(adsbygoogle = window.adsbygoogle || []).push({}); Cantor diagonal argument-???

The following eight statements contain the essence of Cantor's argument.

1. A 'real' number is represented by an infinite decimal expansion, an unending sequence of integers

to the right of the decimal point.

2. Assume the set of real numbers in the interval[0,1] (which excludes 0 and 1)

is countably infinite, (can be paired 1 to 1 with the set of integers).

3. Assume a list exists containing all the numbers in that set.

4. The list begins with these sequences (in random order),

.4075501...

.9240732...

.2110208...

.0345678...

.5161705...

.8978675...

.3000333...

5. Select a diagonal sequence from the list as shown, .4215773... and

substitute a different integer for all positions, e.g. (.5326884...), call the altered sequence x.

6. If x is different at the position of intersection with each successive number,

it is not included in the list.

7. If the list does not include x it is incomplete which contradicts statement 3.

8. If a complete list does not exist, statement 2 if false.

Statement 1 is a definition.

Statement 2 is Cantor's postulate.

Statement 3 is his prop.

Statement 4 is a continuation of the assumptions.

Statement 5 is an acceptable algorithm.

Statement 6 is in error.

Cantor assumes in using his algorithm that the number of positions equals the number of elements,

or intentionally omits this fact.

The list is never square because the columns increase at a linear rate and the rows increase at an exponential rate.

The altered diagonal sequence can only be new if it differs in at least one position forallnumbers listed.

By using only a square portion of the list in the algorithm, the diagonal is by definition always incomplete.

You cannot make 10^n comparisons with a diagonal containing n elements,

therefore the algorithm cannot determine statement 6 as true or false, nor the dependent statements 7 and 8.

any comments?

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# Cantor diagonal argument-?

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