PHILIP EHRLICH
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES1
An Essay in the Foundations of Euclidean Geometry
Hilbert’s Axiom of (Arithmetic) Completeness first appeared in his classic
investigation Über den Zahlbegriff (Hilbert 1900a, 183) as a novel means
of distinguishing the ordered field R of real numbers (and continuous
ordered fields more generally) from the remaining Archimedean ordered
fields. Soon thereafter, the axiom was incorporated into the second German
edition of Hilbert’s Grundlagen der Geometrie (1903, 16) along with the
corresponding Axiom of (Geometric) Completeness that had already been
added to the French (Hilbert 1900b, 25) and English (Hilbert 1902, 25)
editions to distinguish “ordinary analytic geometry” from the remaining
models of Archimedean Euclidean geometry. The possibility of providing categorical characterizations of R and continuous Euclidean geometry
by means of Hilbert’s completeness axioms rests upon the fact that with
the exception of isomorphic copies thereof the remaining Archimedean
ordered fields and Archimedean Euclidean geometries admit proper extensions to structures of their respective kinds. That the latter structures are
not merely so extensible, but extensible to isomorphic copies of R and
“ordinary analytic geometry” respectively, was already known to Hilbert
when he wrote the Grundlagen, explicitly stated therein for the geometrical case (Hilbert 1899, 39; 1902, 55–56; 1903, 38–39; 1971, 58–59), and
propounded for the arithmetic case in a related work (Hilbert 1904a, 185;
1904b, 138) that was subsequently incorporated into the Grundlagen itself
(Hilbert 1909, Anhang VII, p. 279; 1930, Anhang VII, p. 261).
In a recent investigation (Ehrlich 1995), the author provided a historically sensitive overview of the completeness and embedding properties of the
Archimedean complete ordered number systems introduced by Hans Hahn
in his great pioneering work Über die nichtarchimedischen Grössensysteme
(1907). We noted that, just as Hilbert’s completeness and embedding properties make R a particularly revealing forum for the comparative study of
Archimedean ordered fields, the corresponding completeness and embedding properties of Hahn’s ordered number fields provide an analogous
forum for the comparative analysis of ordered fields more generally. In
the pages that follow, we will further develop the analogy by showing that
Synthese 110: 57–76, 1997.
c 1997 Kluwer Academic Publishers. Printed in the Netherlands.
58
PHILIP EHRLICH
just as Hilbert was able to transfer certain aspects of his completeness and
embedding properties for R to obtain important insights into the structure and variety of Archimedean Euclidean geometries, the corresponding
completeness and embedding properties for Hahn’s ordered number fields
can be similarly exploited to obtain a rich understanding of the spectrum
of models of Euclidean geometry more generally.2
To keep the paper relatively self-contained, a good portion of the discussion will be expository in nature. In Section 1, we will set the algebraic
stage with a brief discussion of Hahn’s number systems and the embedding and completeness theory inspired by them, leaving it to the reader
to consult (Ehrlich 1995) and the literature cited therein for more detailed
discussions of the theory. Following this, in Section 2, we will begin
with some remarks about classical Cartesian geometry and the models of
elementary Euclidean geometry more generally. Since Tarski’s system P
(Schwabhäuser, Szmielew, and Tarski 1983) provides a particularly simple
and elegant setting for a discussion of these models, we will employ his
system in place of the more complicated equivalent framework of Hilbert.
Against this backdrop, geometrical counterparts for the central algebraic
notions from Section 1 will be introduced and classical Cartesian geometry
will be exhibited as the simplest special case of an entire proper class of
Archimedean complete models of P whose embedding and completeness
properties generalize the familiar analogs for Cartesian geometry. As an
illustration of this, we will show that Hilbert’s classical continuity conditions can be replaced by a more general pair of conditions that lead to
categorical characterizations of the various Archimedean complete models
of P . We will also find that Bernays’ Axiom of Line Completeness – the
continuity condition which has replaced Hilbert’s completeness axiom in
the later editions of the Grundlagen der Geometrie – also admits a natural Archimedean complete generalization that leads to a generalization of
the categoricity theorem familiar to readers of Hilbert’s geometry today.
Following this, we will isolate the Archimedean complete models of the
two extensions of P that are prominent in many contemporary discussions of the foundations of Euclidean geometry, namely, the historically
rooted geometry of elementary constructions and Tarski’s system of complete elementary geometry. Finally, since many authors prefer to employ
Dedekind’s continuity condition in place of the continuity axioms due to
Hilbert and Bernays, in Section 3 we will introduce a natural geometrical analog of the author’s recent Archimedean complete generalization of
Dedekind’s continuity condition for ordered fields (Ehrlich forthcoming)
and show how it can play a similar role in characterizing the Archimedean
complete models of P .
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
59
1. ARCHIMEDEAN COMPLETE ORDERED FIELDS
Let G be an ordered abelian group written additively. The absolute value
of a 2 G, written jaj, is defined as the greatest member of fa; ag. If a
and b are members of G f0g, then a is said to be Archimedean equivalent
to b if there are positive integers m and n such that mjbj > jaj and
njaj > jbj. Archimedean equivalence partitions the elements of G f0g
into disjoint classes called Archimedean classes. The term “Archimedean
class” is intended to indicate that within a given class the Archimedean
condition holds. If a, b 2 G are not Archimedean equivalent, then a
is said to be infinitesimal (in absolute value) relative to b if jaj < jbj.
In accordance with these conventions, 0, which is not a member of any
Archimedean class, is infinitesimal (in absolute value) relative to every
other member of G:3 Following tradition, henceforth we will denote the
Archimedean class containing a 2 G f0g by [a].
