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Topic: § 432 The complete list is not a square
Replies: 10   Last Post: Feb 18, 2014 4:48 AM

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Registered: 10/18/08
§ 432 The complete list is not a square
Posted: Feb 13, 2014 4:01 PM
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Cantor's diagonalization argument.
Ok, I've seen this proof countless times. And like I say it's logically flawed because it requires the a completed list of numerals must be square, which they can' t be.
First off you need to understand the numerals are NOT numbers. They are symbols that represent numbers. Numbers are actually ideas of quantity that represent how many individual things are in a collection.
So we aren't working with numbers here at all. We are working with numeral representations of numbers.
So look at the properties of our numeral representations of number:
Well, to begin with we have the numeral system based on ten.
This includes the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
How many different numbers can we list using a column that is a single digit wide?
Well, we can only list ten different numbers.
Notice that this is a completed list of all possible numbers. Notice also that this list is not square. This list is extremely rectangular. It is far taller than it is wide.
Let, apply Cantor's diagonal method to our complete list of numbers that are represented by only one numeral wide.
Let cross off the first number on our list which is zero and replace it with any arbitrary number from 1-9 (i.e. any number that is not zero) [...]
Ok we struck out zero and we'll arbitrary choose the numeral 7 to replace it.
Was the numeral 7 already on our previous list? Sure it was. We weren't able to get to it using a diagonal line because the list is far taller than it is wide.
Now you might say, "But who cares? We're going to take this out to infinity!"
But that doesn't help at all.
We can already see that in a finite situation we are far behind where we need to be, and with every digit we cross off we get exponentially further behind the list.
Taking this process out to infinity would be a total disaster.
[Divine Insight, Why Cantor's Diagonalization Proof is Flawed (28 June 2012)]
Thanks to Albrecht Storz for the hint to this source

{{Usually matheologians confuse infinities. A potentially infinits list is always a square up to every n. But the presence of the antidiagonal cannot be excluded. Then they switch infinities. Nothing new. But necessary to mention it over and over again in order to protect newbies from falling into that trap.}}

Regards, WM

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