[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

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 
 your last sentence does not follow logically -- it has strong 
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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