[FOM] Sharp mathematical distinction between potential and actual infinity?

[Harvey Friedman]

Reply to Chow.

On 9/26/03 10:49 AM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> However, I might change my mind if someone could demonstrate a sharp
> distinction between potential and actual infinity.  The distinction
> seems to have pretty much evaporated in modern mathematics, and it
> seems that only the philosophers still talk about it.  Or am I just
> underinformed?  Is there, for example, a way of drawing a clear
> mathematical distinction between potential and actual infinity that
> blocks the move from skepticism-towards-N to skepticism-towards-rules?
> 

I don't know what Chow has in mind here, but there are some very interesting
formal systems that purport to be based on potential rather than actual
infinity. 

Under the usual view, one makes these distinctions.

1. Potential infinity, where one does not accept quantification over all
natural numbers as meaningful.

2. Potential infinity, where one accepts quantification over all natural
numbers as meaningful, but one does not accept the existence of infinitary
objects.

3. Actual infinity, where one accepts the existence of infinite sets of
natural numbers, and quantification over all natural numbers as meaningful,
but one does not accept quantification over all sets of natural numbers as
meaningful.

4. Actual infinity, where one accepts the existence of infinite sets of
natural numbers, and quantification over all natural numbers as meaningful,
and one also accepts quantification over all sets of natural numbers as
meaningful. 

5. Beyond.

There is an elaborate set of formal systems, with rather robust properties,
associated with all four of these attitudes.

Harvey Friedman