[ExI] There are Infinitely Many Numbers, But Each of them is Finite (reply to Bryan Bishop)

[Bryan Bishop]

Hi Lee, glad you liked the bookmarks. When you said "on to the meat,"
I was hoping you had found the latest cultured meats section, but now
that I look at the page I realize I did not include that subject in
the latest synch.

While I am mentioning it, here's my overview of that meaty subject:
http://science.slashdot.org/comments.pl?sid=239081&cid=19583471

Mathematics, however, is equally meaty and interesting. I am partly
embarrased that you were able to find some of the Cantor refutation
links hidden in the deep depths of those categories. I had added them
in a rush, and have slowly been purging them from my collection ever
since.

As I agree with your refutation of the refutation, I figure I better
just add to the discusion by linking over to another page that still
rings in my head:

http://science.slashdot.org/comments.pl?sid=235713&cid=19227871
> In Applied Cryptography he outlines an interesting little computation to
> demonstrate why this is. Suppose you had a computer that contained a
> 256-bit register that was maximally efficient, meaning that toggling a
> bit required exactly one quantum of energy. Since smaller units of
> energy don't exist, you can't do better than that[*]. With that
> assumption, you can calculate how much energy it would take to cycle
> your 256-bit counter through all possible states. Schneier calculates
> that if you could capture all of the energy from a typical supernova
> and run your counter on that, you could count from 0 all the way up
> through about 2^219. So you'd need about 130 billion supernovas to run
> your counter through all of its 2^256 possible states.

And just some janitorial work:

> I perused the cryonics section's titles. It all looked pretty familiar (I've
> been signed up for nearly 20 years, and many people here are signed
> up with either Alcor or CI.)  Is there something in particular about it
> that you wanted to discuss?

Not in particular. I was figuring that adding to the discussion re:
probability and changing from one moment to the next had strong
correlations to cryonics and ultracold vacuum physics because as you
get colder, less and less actions take place. (And quantifying action
is a tough, I hear.)

And I have no idea if this makes for enough relevance to the previous
subject line that you offered in the previous message. I'll make an
assumption and shoot.

- Bryan

On 7/15/07, Lee Corbin <lcorbin at rawbw.com> wrote:
> Hi Bryan,
>
> Welcome to the list!  By the way, one really should change the subject
> line whenever they're a significant change of subject. If you've perused
> our archives at all, you'll see the pattern, and see how (just to help an
> archive reviewer) sometimes the construction
>
>         [Exi] <newSubjectLine> (was <old subject line prefix>)
>
> is used, except the [Exi] part is automatically generated and need
> not be added by you.
>
> When there is *no* connection, then make *no* "(was..." construction).
>
> I perused the cryonics section's titles. It all looked pretty familiar (I've
> been signed up for nearly 20 years, and many people here are signed
> up with either Alcor or CI.)  Is there something in particular about it
> that you wanted to discuss?
>
> Also, your list of essays and references looks like a non-trivial subset of
> the entire web!  A list, evidently (so far as I have had time to peruse it)
> reflecting good taste!
>
> Ahem, on to the meat.  One entry you list is
>
> http://www.marknielsen.net/Math/Counting/Diagonalization/Diagonalization_Disproof2.html
>
> which is a fun construction that produces a list of Very Rational numbers,
> where I am defining Very Rational (Decimal Fractions) to be those that
> terminate after a finite number of digits.  The list you create begins
>
> 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.11, 0.21, 0.31, 0.41, 0.51, 0.61, 0.71, 0.81, 0.91, 0.02, 0.12, 0.22, 0.32,
> 0.42, 0.52, 0.62, 0.72, 0.82, 0.92, 0.03, 0.13, 0.23, 0.33, 0.43, 0.53, 0.63, 0.73, 0.83, 0.93,
>
> The problem is that all these *are* indeed rational, but that the *infinite* string
> produced by the diagonalization is *not* rational.  So of course it does not
> appear on the list.
>
> The Step 5 of the ISSUES SECTION illustrates the error. It reads
>
> Step 5:  "Also, the sequence 0.33333... and so on is in my set D. There is no single entry which matches 1/3, but if can accept that
> a sequence 0.33333... is equal to 1/3, then after my list has been generated, 1/3 is in the list. --- I think."  is mistaken.  All
> the numbers
>
>       0.3
>       0.33
>       0.333
>       0.3333
>       0.33333
>       0.333333
>       ...
>
> is in the list, but the writer apparently does not understand that the *infinite* string .333333333...
> does not anywhere appear on the list!
>
> Lee