[FOM] Potential and Actual Infinity

Paul Budnik 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 
generalized further.

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).

Paul Budnik

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