On 26 Jan 2015, at 23:54, John Clark wrote:
On Mon, Jan 26, 2015 Bruno Marchal <marc...@ulb.ac.be> wrote:
> We cannot generate algorithmically a random sequence, but we can
generate algorithmically all random sequence, thanks to the fact
that the in the sequence
0
and
1
we already generate the correct digit "0" of the 2^aleph_zero random
sequences beginning by 0, and the correct digit "1" of the other half.
Then we proceed,
00
01
and
10
11
and we continue in that way, we generate in that way all finite
initial segments of all sequences.
Well sure, it would be easy to write a program to generate every
possible sequence of digits of FINITE length, but If you give me a
list, any list, of infinitely many infinite numbers I know it does
not contain them all because I can always use Cantor's diagonal
argument to generate a number that is not on your list.
Finiteness plays no role here. I can generate individually all
infinite sequences, and you will not been able to use Cantor's
diagonal to find a sequence which is not individually generated. If
you believe so, give me the infinite binary sequence not generated by
my procedure above.
Cantor brought the contradiction by assuming there is a bijection
between N and the set of infinite binary sequences (say).
The procedure that I use does not assume such bijection. On the
contrary, as I said explicitly each finite sequence of digits
generated at any time is admitted as being the initial segment of a
continuum (uncountable sequence). The "01" appearing above is supposed
to be an initial segment of one sequence, but of a continuum of
sequences, right at the start.
All you need to accept is that "generating an infinite sequence" means
generating all its initial finite segment, perhaps among other
things, like
3
31
314
3141
31415
...
generates the decimals of Pi (despite at each finite instant only a
small portion of the decimals of Pi are generated).
If I was pretending that I can generate all sequences individually,
and without generating infinitely many other sequences, then you could
use Cantor to refute my claim.
What I say is very easy, and sometimes exploited in recursion theory,
or in intuitionist analysis (Brouwer Fan theorem). It plays some role
in the UDA too. There is no problem with Cantor.
Bruno
John K Clark
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it,
send an email to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
