Re: Eggs is eggs - another bash at Cantor's diagonal theorum
- From: Virgil <Virgil@xxxxxxxxx>
- Date: Sat, 05 Apr 2008 12:58:20 -0600
In article
<30055829.1207397642582.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Andrew Hood <sniq_e@xxxxxxxxx> wrote:
Hi All,
I have been trying to believe Cantor's diagonal theory since my Grandad
showed me it when I was about 10 years old. Somehow, it just doesn't wash
for me. Recently, I think that I have been able to describe why and would
love to either have confirmation or intelligent refutation - please, do take
my argument to pieces if it is not correct - I'd love to know why!
Allow me to consider first the case of Cantor's diagonal where we ask to
represent all of the real numbers between 0 and 1 in binary form, assuming
the preceding "0.":
Allow me to consider first of all the possibilities with two decimal places:
0.00
0.01
0.10
0.11
And we need four rows... with three decimal places, eight rows... and so on.
Now, it seems to me that the taking of a diagonal is only possible when the
table is square (or for Cantor's argument to work, the number of columns must
be equal to or greater than the number of rows); Cantor's table is clearly
not square, since by increasing the number of columns (therefore the number
of decimal places) by one, we double the number of rows.
The only possible reconciliation to this is to claim that all infinites are
equal (the opposite of what Cantor claims to prove); particularly, looking at
the number of rows and columns: r = 2^c, we find that Cantor and his
followers would agree that these infintes are different (where there might be
argument about r=c^2).
All sequences necessarily are countable, and are of equal cardinality.
What Cantor showed was that the set of all reals cannot be arranged into
a sequence.
.
- References:
- Eggs is eggs - another bash at Cantor's diagonal theorum
- From: Andrew Hood
- Eggs is eggs - another bash at Cantor's diagonal theorum
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