DEFINITION 1.1. Let G and G0 be ordered abelian groups where G G0 .
G0 is said to be an Archimedean extension of G if and only if for each
y 2 G0 f0g there is an x 2 G f0g that is Archimedean equivalent to
y; on the other hand, if G admits no proper Archimedean extension, G is
said to be Archimedean complete; furthermore, if an ordered field F has
an Archimedean complete ordered additive group, then F is said to be an
Archimedean complete ordered field.
Since the idea of an Archimedean complete ordered field does not
presuppose that the field is Archimedean, it leaves open the possibility that
there are non-Archimedean, Archimedean complete ordered fields. Hahn
not only discovered that such structures exist, he introduced the following
celebrated construction which isolates them up to isomorphism.
HF0 . Let R be the ordered field of real numbers, G be an ordered abelian
group and R(G) be the set of all formal series of the form
X
r!y
<
where is an ordinal, fy : < g is a descending sequence of elements
of G and r 2 R f0g for each < . The unique such series for which
= 0 (i.e., the empty series) is the 0 of the Hahn field (with exponents
in G) that arises by ordering the elements of R(G) lexicographically and
defining addition and multiplication according to the rules
X
X
X
ay !y + by !y = (ay + by )!y
y2G
y2G
y2G
60
PHILIP EHRLICH
2
3
X
X
X6 X
7
ay !y by !y = 64
a b 75 !y
y2G
y2G
y2G (; )2GG
+ =y
where terms with zeros for coefficients are inserted and deleted as needed.
The reader will notice that R(G) is isomorphic to R when and only
when G = f0g. To formulate the completeness and embedding theorems
for R(G) that sharpen and generalize the familiar results for R we require
the following theorem which has its origins in the work of Baer (1927) and
Krull (1932):
F1 . The class AF = f[a]:a 2 F f0gg of Archimedean classes of an
ordered field F constitutes an ordered abelian group when order and multiplication are defined for all [a], [b] 2 AF by the conditions: [a] [b] if
jaj is infinitesimal relative to jbj and [a][b] = [ab].
In virtue of F1 , we will say that F is an ordered field of Archimedean
type G if the ordered abelian group of Archimedean classes of F is isomorphic to some ordered abelian group G. When this terminology is adopted,
the Archimedean complete generalizations of the embedding and completeness theorems for R assume the following forms:
HF1 : EMBEDDING THEOREM. If F is an ordered field of Archimedean
type G, then R(G) is an Archimedean extension of an isomorphic copy of
F.
HF2 : COMPLETENESS THEOREM. R(G) is up to isomorphism the
unique Archimedean complete ordered field of Archimedean type G:4
2. ARCHIMEDEAN COMPLETE MODELS OF P
Hilbert’s axiomatization of Euclidean geometry is formulated in terms of
eight primitive notions and a number of defined terms. As primitive, he
takes the individual notions of point, line, and plane, an incidence relation
between points and lines, another such relation between points and planes,
a ternary betweenness relation on points, and a pair of congruence relations,
one for angles and another for segments where an angle is defined as a
pair of half-rays in a plane that emanate from a common point and a
segment is defined as a set of distinct points A and B on a line which
may be denoted by AB or BA. As a result, in Hilbert’s theory, models
of Euclidean geometry emerge as rather intricate three-sorted structures
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
61
in which, for example, one must distinguish between a line and the set of
points on a line, and a plane and the set of points on a plane. For these and
other reasons, Hilbert’s system, which played such an important historical
role in exposing many of the fundamental relations between number and
Euclidean magnitude, does not provide the most convenient forum for a
precise discussion of them.5
Tarski’s system P , on the other hand, provides an especially convenient
starting point for a discussion of these issues. The descriptive simplicity of
its models emerges from the fact that only points are treated as individuals
and the only predicates employed in the axioms are a ternary predicate B
(where ‘Bxyz ’ is read y lies between x and z [the case when y coincides
with x or z not being excluded]) and a quaternary predicate (where ‘xy zu’ is read x is as distant from y as z is from u). Over the years, Tarski’s
system of axioms for P has undergone substantial evolution, culminating
in the following axiomatization from (Schwabhäuser, Szmielew, and Tarski
1983):6;7
A1 . Reflexivity Axiom For Equidistance.
ab = ba.
A2 . Transitivity Axiom For Equidistance.
ab pq ^ ab rs ! pq rs.
A3 . Identity Axiom For Equidistance.
ab cc ! a = b.
A4 . Axiom of Segment Construction.
9x(Bqax ^ ax bc).
A5 . Five-Segment Axiom.
a 6= b ^ Babc ^ Ba0b0 c0 ^ ab a0 b0 ^ bc = b0 c0 ^ ad a0 d0 ^ bd b0 d0 ! cd c0 d0.
A6 . Identity Axiom For Betweenness.
Baba ! a = b:
A7 . Pasch’s Axiom.
Bapc ^ Bbqc ! 9x(Bpxb ^ Bqxa):
A8 . Lower Dimension Axiom.
9a9b9c(:Babc ^ :Bbca ^ :Bcab):
A9 . Upper Dimension Axiom.
p 6= q ^ ap aq ^ bp bq ^ cp cq ! Babc _ Bbca _ Bcab:
A10. Euclid’s Parallel Axiom.
Badt ^ Bbdc ^ a 6= d ! 9x9y(Babx ^ Bacy ^ Bxty):
Since P is known to be equivalent to Hilbert’s system of axioms less the
Axioms of Completeness and of Archimedes’, to obtain a Tarskian analog
62
PHILIP EHRLICH
of Hilbert’s system, we need only supplement P with the following two
assertions.
