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Re: Mathelogy S 011
Posted:
May 20, 2012 3:26 PM
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On 20 Mai, 20:21, "dilettante" <n...@nonono.no> wrote:
> Doron's sentence may or may not be serious, but it is a punchline whether he
> knew it or not. It declares itself to be meaningless.
>
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> news:b02a9210-1c47-499b-865c-f6eaa131910f@e20g2000vbm.googlegroups.com...
> On 20 Mai, 16:29, "dilettante" <n...@nonono.no> wrote:
>
> >> Still can't quite tell if you realize that the paper you linked to and
> >> quoted, even though it touches on some interesting mathematics, was
> >> intended
> >> primarily as a joke. The sentence you quote, 'Apriori, every statement
> >> that
> >> starts "for every integer n" is completely meaningless' is the punch
> >> line.
> >You allude to Wittgenstein's view of set theory? For if one person can
> >see it as a paradise of mathematicians, why should not another see it
> >as a joke? [Ludwig Wittgenstein, Matheoogy 012]
> >No, Doron's sentence is serious and true. Of course there are not
> >*all* numbers. Ask anybody to tell you where these "all numbers" are.
> >Ask anybody to name a natural number that would require more than
> >10^100 bits. It's simply as impossible as to write numbers with more
> >than 10 different digits on your pocket calculator.
>
> What does it mean for a natural number to "require N bits"?
A binary string that is not defined by a certain definition like
"take a one and then three zeros"
or
"take the first ten digits of the binary expansion of 1/3"
must be written with bits like
1000
or
0.010101010.
If there is no shorter definition is available, you need bits.
If more than 10^100 bits are required, then the number has a due
complexity. (Related information can be found under "Kolmogorov
complexity.)
> Are there any
> natural numbers that require more than 10000 bits? If so, what is the
> smallest one?
There are natural numbers that require more than 100 bits. A
comfortable way to find the smallest one, is to list all smaller ones.
But it is tedious. Perhaps there are better means to answer your
question.
A simple model is the pocket calculator. Check the smallest number
that cannot be displayed.
Regards, WM
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