# [FOM] From theorems of infinity to axioms of infinity

MartDowd at aol.com MartDowd at aol.com
Thu Mar 14 13:01:58 EDT 2013

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Dr. Ferreiros,

Regarding point 1, according to previous postings infinity was added by
Zermelo because it was one axiom which conveniently dealt with infinite sets
within the framework of the other axioms.  The founders of set theory
realized the "need" for such an axiom, to make set theory powerful enough to
formalize mathematics.

Regarding point 2, I'm not an expert, but there is a thriving body of work
on how much of classical mathematics can be formalized in PA, and also
subsystems of second order arithmetic,  This is a specialized topic,  though,
because infinity and power set give the cumulative hierarchy, which  allows
formalizing the real numbers without any of the "tricks" needed to  formalize
in weaker systems.

Regarding point 3, the existence of the power set of omega, and its use in
constructing the real numbers, is a fundamental example of the use of set
theory, which increases our confidence in it.  There are other basic
examples of its use in lower levels of the cumulative hierarchy, such as  function
spaces, which I believe were a motivation for Cantor.

In sum, set theory is an example of the maxim, "if it works don't fix  it".

In a message dated 3/13/2013 10:46:31 A.M. Pacific Daylight Time,
josef at us.es writes:

1. Sure, the question may be raised in that context, although the "must" in
philosophical  presuppositions.
2. 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 infinity.
3. 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.
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