A. ARCHIMEDEAN AXIOM. Whenever Ba0a1c, there is a finite set of
points a0 , a1 ; : : : ; an such that a0 a1 = ai ai+1 and Bai 1 ai ai+1 for all
1 i < n, and Ba1 can .
C . AXIOM OF COMPLETENESS. The structure consisting of the col-
lection of points together with the betweenness and equidistance relations
defined on it admits no proper extension to a model of P [ fAg.8
The conception that bridges the gap between the domains of number
and Euclidean magnitude is, of course, the classical notion of a Cartesian
space over an ordered field F , henceforth C2 fF g. In Tarski’s framework,
this familiar concept and the categoricity theorem for P [ fA; C g which
may be formulated in terms of it assume the following simple forms.
DEFINITION 2.1. A Cartesian space over an ordered field hF; +; ; i is
a structure hAF ; BF ; F i where the relations BF and F are defined on
AF = f(x1 , x2 ) : x1, x2 2 F g by the stipulations:
BF xyz if and only if there is a 2 F for which 0 1 and
y x = (z x ) for = 1, 2;
xy F uv if and only if
2
X
x y )2 =
(
=1
2
X
=1
u v )2 :
(
CATEGORICITY OF P [fA; C g. C2 fRg is up to isomorphism the unique
model of P [ fA; C g.
As we mentioned in the Introduction, in addition to generalizing the
completeness theorems for C2 fRg, we will be concerned with extending
the
EMBEDDING THEOREM FOR C2 fRg. C2 fRg is a model of P [ fAg
that is an extension of an isomorphic copy of every model of P [fAg. Any
other model of P [ fAg having this property is isomorphic to C2 fRg:9
To obtain Archimedean complete generalizations of these important
results we require counterparts for the concepts of Archimedean extension,
Archimedean completeness, and Archimedean type that are appropriate for
models of P . For the concept of Archimedean type, in particular, we require
the construct that emerges from the following familiar result:
THE FUNDAMENTAL LEMMA OF CARTESIAN GEOMETRY. If L is
the line in a model M of P containing the distinct points o and e, then by
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
63
letting o and e serve as the zero and unit, respectively, and by appealing to
the Euclidean theory of proportions which can be derived from P , one can
define operations +L and L on, and a relation L between, any two points
L (M ) = hL; + ; ; ; o; ei is
of L so that the resulting structure Foe
L L L
M
an ordered field for which the following is true: A model M 0 of P is
isomorphic to M if and only if M 0 is isomorphic to a Cartesian space over
L (M ).
an isomorphic copy of Foe
L (M ), or any
In virtue of the Fundamental Lemma, the ordered field Foe
isomorphic copy thereof, may be said to be the characteristic field of M .
Being an ordered field, the characteristic field of M has an Archimedean
type; it is this that motivates the following definition and the subsequent
generalization of the Archimedean axiom.
DEFINITION 2.2. A model M of P will be said to be of Archimedean
type G if its characteristic field is of Archimedean type G.
AG . AXIOM OF ARCHIMEDEAN TYPE G. The characteristic field
of the structure consisting of the collection of points together with the
betweenness and equidistance relations defined on it is of Archimedean
type G.
The most general description of the characteristic fields of models of P
is given by the following classical result whose roots lie in the Grundlagen:
CFP . The characteristic fields of models of P arepprecisely the Pythagorean ordered fields, i.e. the ordered fields in which a2 + b2 is a member of
the field whenever a and b are members of the field.
Plainly then, in virtue of the above, we have the following
REPRESENTATION THEOREM FOR P [ AG . M is a model of P [ AG
if and only if M is isomorphic to a Cartesian space over a Pythagorean
ordered field of Archimedean type G.
There are a variety of alternative approaches one may adopt to introduce
the requisite geometrical counterparts for the group-theoretic conceptions
of Archimedean extension and Archimedean completeness. The definition
employed below makes use of the concept of a segment – a non-ordered
pair of distinct points – in a model of P and is motivated by the following
elementary result:
PROPOSITION 1. If G and G0 are ordered abelian groups where G G0 ,
then G0 is an Archimedean extension of G if and only if for each y > 0 in
G0 there is an x > 0 in G and a positive integer n such that x y nx:10
64
PHILIP EHRLICH
DEFINITION 2.3. If M and M 0 are models of P where M M 0 , then M 0
will be said to be an Archimedean extension of M if for every line segment
ab in M 0 there is a finite set of points x0, x1; : : : ; xn in M and a point y in
M 0 such that x0x1 xi xi+1 and Bxi 1xi xi+1 for all 1 i < n, Bx1yxn
and x0 y ab; on the other hand, if M admits no proper Archimedean
extension, M will be said to be Archimedean complete.
Now suppose GL
o (M ) is the ordered additive group of the characteristic
L
field Foe (M ) of a model M of P . The construction of GL
o (M ) does not
make use of or e. Moreover, if M and M 0 are models of P where M M 0
L (M ) F L0 (M 0 )
and L0 is the unique line in M 0 that extends L, then Foe
oe
L0
0 11
and, consequently, GL
o (M ) Go (M ): In virtue of this, Proposition 1,
and the fact that for every line L and segment ab of M there are points
c and d of L for which ab cd, the validity of the following result is
apparent.
