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

If we're talking about, say, Z set theory, any one of 'actually
infinite', 'potentially infinite', and 'unboundedly finite' would be
meaningful in the theory itself only if you either added it as a
primitive and gave additional axioms or defined it from what's already
primitive or previously defined.

I recently read a post in another forum in which the poster claims that
certain set theoretical proofs have "hidden" axioms. The post author
claims that set theory has a "hidden" axiom that infinite sets are
actually infinite. That is ludicrous and can only be said by someone
who has not a clue as to what an axiomatization is or who is hoping to
mislead anyone who would be misled.

Set theory does not have primitive predicate symbols for 'actual',
'potential', 'actually infinite' or 'potentially infinite'. So of
course set theory does not state any axioms that use those. To claim
that this implies that set theory has a "hidden" axiom is nonsense. Not
'nonsense' in the sense of something that a rational person would
disagree with, but rather 'nonsense' in the sense of truly not making
sense.

But one can make a theory that has primitives for 'actual' or
'potential'. Yet, those will have meaning only through axioms that make
use of them. If axioms are not provided, then the rubric is not
mathematically signficant and it is sophistry to pretend otherwise.

MoeBlee

.

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