On Hardy fields and models of  |
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| April 2022 |
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I write 
for the language of ordered exponential
rings, for the the real ordered exponential
field, and 
for its elementary theory. Recall
that is model complete in 
by Wilkie's theorem, and that is o-minimal.
I start with a few preliminary results that I actually didn't know were true in general.
Lemma 
be an infinite cardinal and let 
be a first-order language. An -structure
is -saturated
if and only if for all subsets 
with 
, all 
-types
over 
are satisfiable in
.
Proposition be an infinite cardinal. Let 
be an
-minimal structure with such
that 
is a dense linear order without endpoints.
Assume that for all subsets 
with cardinality
and with 
,
there is an 
with 
.
Then is -saturated.
Corollary be an infinite uncountable cardinal. The ordered
exponential field 
of surreal numbers of length /
birth day 
with Gonshor's exponential function is
-saturated.
Proof. The underlying ordered set is -saturated by definition of surreal numbers and
by a simple fact (found in [1, Chapter 1]): if 
and 
are sets of surreal numbers
with , then the simplest
number 
with
has birthday 
.
We also know, by a result of Ehrlich and van den Dries [2],
that 
and that 
can be
expanded into a model of 
.
Thus 
is o-minimal.
Corollary elementarily embeds into a for large enough .
I was asking myself the following questions:
Question 
be a Hardy field containing 
. Assume that is real-closed
and closed under 
and 
. Is 
an elementary expansion
of ?
Question be a Hardy field containing
and closed under . Does embed into an elementary extension of ?
Lou found answers to those questions, which I next explain.
In order to answer the first question, in the negative, we require three objects:
We write 
for Hardy's field of
logarithmico-exponential functions, i.e. germs at 
that can be obtained as compositions of 
, 
,
and semialgebraic functions 
.
In other words, this is the closure of the field 
of rational functions under real closure, and
, which Hardy showed to be a
Hardy field.
Let 
denote the Hardy field of germs at of -definable
functions 
, allowing
parameters. Note that given a positive infinite germ 
, its functional inverse 
is also definable with parameters in ,
so 
.
Let 
be non-standard, i.e. a proper elementary
extension of . Fix an 
with 
.
This exists since 
is the largest archimedean
ordered field.
Proposition 
which commutes with
-definable functions 
/ 
with parameters in and sends the germ 
of the
identity function onto 
.
Proof. Cheking all details is a bit tedious but I think
this is a well-known result. I just give the definition. Fix a
representative 
of a germ in . There is a defining formula 
for with parameters 
and
a real number 
such that for all 
, the number 
is unique
with 
. Let 
be a second defining formula with parameters 
satisfying the same relation, with respect to the same germ, for a
possibly distinct 
. In
particular, we have
 |
(1) |
Recall that is o-minimal, so 
is an elementary embedding. We have 
,
so by (1), the unique element 
of
with 
is also the unique
element of satisfying 
. We define 
to be that
element .
Similar arguments show that 
commutes with -definable functions , whence in particular that it is an embedding
in , hence an elementary
embedding by model completeness.
Proposition -embedding of into .
Proof. Assume for contradiction that there is such an
embedding 
and write 
. The field is contained in
, and since
must commute with semialgebraic functions as well as with and , we have
. For the same reasons, we
have 
. The function is injective, so 
must coincide
with , whence in 
. But this is known to be false: for instance
it is a theorem of van den Dries, Macintyre and Marker that the germ
of the functional inverse of 
does not lie in .
This raises a question:
Question ? Is there an H-closed Hardy field which is a model
of ?
The answer to the second question is positive. In order to -embed 
into an
elementary extension of , it
is enough, since 
and by model completeness, to
construe it as a substructure of a model of 
.
Proposition be a Hardy field containing
and closed under . Then embeds into an elementary extension of .
Proof. We need to prove that is
a model of the universal theory 
of . Consider a universal formula 
where 
is a boolean combination of atomic -formulas, hence, up to equivalence
modulo the theory of rings, of exponential-polynomial equations,
inequations, inequalities... We can assume that
is in disjunctive conjunctive form
where each 
, each symbol 
is among 
,
and 
, and 
are finite sets.
Assume that 
is valid in
and let 
be representatives of germs in . Since 
is
finite, there are a cofinal subset 
and an 
for which we have
whenever 
. Since is a Hardy field, the sign of each function
for 
is stationnary. So we actually have 
for all sufficiently large 
. This means that 
,
whence in particular that 
.
Therefore is valid in . This shows that 
embeds
into a model of .
Corollary closed under embeds into
as an ordered exponential field.
Proof. We first embed into a
Hardy field 
which is closed under . This is just done by closing 
under as explained in Lou's lecture. Then embed
into a model of using
Proposition 4, and conclude with Corollary 2.