LEMMA 1. If M and M 0 are models of P where M M 0 , o and e are
distinct points on a line L in M and L0 is the unique line in M 0 that extends
L, then M 0 is an Archimedean extension of M if and only 0if FoeL0 (M 0 )
L (M ) if and only if GL (M 0 ) is an
is an Archimedean extension of Foe
o
L
Archimedean extension of Go (M ); and, so, if M is a model of P , then M
L (M ) is Archimedean complete
is Archimedean complete if and only if Foe
if and only if GL
o (M ) is Archimedean complete.
Thus far, nothing we have said ensures that there are any Archimedean
complete models of P other than isomorphic copies of C2 fRg. On the other
hand, it follows from HF2 together with Lemma 1 and Definitions 2.2 and
2.3 that M is an Archimedean complete model of P of Archimedean
type G if and only if M is a model of P whose characteristic field is
isomorphic to R(G). Moreover, by the Fundamental Lemma, every model
of P whose characteristic field is isomorphic to R(G) is isomorphic to
C2fR(G)g. As a result, to establish the existence (up to isomorphism) of
a unique Archimedean complete model of P of Archimedean type G, it
only remains to show that R(G) is the characteristic field of some model
of P . But that R(G) is indeed such a field is evident in light of CFP and
the following result due to Prestel (1974, 101):
HF3 . Every Hahn field is a Pythagorean ordered field.12
In virtue of the above, it is now clear that to obtain a categorical axiomatization of Archimedean Complete Euclidean geometry of Archimedean
type G, one need only supplement P [fAG g with the following assertion:
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
65
AC . AXIOM OF ARCHIMEDEAN COMPLETENESS. The structure
consisting of the collection of points together with the betweenness and
equidistance relations defined on it admits no proper Archimedean extension to a model of P .
More specifically, we have said enough to prove the following revealing
formulation of the
CATEGORICITY OF P [ fAG ; AC g. C2 fR(G)g is (up to isomorphism)
the unique model of P [ fAG ; AC g.
Having established the desired generalization of the categoricity theorem for P [ fA; C g, we will now direct our attention to generalizing the
embedding theorem for C2 fRg. For this purpose, it is convenient to have
available
LEMMA 2. If C2 fF g is a model of P and F 0 is a Pythagorean ordered
field such that F 0 F , then C2 fF 0 g also is a model of P and C2 fF 0 g C2fF g. Moreover, if F is an Archimedean extension of F 0 , then C2 fF g
is an Archimedean extension of C2 fF 0 g.
Proof. The first half of Lemma 2 is an elementary consequence of
Definition 2.1 and the Fundamental Lemma. To establish the second half,
let a, b 2 C2 fF g where a 6= b and let x0 = (0; 0) where 0 is the additive
identity of F . Then x0 = (0; 0) 2 C2 fF g and there is a unique point
y = (0; ) 2 C2fF g such that x0 y ab where > 0. Moreover, since
F is an Archimedean extension of F 0 , it follows from Proposition 1 that
there is a > 0 in F 0 and a positive integer n such that < < n .
Hence, to complete the proof, it only remains to point to Definition 2.3
and note that there are points x0 = (0; 0), x1 = (0; ); : : : ; xn = (0; n )
in C2 fF 0 g such that x0 x1 xi xi+1 and Bxi 1xi xi+1 for all 1 i < n,
Bx1yxn and x0y ab.
EMBEDDING THEOREM FOR C2 fR(G)g. C2 fR(G)g is a model of
P [ fAG g that is an Archimedean extension of an isomorphic copy of
every model of P [ fAG g. Any other model of P [ fAG g having this
property is isomorphic to C2 fR(G)g.
Proof. Let M be a model of P [ fAG g. Then the characteristic field
of M is of Archimedean type G and, by HF1 , R(G) is an Archimedean
extension of an ordered field F that is isomorphic to the characteristic field
of M . Moreover, by the Fundamental Lemma, C2 fF g is isomorphic to M ;
and, by Lemma 2, C2fR(G)g is an Archimedean extension of C2fF g. Now
suppose M is a model of P [ fAG g that is an Archimedean extension of
an isomorphic copy of every model of P [ fAG g. If M is not isomorphic
to C2 fR(G)g, then M must be a proper Archimedean extension of an
66
PHILIP EHRLICH
isomorphic copy of C2 fR(G)g. But this is impossible, since C2 fR(G)g is
Archimedean complete; and, so, M is isomorphic to C2 fR(G)g.
As we already mentioned in the Introduction, in the later editions of the
Grundlagen Hilbert’s completeness axiom is replaced by a line completeness axiom due to Bernays.13 The line completeness condition is stated just
like the completeness axiom except that attention is directed to an arbitrary
line in space as opposed to space itself and the reference to Hilbert’s analog
of P is replaced by a reference to what Bernays calls “the fundamental
properties of line order and [segment] congruence” (Hilbert 1956, p. 30;
1971, p. 26).
Although Bernays (who edited the later editions of the Grundlagen)
never says as much, his set of “fundamental properties” is equivalent to
Hilbert’s original set of axioms of line order and segment congruence less
the presumed density of the points on a line. As is well known, the analytic
representations of this axiom set are defined on ordered abelian groups.
Accordingly, whereas Hilbert’s Completeness Axiom isolates “ordinary
analytic geometry” (and its various isomorphs) by limiting the models of
Hilbert’s other axioms to those whose characteristic fields admit no proper
extension to an Archimedean, Pythagorean ordered field, Bernays’ Line
Completeness Axiom serves the same function by limiting the models to
those whose characteristic fields (considered as Archimedean ordered additive groups) admit no proper extension to an Archimedean ordered abelian
group.14 It is the fact that the characteristic fields of Archimedean complete
models of Euclidean geometry of Archimedean type G coincide with the
Pythagorean ordered fields of Archimedean type G whose ordered additive groups admit no proper Archimedean extension to an ordered abelian
group which makes possible an Archimedean complete generalization of
Bernays’ categoricity theorem. To obtain such a generalization appropriate
for models of P [ fAG g, we require a suitable analog of Bernays’ set of
“fundamental properties”.
