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From: George Greene on 7 Jul 2010 13:18
On Jul 6, 3:23 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> C10 = 0.12345678910111213141516...
>
> It contains pi, segmented.
Well, gee,
..012345678901234567890123456789......
repeating forever ALSO contains Pi, segmented, OR ANY OTHER NUMBER,
SEGMENTED, if you will allow a segment-length OF 1, DUMBASS.

> You have to stop thinking of infinity
> as a really long string for a moment,

WE never STARTED thinking of "infinity" as that.
A really long string is an infinite SUBSET.
A REAL is represented as a really long string because it's an infinite
SUBSET.

> because there's no infinite string of pi inside root 2 or e etc.

OF COURSE there is, SEGMENTED.

> but the further you go along the string the
> longer and longer sequences are found.

SO WHAT??
NOBODY HAS EVER DISPUTED THIS!
You can take a list OF FINITE strings and have it be true that
the further down it you go, the more OF EVERY real or string hs
occurred in it!

This simply does not have anything to do with the existence of
infinity.
Infinity exists because there are an infinite number of natural
numbers.
Higher infinities exist because if those natural numbers form a set,
then it has a powerset, and the powerset IS ALWAYS bigger!

From: Dan Christensen on 7 Jul 2010 13:36
On Jul 6, 3:23 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> > does Champernowne's constant contain pi?  It does contain all finite digit
> > sequences of pi.
>
> > - Tim
>
> C10 = 0.12345678910111213141516...
>
> It contains pi, segmented.
>
> You have to stop thinking of infinity as a really long string for a moment, because there's
> no infinite string of pi inside root 2 or e etc.  but the further you go along the string the
> longer and longer sequences are found.
>
> -----------------------------------------------
>
> 1/ there's no infinite sequence of pi's digits within C10  (every finite starting point has a finite ending point)
> 2/ as the length of C10 digit expansion -> oo, the consecutive number of digits of pi -> oo
> 3/ the length of C10 digit expansion is oo
> 4/ the consecutive number of digits of pi = oo  (3) -> (2)
>
> CONTRADICTION (1) & (4)
>
> THEREFORE no limit exists as the length of digit expansions (of any real) -> oo
>
> GENERALIZATION no limit exists as the length of sequences (of any type) -> oo
>
> INFERENCE there is no oo
>

Sorry, I can't follow your proof, but don't you think there might be
something wrong with your system of axioms, whatever they may be?
Infinity is no big deal. In natural number arithmetic for example, we
just assume that if you add 1 to any number, the result will be
another, larger number. Do you really think it is necessary to specify
some arbitrary maximum number?

Dan
Download my DC Proof software http://www.dcproof.com
From: Curt Welch on 7 Jul 2010 14:36
George Greene <greeneg(a)email.unc.edu> wrote:
> On Jul 6, 3:23=A0pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:

> Infinity exists because there are an infinite number of natural
> numbers.
> Higher infinities exist because if those natural numbers form a set,
> then it has a powerset, and the powerset IS ALWAYS bigger!

You have to be careful in such debates to understand what "exists" means
and how you are using it.

You can't prove that "infinity" exists by referencing the natural numbers
if you can't first prove that the natural numbers "exist".

There is physical existence which is the foundation of all types of
existence. But math isn't about simple physical existence. It's about
words and their definitions. It's a study of formal langauge and what can
be _said_ using a formal language. It's not about physical existence
beyond the physical existence of the langauge.

Infinity as a physical object can be argued not to exist. But as a word in
a formal language, it does exist. And in Math, if the word exists, and
it's associated definition exists, then it exists in math. And we can,
very easily, use words to define what we mean by infinite - so the concept
does exist, even if it can't be used to label a physical object.

It exists as a procedure which has no way to halt. (10 goto 10).

The self referencing power of language then allows us to define an infinite
number of different procedures which all never halt. And in that self
reference, we defined (aka brought into existence) yet another type of
infinity.

But just becuase we can use the power of language to define these concepts,
doesn't mean they are valid labels for real aspects of our physical
universe. Math however is not about that. It's just about exploring what
can be _said_ with a formal langauge. And with a formal langauge, we can
define infinity, and higher orders of infinity. They exist in the language
of mathematics, but may or may not, exist in any sense in the physical
universe. You have to keep those two types of existence separate when
studying and understanding mathematics.

--
Curt Welch http://CurtWelch.Com/
curt(a)kcwc.com http://NewsReader.Com/
From: Aatu Koskensilta on 7 Jul 2010 14:31
curt(a)kcwc.com (Curt Welch) writes:

> But math isn't about simple physical existence. It's about words and
> their definitions. It's a study of formal langauge and what can be
> _said_ using a formal language.

No it's not.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: |-|ercules on 7 Jul 2010 18:30
"Aatu Koskensilta" <aatu.koskensilta(a)uta.fi> wrote ...
> curt(a)kcwc.com (Curt Welch) writes:
>
>> But math isn't about simple physical existence. It's about words and
>> their definitions. It's a study of formal langauge and what can be
>> _said_ using a formal language.
>
> No it's not.

A counter example?

Herc
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Prev: EINSTEIN'S ABUSE OF TIME
Next: NP+complete-problem navigation, search In computational complexity theory, the complexity class NP-complete (abbreviated NP-C or NPC), is a class of problems having two properties: * It is in the set of NP (nondeterministic polynomial time) pr