Re: Infinity and indefinite extensibility

David Cressey wrote:

There is probably some standard terminolgy for what you are about to read.
I don't know what that standard terminology is. Sorry about that.

I think it's worthwhile, in the discussion of infinite domains and finite
state implementations, to distinguish between "infinite" and "indefinitely
extensible".

For example, the familiar decimal place value notation system for natural
numbers is indefinitely extensible. That is, there is no such thing as the
largest natural number that the scheme can represent. If a number can be
represented in decimal, its successor can be represented as well. And so
on.

That, of course, assumes infinite space to write on. However, however large our universe is, we can always imagine a larger one where we could write a larger number using the notation.


However, every number that can be written in decimal is finite. And every
set of natural that has been formed by listing the elements is a finite
set. And this will remain true until the twelth of never. (yuk, yuk).

You can't represent "infinity" with natural numbers, but there is no upper
bound to the range of natural numbers.

Actually, infinity is the upper bound. That's what infinity is. It is the bound that numbers approach as they grow arbitrarily large. We represent it using the lazy eight.

It's value is not defined in the real numbers but establishes a boundary for the set of reals.

Think back to highschool and statements like "The limit of 1/x as x approaches infinity". The limit, of course, is zero because "as x approaches infinity, 1/x approaches 0". 1/0, of course, is undefined.


It seems to me that this distinction would be revelant over in the
"impossible database design" topic.

As I already pointed out over there, whether and how often intervals of two time types overlap is a matter of harmonics. The limits of a finite computer only determine the number of beats one gets before one runs out of representation.
.

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