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

[Dean Buckner]

Alexander -

You've set up a very powerful straw man, in describing a process that could
never terminate, as though a set theorist ever would claim otherwise.  We
suppose there is a machine or process that generates states.  As follows

(a)  some states have already completed
(b) others haven't
(c) but each state will be completed at some time t.
(d) every state when it completes, will generates yet another state to be
completed
(e) thus there is no "terminating state", i.e. a state that generates no
further completeable states

Note c) is not to be confused with "there is some time t, at which every
state will have been completed"!  There is no such time, and in that sense,
the process is uncompletable

No set theorist would disagree with this.  But a set theorist would assert
there is or could be a set S, such that every state that has or will be
generated by the machine, is in the set.  Every state will be generated at
some point, and every state is in the set.  Even though there is no time t
at which every state in the set is complete, you can still have things that
will be completed within the set.

By "extra-logical" I mean "extra-set-theoretical".  Set theory consists of a
small number of axioms, terms and derivation rules.  To take on set theory
itself, you have to show where its logic breaks down.  There is no gross or
obvious point where it does.   To quote Harvey Friedman"Are you talking
about set theory or not?  And if not, what are you talking about"?  (A point
he originally argued against me).

My view is that set theory is wrong, in the sense it does not capture our
ordinary intuitions about sets and things.  But this requires a theory about
the syntax, semantics and fundamental assumptions underlying our "folk
conceptual scheme".

>From which I'm still far off.

Dean





Dean Buckner
London
ENGLAND

Work 020 7676 1750
Home 020 8788 4273