[FOM] 619: Nonstandardism 1

[Sam Sanders]

Dear Harvey,

> Nonstandard analysis, in its modern sense, is normally credited to A.
> Robinson's ultrapower construction, in a successful effort to give a
> rigorous foundation for the original presentation of the calculus.
> Since then, several main things have emerged.

So Nelson claimed that his Internal Set theory is just an alternative approach to Robinson’s NSA, 
i.e. IST is not fundamentally different from the ultrafilter construction.  I tend to 
agree with this claim, especially as models from Robinsonian NSA provide models of IST.    

The only crucial difference between the Robinson and Nelson approach is indeed
the concept of “external sets” which we should discuss in more detail. 

Technically, the collection { x : st(x) AND A(x)} cannot be formed in IST as the defining formula “st(x) AND A(x)” is external (involves ’st’).  
In Robinsonian NSA, this collection does make sense, giving rise to two different views, as you describe rather nicely as follows:

>  Prima facie
> metamathematically, (I am assuming here that we are talking about the
> usual classical logic framework), there is no difference between
> thinking of the standard objects as the real, ground model of ordinary
> reality, and the wider nonstandard objects as a primitive or
> constructed enlargement, or having the standard objects be part of
> ordinary reality

Exactly.  This should be compared to (my interpretation of) Nelson’s view, to be found in his original paper introducing IST:

*Every specific object of conventional mathematics is a standard set. *  

Intuitively speaking, the notion of “objects of conventional math” (versus "purely logical objects") is vague, ever-changing, informal, cannot be formed, etc.
The standard sets are meant to describe conventional math, and for this reason (and many others), we cannot define external sets in IST.  
In other words, “standard" is meant to reflect “conventional math” and since we cannot provide a sharp distinction between “conventional math” and “purely logical statement”, 
we cannot use comprehension on formulas mentioning “standard”.  

As you suggested previously, this interpretation can be seen to reflect Nelson’s finitist views.  

> . HOWeVER, the internal approach, as normally
> practiced, closes off the possibility that the standard part is a
> legitimate object of the larger, real universe (violation of
> induction), whereas there is nothing to prevent an outer construction
> on top of reality. THIS ISSUE of the (IN)COMPATIBILITY OF TWO
> UNIVERSES will be the main thrust of this POSTING.

One should not exaggerate the incompatibility:  Most arguments in either flavour of NSA do 
not require external sets *in an essential way*.   An exception is the 
Loeb measure (which does use external sets), as I pointed out previously.

A good analogy is perhaps “complex vs real number” and “external vs internal sets”:    

The complex numbers simplify real analysis greatly in many places, but one can 
also do parts of real analysis while avoiding talking about complex numbers.  

Similarly, the external sets (perhaps) make NSA more palatable, but one can do 
NSA while avoiding talking about external sets, i.e. in the internal framework.   

> 5. In addition, there has emerged a kind of constructive nonstandard
> analysis, where the underlying logic is intuitionistic. This seems to
> necessarily close off the A. Robinson style direct construction
> approach,

So it is possible to build a nonstandard model (using filters weaker than a free ultrafilter)
inside Martin-Loef Type Theory (See the work of Erik Palmgren).  

> and delve directly into either Constructive Nonstandardism,
> or a web of proof theoretic transfer and conservative extension
> issues.

Well, there are many systems with various conservation results, 
but a few major ones tend to stick out (like the Big Five in RM).  

> In this posting, I want to focus on one specific thing. I think that
> this may the reason why Goedel was so enthusiastic about the potential
> power of nonstandard analysis, suggesting that it may allow us to do
> new things. He certainly(?) did not anticipate this development below,
> in the normal sense of the word, but I think he kind of vaguely
> intuited it, somehow sensed it.

I have to think about this before answering.

Best,

Sam