Re: Why isn't there a 1 to 1 correspondence between the real numbers and the natural numbers?

On 2008-04-19 15:46:20 -0300, Lou Pecora <pecora@xxxxxxxxxxxxxxxxxx> said:

In article
<eda17d9f-ebd8-4f03-87af-dde056d9cfb8@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Dave Dodson <dave_and_darla@xxxxxxxx> wrote:

On Apr 19, 9:42 am, Zanthius <zanthius of d...@xxxxxxxxx> wrote:
On 19 Apr, 15:19, Dave Dodson <dave and da...@xxxxxxxx> wrote:

Well, you can't see very far. While some numbers are defined as
infinite series or limits of infinite sequences, pi is not one of
them. The definition of pi has no mention of an infinite series. The
fact that it is irrational and transcendental merely means that we
can't represent its value numerically.

There are many "definitions" of pi. If your are into that sort of things
you can give a large variety of infinite series that sum to pi. They
usually show up as some interesting series = multiple of pi but that
is easily fixed. When friends of mine were studying number theory I
recall things that were "curious sum over primes yields pi/6" and such.
They are not really worth looking up.

One can discuss which of the many definitions are truly fundamental and
which are mathematical tricks that rely on the fundamental ones. My
personal choice is based on derivatives of suitably normalized (viz
period one) trigonometric functions. The relationship to circumferences
of circles then becomes a mathematical curiosity. In many systems of
definitions and theorems you can construct a system with the same facts
but with a different assignment of which are definitions and which are
theorems. Makes for an interesting advanced course that goes through some
standard topic in a different order, typically with theorems and definitions
interchanged. No one claimed that pure mathematics was instantly useful!

So, pi is a number that cannot be represented numerically.

That is correct. In fact, most numbers cannot be represented
numerically.

Dave

Hmmm... you are jogging my memory (correctly, I hope). I think the
situation is even more complex. Most real numbers cannot be expressed
with an algorithm that is more compact, i.e. of less complexity, than
the number itself. So what we think of as typical irrational numbers
like e and pi are exceptions (of measure zero, in a sense) in that they
can be expressed in terms of finite algorithms.

Maybe someone more knowledgable can add more or correct what I said,
although I think it's basically right.

Kolmogorov randomness is that there is no compact algorithm. That most
transcendentals are Kolmogorov random is a plausible result. Someone
who has actually studied algorithmic complexity/randomness would be
able to quote the relevant results.



.

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