On Jul 15, 4:19 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 15 Jul., 22:27, Virgil <vir...@ligriv.com> wrote: > > > In article > > <afe01c41-f3f8-4ff7-8898-da7ebb952...@g2g2000vbl.googlegroups.com>, > > > > Potential infinity: ? n ? m : m > n > > > Actual infinity: ? m ? n : m > n > > > Potential infinity: There cannot be any set which is a superset of all > > finite initial segments of the naturals. > > > Actual infinity:There is a set which is a superset of all > > finite initial segments of the naturals, the union of them > > Completely correct!
Those are principles, not definitions.
They don't define 'x is a potentially infinite set' nor 'x is an actually infinite set'.
(1) "There is a set which is a superset of all finite initial segments of the naturals, the union of them"
ExAy(y is a finite set of natural numbers -> y is a subset of x)
In Z set theory, that's just equivalent to the axiom of infinity.
(2) "There cannot be any set which is a superset of all finite initial segments of the naturals."
~ExAy(y is a finite set of natural numbers -> y is a subset of x)
That's just equivalent to the negation of the axiom of infinity.
So the "potential" vs. "actual" offered here is nothing more than
the axiom of infinity vs. the negation of the axiom of infinity.
No light has been shed on what it means to say that a set is "potentially infinite" but not "actually infinite".
