[FOM] As to strict definitions of potential and actual infinities.

[Dean Buckner]

AZ wrote

>I believe that all the axiomatic set theories break down the
>classical logic and the classical mathematics in the following point.
> We have [...] the following two opposed axiomatic statements.

> ARISTOTLE'S AXIOM. All infinite sets are potential.
> CANTOR'S AXIOM. All sets are actual (since all finite sets are
actual by definition).


Let's be clear, are we arguing about what Cantor is supposed to have said or
meant, in which case it's merely a textual argument, belonging to the
history of maths?  Or are we talking set theory?  The words "potential" and
"actual" do not occur in set theory.  Leaving out all but the essential
details, we suppose there exists a set S such that

1.  0 in S
2.  (E S) (x) [ x in S --> x u {x} in S ]

and there is no contradiction.  You could "interpret" it as saying that the
series of x's, ie. 0, {0}=1, {0,1} = 2 is such that it will never terminate,
so that in that sense it is "incomplete" or amounts to a potential infinity.
But in the same breath you have to use the word "every", you have to make a
universally quantified statement.  You have stipulated something about
_every_ member of that series without exception.  And if that's OK, what's
wrong with stipulating the existence of an object - a set - such that
"every" member of the series is "in" that set?  We're not saying that S is
one of its members, so that it cannot be "completed".  We're saying S
contains all of its members.

All you have to buy is the idea of a set.  That is the only bit I have
difficulty with.


"A set having three members is a single thing wholly constituted by its
members but distinct from them.  After this, the theological doctrine of the
Trinity as "three in one" should be child's play."  (Max Black, _Caveats and
Critiques_)





Dean Buckner
London
ENGLAND

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