Re: Cantor Confusion


mueckenh@xxxxxxxxxxxxxxxxx wrote:
William Hughes schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
William Hughes schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
Virgil schrieb:

(It is contained in the union of all lines, but the
union of all lines is not a line)

That is a void assertion unless you can prove it by showing that
element by which the union differes from all the lines.

Not quite. In order to achieve that the diagoal is not in any linem all
that is required is:
Given any line there is an element of the diagonal not in THAT line.
It is not requires that:
There is an element of the diagonal that is not in any line.


For linear sets you cannot help yourself by stating that the diagonal
differs form line A by element b and from line B by element a, but a is
in A and b is in B. This outcome is wrong.

Therefore your reasoning "there is an element of the diagonal not in
THAT line. It is not required that: There is an element of the diagonal
that is not in any line." is inapplicable for linear sets. You see it
best if you try to give an example using a finite element a or b.


In every finite example the line that contains
the diagonal is the last line.

Every example with natural numbers (finite lines) is a finite example.

Your claim is that there is a line which contains the diagonal.

Because a diagonal longer than any line is not a diagonal.

Call it L_D. Question: "Is L_D the last line?"

There is no last line

Then, there is a line that comes after L_D.

Therefore :L_D does not contain every element
that can be shown to exist in the diagonal.

All elements that can be shown to exist in the diagonal can be shown to
exist in one single line.


Call it L_D

L_D contains a largest element. n.

L_D is not the last line, so there is
a line with element n+1,

Element n+1 can be shown to exist in the diagonal.

- William Hughes

.

Relevant Pages

  • Re: Cantor Confusion
    ... That is a void assertion unless you can prove it by showing that ... element by which the union differes from all the lines. ... For linear sets you cannot help yourself by stating that the diagonal ...
    (sci.math)
  • Re: Cantor Confusion
    ... That is a void assertion unless you can prove it by showing that ... element by which the union differes from all the lines. ... For linear sets you cannot help yourself by stating that the diagonal ...
    (sci.math)
  • Re: Cantor Confusion
    ... element by which the union differes from all the lines. ... For linear sets you cannot help yourself by stating that the diagonal ... so the antecedent cannot be true. ...
    (sci.math)
  • Re: Cantor Confusion
    ... That is a void assertion unless you can prove it by showing that ... element by which the union differes from all the lines. ... For linear sets you cannot help yourself by stating that the diagonal ...
    (sci.math)
  • Re: Cantor Confusion
    ... element by which the union differes from all the lines. ... For linear sets you cannot help yourself by stating that the diagonal ... Every element that can be shown to belong to the diagonal can be shown ...
    (sci.math)
Loading