprove my argument. But is it so?

Comments invited

-Steve

[[User:69.209.148.136|69.209.148.136]] 01:10, 24 May 2006 (UTC)

:So this is an objection that occurs to lots of people from time to time. Many of them come to the conclusion that they've discovered a simple error that more than a century of mathematicians have somehow missed; I have to say it's refreshing to see your quite different attitude.

:The thing to keep in mind is that the argument applies to ''every'' enumeration. Yes, if you start with an incomplete enumeration, the argument will find some you forgot to include. But, if a complete one exists, then why start with an incomplete one? Just throw the complete one in from the start.

:But of course if you did, then the argument would give you a contradiction. So you can't. But if it existed, you could. So it doesn't exist. --[[User:Trovatore|Trovatore]] 03:48, 24 May 2006 (UTC)

Suppose that you were correct. By the same reasoning, let L be a finite list of integers. There is an integer not on the list, so add it. Continue this process, generating new lists. The problem is that after doing this infinitely many times, the list is no longer finite. In your case above, if you could some how carry out the process; it still does not mean that the list will remain countable. On an aside, the Baire Category Theorem can be used to show that the reals are not equipotent the integers, hence, even if Cantor did make an error; the reals are still nondenumerable.[[User:Phoenix1177|Phoenix1177]] ([[User talk:Phoenix1177|talk]]) 05:03, 2 January 2008 (UTC)