Let LB = fA1 ; : : : ; A7 ; :A8 g:15 As a glance at A8 indicates, :A8
is equivalent to the assertion: Babc _ Bbca _ Bcab for all a, b and c.
Accordingly, since in a model of P the set of all points collinear with
distinct points a and b is equal to fx: Babx _ Bbxa _ Bxabg, :A8
assures us that all of the points being considered lie on the same line. In
virtue of this and some remarks in the preceding paragraph, to establish
that LB has the desired properties, we need only mention the following
theorem which is essentially due to Tarski (1959, p. 21 [note 5]):
CGL . The models of LB are (to within isomorphism) precisely the 1Dimensional Cartesian spaces defined over some ordered abelian group
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
67
hG; +; i i.e., the structures hG; BG; G i where the relations BG and G
are defined on G by the stipulations:
BG xyz if and only if x y z or z y x;
xy G uv if and only if jy xj = jv uj:
If M is a model of P , the restriction of M to the points lying on an
arbitrarily specified line of M is, of course, isomorphic to a 1-dimensional
Cartesian space over the ordered additive group of the characteristic field of
M . Furthermore, the concepts of Archimedean extension and Archimedean
completeness introduced in Definition 2.3, while defined for models of P ,
can be readily adapted to apply to models of LB . In virtue of this, CGL and
the preceding remarks, Archimedean complete generalizations of Bernays’
line completeness axiom and categoricity theorem that are appropriate for
models of P [ fAG g may therefore be stated as follows.
ACL . ARCHIMEDEAN COMPLETENESS OF THE LINE. A substructure consisting of the collection of points on a line together with the
betweenness and equidistance relations restricted thereto admits no proper
Archimedean extension to a model of LB :16;17
CATEGORICITY OF P [ fAG ; ACL g. C2 fR(G)g is up to isomorphism
the unique model of P [ fAG ; ACL g.
To draw this section of the paper to a close, it now only remains to
isolate and briefly comment on the Archimedean complete models of the
two distinguished extensions of P that we alluded to in the Introduction.
For this purpose, we will let P 0 = P [ fLC g and P = P [ fA11 g,
whereby LC and A11 we mean the following familiar conditions:
LC . The Line-Circle Axiom.
9x(da da0 ^ dc dc0 ^ Bdac ^ Babc ! db dx ^ Ba0xc0 ):
A11. Tarski’s Continuity Schema.
All sentences in the language , B of the form:
9a8x8y((x) ^ (y) ! Baxy) ! 9b8x8y((x) ^ (y)
! Bxby)
68
PHILIP EHRLICH
where and are first order formulas, such that the variables a, b, y do
not occur in , and the variables a, b, x do not occur in .
LC is a formulation, appropriate to P , of Euclid’s tacit assumption that
a line containing two points, one inside and one outside a given circle,
always intersects the circle. As is well known, P 0 is closer in spirit than
P to the geometry of Euclid since LC is not a consequence of P and the
models of P 0 coincide with the models of P in which one can perform all
constructions requiring only a finite number of operations with straightedge
and compass used solely in the following classical Euclidean sanctioned
ways:
EC1. Through two distinct points, draw the line determined by them;
EC2. About a point designated as center, draw a circle whose radius is
equal to the distance between two given points.
A11 , on the other hand, is entirely modern in conception; roughly speaking, it is the infinite set of assertions obtained from the following formulation of the Dedekind continuity condition
8X 8Y (9a8x8y(x 2 X ^ y 2 Y ! Baxy)
! 9b8x8y(x 2 X ^ y 2 Y ! Bxby))
by replacing the explicit references to arbitrary sets of points X and Y
with implicit references those sets which are definable in the language ,
B by first order means.
The analogs of CFP for P 0 and P are given by the following wellknown results, the second of which originates with Tarski:
CFP 0 . The characteristic fields of models of P 0 are precisely the
p Euclidean
ordered fields, i.e. the (Pythagorean) ordered fields in which a is always
a member of the field whenever a is a positive member of the field.
CFP . The characteristic fields of models of P are precisely the real-
closed ordered fields, i.e. the Euclidean ordered fields in which every
polynomial equation of odd degree with coefficients in the field has a
solution in the field.
The ordered field of real numbers is of course a real-closed ordered
field. This being the case, C2 fRg is a model of P 0 as well as P . Unlike
C2fRg, however, Archimedean complete, non-Archimedean models of P
need not be models of P or even models of P 0 . Precisely when these
theories are modeled by such structures is encapsulated by the following
two results.
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
69
(i) A model of P [ fAG ; AC g (or alternatively P [ fAG ; ACL g) is a
model of P 0 if and only if G is 2-divisible, i.e., for all x 2 G there is
a y 2 G such that x = 2y .
(ii) A model of P [ fAG ; AC g (or alternatively P [ fAG ; ACL g) is a
model of P if and only if G is divisible, i.e., for all x 2 G and all
positive integers n, there is a y 2 G such that x = ny .
To see that (i) and (ii) are indeed the case, one need only appeal to CFP 0 and
CFP , the categoricity theorems for P [fAG ; AC g and P [fAG ; ACL g,
and the following two algebraic results:
HF4 . (Prestel and Ziegler 1975) R(G) is a Euclidean ordered field if and
only if G is 2-divisible;18
HF5 . (Krull 1932) R(G) is a real-closed ordered field if and only if G is
divisible.
