[FOM] Potential and Actual Infinity
paul at mtnmath.com
Tue Mar 17 13:42:59 EDT 2015
Potential infinity is a philosophical concept which is usually defined
in a way that requires time as a fundamental assumption. For example the
Wikipedia entry on Actual Infinity says "This is contrasted with
*potential infinity*, in which a non-terminating process (such as "add 1
to the previous number") produces an unending "infinite" sequence of
results, but each individual result is finite and is achieved in a
finite number of steps." Or consider Aristotle's definition from the
same article "/Potential infinity/ is something that is never complete:
more and more elements can be always added, but never infinitely many."
Quantifying over the integers and a recursive relations (PI_1) is
equivalent to asking if an easily defined Turing Machine (TM) has an
unbounded number of outputs. Generalizations of this question can be
defined for Pi_n. With the aid of recursive ordinal notations it can be
For more about this approach see "Objective mathematics in a finite
unbounded universe" (www.mtnmath.com/math/objMath.pdf) and "Generalizing
Kleene’s O ..." (www.mtnmath.com/ord/kleeneo.pdf).
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