[FOM] a definable nonstandard model of the reals
Philip Ehrlich
ehrlich at oak.cats.ohiou.edu
Tue Nov 25 14:04:50 EST 2003
In response to a comment of mine to John Baldwin, Dave Marker wrote:
Philip Ehrlich wrote:
>Since Conway's ordered field No is a nonstandard model of the reals,
>doesn't the following characterization of No I published back in
>1988* provide a positive answer to your recent question about a
>definable nonstandard model of the reals?
>No is (up to isomorphism) the unique real-closed field that is an
>eta_On ordering, i.e., No is (up to isomorphism) the unique
>real-closed field such that for all subsets L and R of the field
>where every member of L precedes every member of R , there is an
>element y of the field lying strictly between those of L and those of
>R.
>Of course, No is a proper class, not a set. Is that a problem for you?
This gives some kind of cannonical real closed field, but, as I understand
it, the problem is to look for a nonstandard model of the reals where we
have predicates for all subsets of R^n.
If we are only looking at the ordered field structure, there are many
natural nonstandard models. For example, the real closure of R(t) where
t is a positive infinite element. This could also be described
algebraically as the the field of algebraic Puiseux series (or we
could look at the larger models of formal or convergent Puiseux series).
While there is a notion of "integer" in these models, it is very poorly
behaved (unless you want the square root of 2 to be rational).
Dave Marker
Dear Dave,
I went back and re-read John's question, and I think I actually did
answer it. [If not, my apologies to all]. On the other hand, there is
another interesting question which I take it you are raising (at
least implicitly); namely, can one do non-standard analysis in No
(suitably expanded)? If I am not mistaken, John attended a talk I
gave this summer in Helsinki in which I pointed out as part of a more
general theme regarding No that the answer is "yes"! However, if I
understand your remarks correctly, you seem to suggest the answer
should be "no" because No has the wrong sort of "integers". I will
consider these issues in turn.
To begin with, the fact that No can indeed be employed as the basis
of a non-standard approach to analysis is a consequence of some
lovely classical work of Keisler together with the characterization
of No I stated in my response to John, namely:
No is (up to isomorphism) as the unique real-closed field that is an
eta_On ordering of power On.
Let me explain.
In his monograph "Foundations of Infinitesimal Calculus [1977]"
Keisler provided the following Axioms for Hyperreal Number Systems
where R is the ordered field of reals.
Axiom A. R is a complete ordered field.
Axiom B. R* is a proper ordered field extension of R.
Axiom C. (Function Axiom). For each function f of n variables there
is a corresponding hyperreal function f* of n variables, called the
natural extension of f. The field operations of R* are the natural
extensions of the field operations of R.
Axiom D. (Solution Axiom). If two systems of formulas have exactly
the same real solutions, they have exactly the same hyperreal
solutions.
Following his introduction of Axioms A-D, Keisler makes the following
intriguing observation.
"The real numbers are the unique complete ordered field. By analogy,
we would like to uniquely characterize the hyperreal structure (R,
R*, *) by some sort of completeness property. However, we run into a
set-theoretic difficulty; there are structures R* of arbitrary large
cardinal number which satisfy Axioms A-D, so there cannot be a
largest one. Two ways around this difficulty are to make R* a proper
class rather than a set, or to put a restriction on the cardinal
number of R*. We use the second method because it is simpler."
Motivated by the above, Keisler goes on to introduce the following
Saturation Axiom and the subsequent theorem:
Axiom E. Let S be a set of equations and inequalities involving real
functions, hyperreal constants, and variables, such that S has a
smaller cardinality than R*. If every finite subset of S has a
hyperreal solution, then S has a hyperreal solution.
Keisler's Theorem: There is (up to isomorphism) a unique structure
(R, R*, *) such that Axioms A-E are satisfied and the cardinality of
R* is the first uncountable inaccessible cardinal.
In the above theorem, R* is, of course, (up to isomorphism) the
unique real-closed field that is an eta_alpha ordering of power
aleph_alpha where aleph_alpha is the first uncountable inaccessible
cardinal. The real-closed nature of the field follows from Axioms A-D
and the eta_alpha property follows from the Saturation Axiom.
To prove the theorem Keisler makes use of a limit ultrapower together
with a superstructure embedding.
Like Keisler -- though he never says as much -- I was well aware that
to establish a proper class analog of his just-stated Theorem -- a
possibility Keisler already hinted at in above quoted remark -- R*
would have to be a real-closed field that is an eta_On ordering of
power On, i.e., R* would have to be isomorphic to No. I was also
aware that while Keisler's proof could not be extended for proper
classes in NBG (with Global Choice) it could be carried out, for
example, in Ackermann's Set (with Global Choice), a conservative
extension of NBG (with Global Choice) which permits the construction
of the power class of a proper class. That is, within Ackermann's
Set (with Global Choice) one can prove:
Theorem: There is (up to isomorphism) a unique structure (R, R*, *)
such that Axioms A-E are satisfied and the cardinality of R* is the
power of On. Moreover, R* is isomorphic to No.
Some time ago, against this backdrop, I wrote to Keisler to ask if
this is what he had in mind. He informed me it was not, and outlined
a simple and clever alternative to the superstructure approach for
constructing such a model (R, R*, *) that is carried out in NBG (with
Global Choice) supplemented with an extra symbol for a given
well-ordering of the set-theoretic universe V. Since the construction
is Keisler's rather than mine, I hope you will understand why I do
not feel at liberty to share it with you. On the other hand, Keisler
has expressed an interest in updating his beautiful paper "The
Hyperreal Line" for a second edition of my "Real Numbers,
Generalizations of the Reals, and Theories of Continua, Kluwer
Academic Publishers, 1994"and I am planning to ask him to include the
outline therein.
Let me now turn to your argument that seems to suggest that
non-standard analysis could not be based upon No since No's integers
are inappropriate to serve as a non-standard model of Arithmetic.
Unless I am mistaken, you are referring to No's "omnific integers".
Assuming this to be the case, you are certainly correct in
maintaining that they do not constitute a non-standard model of
Arithmetic (as Conway himself notes [On Numbers and Games, p. 44]).
What I believe is misleading in your remark, however, is the
implication that since No's omnific integers do constitute a suitable
non-standard model of Arithmetic, No contains no such model. As my
previous remarks suggest, in this regard, you seem to be mistaken. On
the other hand, whether or not one could find a simple canonical
characterization of such a subring I do not know, I haven't given it
any thought.
Of course, Dave, I might have completely misconstrued your remarks
and your intentions. If so, I do apologize. In any case, have a great
Thanksgiving!
Phil
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