Re: Poetential infinity


Bill Taylor wrote:

> There was once a lot of debate, and still occasionally is,
> about the difference between "actual infinity" and "potential
> infinity".
>
> It always struck me that, whatever you thought about them,
> at least the defintion of actual inf was reasonably clear,
> but that the same couldn't really be said of potential inf.


The words "potential" and "actual", when applied to infinity, usually
indicate how we reason about infinity.

For those who claim that infinity has only a potential existence, the
only access we have to the infinite is through the notion of limits. So
every property we ascribe to the infinite must correspond to properties
of finite approximations of the infinite. In other words, the infinite
is an abstraction derived from the finite. Constructivism restricts
itself to this notion of infinity.

The idea behind the actual infinite is that we are free to postulate
properties of the infinite which don't correspond to properties of
finite approximations of the infinite, just so long as we are logically
consistent. And of course, this idea was introduced into mathematics by
Cantor.

Hermann Weyl had something interesting to say about all this:

"...classical logic was abstracted from the mathematics of finite sets
and their subsets...Forgetful of this limited origin, one afterwards
mistook that logic for something above and prior to all mathematics,
and finally applied it, without justification, to the mathematics of
infinite sets. This is the Fall and original sin of [Cantor's] set
theory ..." (Weyl, 1946)


> So, it occurred to me that a better name for "potential infinity"
> would be "unboundedly finite".

I don't know why it would be *better*.

.

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