3. ARCHIMEDEAN COMPLETE MODELS OF P (PART II)
As we noted in the Introduction, many authors isolate classical Cartesian
geometry using a geometrical formulation of Dedekind’s continuity condition in place of the completeness axioms due to Hilbert and Bernays. We
will now turn our attention to formulating a generalization of Dedekind’s
axiom that can play an analogous role for Archimedean complete models
of P [ fAG g.
A subset G0 of an ordered set G is said to be convex if every member of
G that lies between some pair of members of G0 is also a member of G0 . If
G0 and G are also ordered groups, then G0 is said to be a convex subgroup
of G. By the breadth of a Dedekind cut (X; Y ) of an ordered abelian group
G, written B (X; Y ), one means the largest convex subgroup G0 of G for
which x + jg 0 j 2 X for all x 2 X and all g 0 2 G0 . Intuitively, the breadth of
(X; Y ) is the hurdle that must be overcome to pass by means of addition
from X to Y . Since G is assumed to be abelian, the breadth of (X; Y ) is
also the largest convex subgroup G0 of G for which y jg 0 j 2 Y for all
y 2 Y and all g0 2 G0; that is, B (X; Y ) is also the hurdle that must be
overcome to pass by means of subtraction from Y to X .
Following convention, for each convex subgroup G0 of G, G/G0 will
denote the ordered factor group of G modulo G0 , i.e., G/G0 = fx + G0 :x 2
Gg where x + G0 = fx + g0 : g0 2 G0 g and sums and order are defined by
the rules:
a G0)+ (b + G0 ) = (a + b) + G0 ;
( +
70
PHILIP EHRLICH
a + G0< b + G0 if a < b and b a 2 G0 :
Also following convention, if X G, we will denote fx + G0 : x 2 X g G=G0 by X + G0.
As the reader will recall, a Dedekind cut (X; Y ) of an ordered abelian
group G is said to be continuous if X has a greatest member or Y has a
least member but not both. R is, of course, up to isomorphism the unique
the ordered abelian group in which every Dedekind cut is a continuous cut.
To obtain an Archimedean complete generalization of this important result,
one therefore requires a suitable generalization of the idea of a continuous
cut of an ordered abelian group.
DEFINITION 3.1. A Dedekind cut (X; Y ) of an ordered abelian group G
will be said to be B (X; Y )-continuous if (X + B (X; Y ), Y + B (X; Y ))
is a continuous cut in G/B (X; Y ).
An ordered abelian group G is said to be continuous if every Dedekind
cut of G is a continuous cut. Generalizing this idea for ordered abelian
groups all of whose Dedekind cuts are B (X; Y )-continuous cuts leads to
DEFINITION 3.2. An ordered abelian group G will be said to be B (X; Y )continuous if every Dedekind cut (X; Y ) of G is B (X; Y )-continuous.
The reader will notice that if (X; Y ) is a Dedekind cut of an ordered
abelian group G, then (X + B (X; Y ), Y + B (X; Y )) is in fact a Dedekind
cut in G/B (X; Y ). Moreover, when G is Archimedean, (X + B (X; Y ),
Y + B (X; Y )) = (X; Y ) and (G=B (X; Y )) = G, since B (X; Y ) = f0g.
Definitions 3.1 and 3.2 are, accordingly, natural generalizations of their
classical counterparts. This being the case, the same can also be said of
our final characterization of Archimedean complete models of P [ AG .
CATEGORICITY OF P [ fAG ; BC g. C2 fR(G)g is (up to isomorphism)
the unique model of P [ fAG ; BC g, whereby BC we mean the assertion:
The characteristic field (of the structure consisting of the collection of
points together with the betweenness and equidistance relations defined on
it) is B (X; Y )-continuous.
Proof. This readily follows from theorems in the previous section and
the following algebraic result.
THEOREM (Ehrlich forthcoming). An ordered abelian group G is Archimedean complete if and only if every Dedekind cut (X; Y ) of G is
B (X; Y )-continuous.
71
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
NOTES
1
This paper is essentially the written version of a talk presented by the author at Boston
University in November of 1993 as part a conference on David Hilbert’s contributions to the
foundations of mathematics. The author wishes to thank the conference’s organizer, Jaakko
Hintikka, for providing me with the opportunity to present the ideas contained herein at
that time. The author also gratefully acknowledges the support provided by the National
Science Foundation (Scholars Award #SBR-9223839) and Ohio University.
2
We wish to emphasize, however, that analogous results can be (and, in a separate paper, will
be) obtained for other geometries including Hyperbolic geometry and Elliptic geometry;
the present paper’s focus on Euclidean geometry is more a reflection of Hilbert’s own use of
completeness and embedding properties in his Grundlagen der Geometrie than an indication
of the range of applicability of Hilbert’s own ideas or of the author’s generalization thereof
contained herein. Similarly, while our results are limited to the 2-dimensional case, they
readily extend to higher-dimensional cases.
3
Several authors employ the following alternative definition which leads to the inclusion
of 0 as an Archimedean class: a G is said to be Archimedean equivalent to b G if
a and n a b . If this convention
there are positive integers m and n such that m b
were adopted, some of the statements in the text would require minor modification.
