[FOM] From theorems of infinity to axioms of infinity

josef at us.es josef at us.es
Wed Mar 13 12:52:54 EDT 2013


Sure, the question may be raised in that context, although the
"must" in your last sentence does not follow logically -- it has strong
philosophical presuppositions. 

My point was a different one: as you
acknowledge yourself, the natural numbers can be analyzed in set theory
without infinity, but the real numbers do require a set theory with

Moreover, I was pointing out that the notion of "all" real
numbers in an interval (e.g. all decimal expansions corresponding to the
unit interval) motivates the power set axiom. 

All the best, Jose F

El 13/03/2013 17:04, Martin Dowd escribió: 

> The existence of
infinite sets is in fact a question which arises with 
> regard to the
natural numbers. ZFC - infinity has the hereditarily finite 
> sets as a
model. It does not have a finite model, analogously to PA, or even 
> Q
(see ). The universe of discourse of arithmetic is infinite and contains
> subsets, whose properties are readily studied using PA. An
axiom must be added to 
> set theory, so that infinite sets are elemens
of the universe of discourse.
> - Martin Dowd
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