[FOM] Potential and Actual Infinity
Joseph Shipman
joeshipman at aol.com
Thu Mar 12 00:30:16 EDT 2015
This discussion seems to be making too much of a simple point. In Peano Arithmetic, which can be equivalently formalized by taking ZF and replacing the axiom of Infinity with its negation, there is no actual infinity. An actually infinite set gives a logically stronger system and we understand nowadays when this is necessary, and exactly how it is stronger (quantifiers are switched so instead of forall n thereexists X (X has at least n elements), we have thereexists X forall n (X has at least n elements).)
Potential infinity gets you quite far, you need actual infinity to get further (and to get to some important things, like the Robertson-Seymour Graph Minor Theorem, you even need uncountable infinities).
The potential/actual distinction is just a type of set/class distinction where we understand that we don't freely quantify over classes. PA respects this because it never requires the domain of quantification to be a completed object--this is a misconception fueled by the translation of the universal quantifier as "for all" rather than "for any",
-- JS
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