4
As we explained in (Ehrlich 1995), HF1 and HF2 have long and complicated histories
that make it difficult to attribute them to any single author. Their roots lie in the work of
Krull (1932), Kaplansky (1942) and Hahn (1907), and by the early 1950s they appear to
have assumed the status of “folk theorems” among knowledgeable field theorists. On the
other hand, it appears to be Paul Conrad who published the first explicit statements of
these “well known results” (1954, 328). Conrad (1954, 327: Theorem 3.3) also provided
a proof of a more general completeness theorem from which HF1 emerges as a special
case; and Conrad (together with Dauns) (1969; Theorem II) later isolated and provided
a detailed proof-sketch of a more revealing formulation of HF1 . Since that time, several
alternative proofs, variations, and strengthenings of HF1 have appeared including those of
Priess-Crampe (1973; 1983, 62–64, 124), Priess-Crampe and von Chossey (1975), Rayner
(1975), and Mourgues and Ressayre (1993).
5
For a good discussion of models of Hilbert’s system, see (Schwabhäuser 1956). Also
see (Eisenhart 1939, 282–292; and Eves 1965, 102–105) for the treatment developed by
Bochner, Church, and Tomkins; for interesting hybrid treatments of the systems of Hilbert
and Tarski, see (Beth 1966, 146–151; and Borsuk and Szmielew 1960) as well as (Hilbert
and Bernays 1934, 4–43) which partially anticipates the treatments of Beth, Borsuk and
Szmielew.
6
Tarski’s work on elementary geometry has a long and interesting history that dates back
to the late 1920s. For the history of Tarski’s system, the reader should consult (Tarski
1948/51; 1959; and Szczerba 1986). Moreover, for references to the vast literature spawned
by Tarksi’s work on geometry, see the relevant papers in (Henkin, Suppes and Tarski 1959)
as well as the bibliographies in (Szmielew 1983) and, especially, (Schwabhäuser, Szmielew,
and Tarski 1983).
7
For the sake of readability, the strings of universal quantifiers that precede A1 –A10 below
have been deleted. The same convention will be utilized in the subsequent formulations of
A, A05, A11 and LC.
8
In its original formulation, the geometric completeness axiom reads as follows where
axiom-groups I–IV are Hilbert’s axioms of incidence, order, congruence, and parallelism
and axiom-group V consists solely of the axiom of Archimedes:
fg
2
j jj j
j jj j
2
72
PHILIP EHRLICH
To the system of points, straight lines, and planes, it is impossible to adjoin other elements
in such a manner that the system thus generalized shall form a new geometry obeying all of
the five groups of axioms I–V; in other words, the elements of the geometry form a system
which is not susceptible of [proper] extension, if all of the stated axioms are preserved.
(Hilbert 1900b, 25)
9
During the course of this century, a number of misleading remarks have appeared in
discussions of Hilbert’s work concerning the relation between the embedding and completeness properties of . As we mentioned in (Ehrlich 1995; note 7), some poorly chosen
words by Hilbert appear to be the source of some of the confusion. Another sort of error
is committed by Carnap and Bachmann (1936, 183, 3/1981, p. 82, 3) in the course of
discussing Hilbert’s completeness condition. In modern terminology, they suggest that: A
model M of a theory T contains an isomorphic copy of every model of T if M is up to
isomorphism the unique model of T that admits no proper extension to a model of T . However, to see that this is incorrect, consider the theory P1 = A1 ; : : : ; A8 ; A9 A09 ; A10 ; A09
A where A09 is the 3-dimensional analog of the 2-dimensional Upper Dimension Axiom
A9 and let C be the completeness condition that results from C by replacing the referA with a reference to P1 . The models of P1 are precisely the models of
ence to P
P2 = A1 ; : : : ; A8; A9; A10 ; A (2-dimensional, non-Archimedean Euclidean geometry)
and the models of P3 = A1 ; : : : ; A8 ; A09 ; A10 ; A (3-dimensional, Archimedean Euclidean geometry). Since (i) no model of P3 can be extended to a model of P2 , (ii) every model
of P2 admits a proper extension to a model of P2 , and (iii) C3
(ordinary 3-dimensional
Cartesian geometry) is up to isomorphism the only model of P3 that admits no proper
C has up to isomorphism only one
extension to a model of P3 , it follows that P1
model, namely C3
, which is a model of P3 . But since no model of P2 is isomorphologically contained in C3
, C3
does not contain an isomorphic copy of every
model of P1 .
The above counterexample to Carnap and Bachmann’s claim naturally suggests the
following result: if a model M of a theory T admits no proper extension to a model of T ,
then M contains an isomorphic copy of every model of T if and only if T has the joint
embedding property, i.e., every pair of models of T can be embedded in a model of T ; in
this case, M is (up to isomorphism) the unique model of T that admits no proper extension
to a model of T as well as the unique model of T that contains an isomorphic copy of every
model of T .
Proof. The nontrivial portion of the result follows from the fact that if A is a model
of a theory T having the joint embedding property, then there is a model M 0 of T and
M 0 and g : M M 0 , the latter of which must be a surjection since
embeddings f : A
g(M ) (which is isomorphic to M ) admits no proper extension to a model of T .
10
This characterization of an Archimedean extension comes from (Holland 1963, 71).
11
We assume here, and henceforth, that one of the standard constructions of characteristic
fields of models of P is selected and employed throughout the discussion. See, for example,
(Schwabhäuser, Szmielew, and Tarski 1983, 143–159).
12
Shortly after Prestel (1974, 101) established the Pythagorean nature of (G) it became
clear to many field theorists that (G) is in fact a paradigm example of what Bröcker (1972)
calls a strictly Pythagorean ordered field or, equivalently, what Elman and Lam (1972) call
a super Pythagorean ordered field. This follows immediately from the Henselian nature of
(G) (which was first established by Krull 1932), the fact that
can be ordered in only
one way, and what is sometimes called the Brown-Bröcker Theorem (cf. Lam 1980, 39,
118). A simpler theoretical basis for the super Pythagorean nature of (G) was provided
R
$ g
f
x
x
f
[f g
f
fRg
:g
g
fRg fRg
!
[f g
_
fRg
!
R
R
R
R
R
73
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
soon thereafter by Lam (1983, 46: Proposition 5.17(2)) who also provided a particularly
simple theoretical basis for HF3 itself (Lam 1983, 28: Theorem 3.16(3)). For rediscoveries
of HF3 , see (Benhissi 1990; Ribenboim 1990; 1992; and Alling 1990).
13
Strictly speaking, there are two different formulations of Bernays’ axiom. The first
formulation appeared in the Seventh Edition of the Grundlagen (Hilbert 1930, 30) and the
second and far more familiar formulation has appeared in every edition since, beginning
with (Hilbert 1956, 30). For a useful discussion of the first formulation, see (Viola 1960).
All of our remarks in the text concern the second formulation.
14
This point is never actually made in the Grundlagen. On the other hand, in a separate
work Bernays proves a closely related result about the positive cone of
considered
as a particular kind of Archimedean ordered abelian semigroup (Bernays 1955, 223).
Presumably, it was the latter result that Bernays took to be the algebraic basis of his then
forthcoming line completeness condition.
15
Alternatively, one may use L0B = A1 ; : : : ; A4 ; A50 ; A6 ; A7 ; A8 whereby A05 we mean
the assertion: Babc Ba0 b0 c0 ab a0 b0 bc b0 c0
ac a0 c0 . Although A05 (which
is a simple consequence of A5 ) is weaker than A5 in the context of P , in the present setting,
given the inclusion of A8 , they are equivalent.
16
In the classical case, ACL may be replaced by the following assertion which has a more
familiar form:
R
^
:
f
^ : g
^ ! CL : COMPLETENESS OF THE LINE. A substructure consisting of the collection of
points on a line together with the betweenness and equidistance relations restricted thereto
admits no proper extension to a model of LB
A:
[f g
CATEGORICITY OF P
[ fA; CLg.
[ fA; CLg: C fRg is up to isomorphism the unique model of P
2
17
English translation of Bernays’ Formulation: “An extension of the set of points on a line
with its order and congruence relations that would preserve the relations existing among
the original elements as well as the fundamental properties of line order and congruence
that follows from Axioms I–III [Hilbert’s axioms of incidence, order, and congruence], and
preserve V,1 [Archimedes’ Axiom] is impossible” (Hilbert 1956, 30).
In his textbook discussion of the Grundlagen, George Martin correctly observes that:
“Anyone who reads the English translations should be aware that some of the ambiguities
have been translated into inconsistencies” (Martin 1975, 114). Although Martin does not
cite specific examples, the statement of the axiom of line completeness provides a case
in point. If one ignores the context, one could reasonably translate Bernays’ formulation
as Unger did in (Hilbert 1971, 26); there one finds in place of the second occurrence of
the term “preserves” in our translation above a second occurrence of the term “from”.
However, since the Archimedean axiom is not included among Bernays’ “fundamental
properties of line order and congruence”, according to Unger’s translation it is not assumed
to be preserved. But as Bernays correctly observes (a few lines following his statement of
the axiom), if the preservation of the Archimedean axiom is not supposed, Hilbert’s system
is rendered inconsistent (Hilbert 1956, 30–31; 1971, 26–27)!
18
Prestel and Ziegler (1975) actually proved the stronger result that (G) is hereditarily Euclidean if and only if G is 2-divisible. For discussions of such fields, see the
paper already cited as well as (Prestel 1975/1984, 102–108; and Lam 1980, 92–94). Also
see the papers by Benhissi, Ribenboim, and Alling cited in note 12 for rediscoveries of HF4 .
R
74
PHILIP EHRLICH
REFERENCES
Alling, N.: 1990, ‘Pythagorean and Euclidean Ordered Valued Fields’, La Société royale
du Canada. Comptes Rendus Mathématique de l’Académie des Sciences (Mathematical
Reports) 12(5), 187–92.
Baer, R.: 1927, ‘Über nicht-archimedisch geordnete Körper’, Sitzungsberichte der Heidelberger Akademie der Wissenschaften 8, 3–13.
Benhissi, A.: 1990, ‘Series Formelles Generalisees Sur Un Corps Pythagoricien’, La Société
royale du Canada. Comptes Rendus Mathématique des Sciences (Mathematical Reports)
12(5), 193–198.
Bernays, P.: 1955, ‘Betrachtungen über das Vollständigkeitsaxiom und verwandte Axiom’,
Mathematische Zeitschrift 63, 219–29.
Beth, E. W.: 1966, The Foundations of Mathematics (Revised Edition), Harper & Row,
Publishers, New York, NY.
Borsuk, K. and W. Szmielew: 1960, Foundations of Geometry, North-Holland Publishing
Co., Amsterdam.
Bröcker, L.: 1972, ‘Über eine Klasse pythagoreischer Körper’, Archiv der Mathematik 23,
405–7.
Carnap, R. and F. Bachmann: 1936, ‘Über Extremalaxiome’, Erkenntnis 6, 166–88.
Carnap, R. and F. Bachmann: 1981, ‘On Extremal Axioms’, History and Philosophy of
Logic 2, 67–85. English translation by H. G. Bohnert of (Carnap and Bachmann 1936).
Conrad, P.: 1954, ‘On Ordered Division Rings’, