Re: Mathematical clarification: Impredicativity in Rosennean parlance

[Calvin Ostrum <***>]

> From: Dr. Aloisius H. Louie 

> PA = Peano axioms (or Peano arithmetic)
> ZF = Zermelo-Franco set theory

Zermelo-Fraenkel set theory, actually

> PA is a formal basis of number theory, and ZFC is a formal
> basis of (naive) set theory.  There is no need to go further
> than this.

What is "naive" about the set theory formalized by ZFC?
What is an example of a set theory that is not naive in
comparison to ZFC, and why is it not naive?

>  We only need to know they are examples of purely
> syntactic formal systems.
 
What are some precisely formulated examples of formal
systems that are not "purely syntactic"?   

> Mathematically, to say a property is "generic"
> means it is the dominant characteristic. 

It doesn't seem to me that the term "the dominant 
characteristic" is any clearer than the term, "generic 
property", that it is meant to explain here.  (Moreover,
the presence of the definite article in one term, but
not the other, seems strange).

> For example,
> in the set of real numbers, irrational numbers are
> generic.  That means if an arbitrary real number is chosen,
> the overwhelming odd (in a precise mathematical sense) is
> that it is irrational. 

Of course, although the "precise mathematical sense" attaches 
not just to the odds, but to the very concept of "choosing an 
arbitrary real number".   In this precise sense, the chance of 
choosing a rational number is zero because the rationals can be 
covered by a set of intervals whose total lengths sum to less than 
epsilon, for any given positive real epsilon. 

But how does this apply to the "mathematics" that is not
"captured" by a formal system?  What are the actual objects?
What is the space? How is a measure supplied for the space?

> This is because we simply cannot practically handle non-terminating
> decimals.

Sure we can.  It is easy to deal with fractions, and we also perform
lots of exact computation with all sorts of irrational numbers like
pi and e all the time. 

>  Neither can computers, for that matter. 

Maple and Mathematica, among others. 

> The fact that a great deal of useful mathematics has been done
> syntactically is precisely the point discussed in the Praeludium
> of LI.

What useful mathematics cannot be done "purely syntactically".
If we look at the mathematics regularly used in physics, chemistry,
all the statistics of the social and behavioral sciences, can we
come up with a list of very much that cannot be quite easily in
principle embedded into ZFC?

> are not useful.  He simply said that we need the "generic parts"
> to do biology (indeed real science).

Why?  For any "real science"?  What about modern physics, 
chemistry, and social and behavioral sciences?  They seem to
get on quite well with mathematics that can be "captured" in 
ZFC.