[FOM] 619: Nonstandardism 1

[Harvey Friedman]

1. REMARKS ON NONSTANDARDISM.

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.

1. An alternative setup by Nelson called IST, where the conceptual
change is that we have the usual set theoretic universe, but add a
predicate for "standard". In a way, this is recasting the original
presentation of the calculus in formal axiomatic terms, using
"standard" as a primitive notion. There is no attempt to get rid of or
explain this notion of "standard". Instead, this is an attempt to
reintroduce an old rejected primitive into mathematics. 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. 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.

2. In connection with 1 and other developments, there has emerged a
subject which I call NONSTANDARDISM, which is a kind of Philosophy of
nonstandard mathematics. In particular, Philosophy of  infinitesimals.
This includes, above all, claims to the effect that these have a
legitimate mathematical status all on their own, independently of
ultra power constructions and the like, and should be added to the
foundations of mathematics as a primitive, sort of like epsilon being
a primitive, or natural numbers being a primitive, or the like. This
also includes claims of superiority in one form or another over the
usual epsilon/delta treatments. Even claims that the abandonment of
infinitesimals in favor of epsilon/delta was an unfortunate step
backward, a mistake that even has damaged the proper development of
mathematics.

3. There has also been put forward the idea that teaching calculus via
infinitesimals is superior in many ways to teaching calculus with
epsilon/delta.

4. There has also emerged applications to ordinary mathematics, where
the nonstandard framework, sitting safely inside the ultra power
construction, simplifies the mathematics enough from the point of view
of some people. Elimination of this is not an issue. The only issue is
just how much it helps, and perhaps, just how much sharper the results
become after the use of nonstandard arguments is removed. I generally
believe that in all cases, when heavy machinery, particularly like
this, is used, one learns a lot by removing it, and one gets sharper
information. Of course, there is one exception to this, and it is
extreme - http://www.cs.nyu.edu/pipermail/fom/2015-September/019128.html
That is the real thing.

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

I plan to take up 1-5 in detail in this series among my numbered postings.

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.

2. COMPATIBLE NONSTANDARDISM

The language we use is epsilon,=, and a unary predicate ST for "being standard".

Under each number, we have the option of choosing any one of the
items, AND the option of choosing NONE of the items. As usual, the
literals are the atomic formulas and their negations.

1. ZFC.

2. Some standard set has a nonstandard element.

3. For every given set of standard sets, there is a standard set whose
standard elements are the elements of the given set.

4. The standard sets form an elementary substructure of the universe,
under epsilon. (scheme).

4'. Every {x in A: (therexists y_1,...,y_k in A)(phi)} is standard,
provided A is standard and phi is a conjunction of literals with at
most the variables x,y_1,...,y_k and standard parameters, not
mentioning ST.  (scheme).

5. The set of all standard sets exists.

Note that 5 brings the notion of standard into line with the normal
set theoretic universe.

Obviously 4 is a lot stronger than 4'. However, we know that 4'
implies the weakening of 4 for delta-0 formulas (bounded quantifiers
allowed).

THEOREM. ZFC + "there exists an extendible cardinal" proves the
existence of a rank satisfying 1-5 above. 1-3,4',5 proves the
existence of a rank satisfying ZFC + "there exists a 1-extendible
cardinal".

In 4', we can require k and the number of parameters in phi to be
fixed small integers. Then the Theorem still holds. How small? I think
something like 3 or 4 quantifiers and 2 parameters, although I haven't
checked this carefully.

**********************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 619th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-599 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2015-August/018887.html

600: Removing Deep Pathology 1  8/15/15  10:37PM
601: Finite Emulation Theory 1/perfect?  8/22/15  1:17AM
602: Removing Deep Pathology 2  8/23/15  6:35PM
603: Removing Deep Pathology 3  8/25/15  10:24AM
604: Finite Emulation Theory 2  8/26/15  2:54PM
605: Integer and Real Functions  8/27/15  1:50PM
606: Simple Theory of Types  8/29/15  6:30PM
607: Hindman's Theorem  8/30/15  3:58PM
608: Integer and Real Functions 2  9/1/15  6:40AM
609. Finite Continuation Theory 17  9/315  1:17PM
610: Function Continuation Theory 1  9/4/15  3:40PM
611: Function Emulation/Continuation Theory 2  9/8/15  12:58AM
612: Binary Operation Emulation and Continuation 1  9/7/15  4:35PM
613: Optimal Function Theory 1  9/13/15  11:30AM
614: Adventures in Formalization 1  9/14/15  1:43PM
615: Adventures in Formalization 2  9/14/15  1:44PM
616: Adventures in Formalization 3  9/14/15  1:45PM
617: Removing Connectives 1  9/115/15  7:47AM
618: Adventures in Formalization 4  9/15/15  3:07PM

Harvey Friedman