Re: Does a potential infinity actually exist?

In article <1187827715.173302.161730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, david petry writes:

It seems that there is a lot of confusion about the notion of a
"potential" infinity.

It seems as if there is an inability to provide a *definition* of
the term "potential infinity".

I tried myself to find a really clear explanation of the concept, and
I was unable to. So I guess it's up to me to explain it.

No, it's up to you to provide a definition of "potential infinity". Quotes
from famous dead people are not definitions. Saying "we could say this
instead of saying that" isn't a defintion. Saying that it's not possible
to write down all of the digits of the decimal representation of sqrt(2)
isn't a definition.

The idea of a potential infinity may be equated with the view of
infinity held by Gauss, Poincare, and Weyl. The following quotes
characterize this view:

So the basic idea, as so eloquently stated by Weyl, is that infinite
sets do not exist in exactly the same sense as the sense in which
finite sets exist, and that mathematics should distinguish between the
two senses.

So, if I prove a result about functions, I need to prove it twice,
once for functions with finite domains and ranges and a second time
for funtions with infinite domains and ranges? Oh, and I guess a
third time for functions with infinite domains and finite ranges.
I don't see how this is a benefit to anybody.

distinction whatsoever between different senses of existence,

"It depends upon what the meaning of 'is' is."

hence it impossible, or nearly so, to even talk about the potential
existence of infinity within the language of classical mathematics.
That seems to explain why there is so much confusion about the notion
of potential infinity.

This seems to explain the continued inability of the Potentialists
to provide definitions of whatever the heck it is that they're talking
about.

The mathematical entities that have an "actual" existence are the
things that we can actually compute. Thus, for example, we can
actually compute an approximation of sqrt(2) accurate to seven digits,
and hence such an approximation actually exists. Likewise, with a
computer, we can compute a hundred digit approximation of sqrt(2), or
even a million digit approximation, and so such approximations
actually exist.

Okay, so we can come up with better and better approximations to
sqrt(2). This is, of course, acknowledgement that the real number
"sqrt(2)" exists as something that can be approximated. Good. We've
established that sqrt(2) exists.

In general, we can compute arbitrarily many digits of
sqrt(2), but we cannot actually compute infinitely many such digits.

Since nobody here (to the best of my knowledge) has claimed that
it is possible to write out "all of the digits of sqrt(2)", this
point is really irrelevant.

To illustrate these points, consider the following two sentences:

1) sum (k=1..oo) 1/2^k = 1

The expression "sum (k = 1..oo)" is really just a short-hand for
"Lim (m -> oo) sum (k=..m)". Which means that it's the same as
the following expression:

2) Ap En Am (m>n) -> |sum (k=1..m) 1/2^k - 1| < 1/p

The two sentences say exactly the same thing, but the first sentence

uses a shorthand that

invokes infinity, while the second doesn't. Most people would agree
that the first sentence is far more readily comprehended,

It's nice to know that you realize this.

and so we
see that we don't actually need infinity, but infinity is indeed a
very useful figure of speech.

Or an abbreviation. I do hope that you're aware of two things:

First, that "Lim (m -> oo)" is just a shorthand or abbreviation for
"An Em (m>n)"

Secondly, that you are not talking about infinite sets, but about limits
as something increases without bound (commonly known as "going to
infinity").

So how would we develop a set theory if we insist that infinity has
only a potential existence? For starters,

we'd start by providing a definition of what this means.


--
Michael F. Stemper
#include <Standard_Disclaimer>
Build a man a fire, and you warm him for a day. Set him on fire,
and you warm him for a lifetime.

.

Relevant Pages

  • Re: Relative Cardinality
    ... No problem with potential infinity. ... > Knowing all the digits of the real numbers can be done with arithmetic ... > particles, in the same order you've established. ... > not have said that your "cardinality" is the same as traditional ...
    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... Infinity is not an integer, ... just a string of digits, ... >possible to prove anything by contradiction. ...
    (sci.math)
  • Re: Relative Cardinality
    ... Which infinity is a number according to Cantor? ... >> particles, in the same order you've established. ... But it is sufficient to encode the information: All digits ... >> cardinality, unless you can show that, for any set S, the cardinality ...
    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... > infinity, you are making some basic assumptions on what an idea is. ... > of the size of an infinite set is a contradiction in itself). ... > So you just reverse the digits in the integer to create the real. ... > this mapping is one to one and covers all the reals in that range. ...
    (sci.math)
  • Re: Relative Cardinality
    ... > No problem with potential infinity. ... How do you express "potential infinity" in the universe? ... >> particles, in the same order you've established. ... But it is sufficient to encode the information: All digits ...
    (sci.math)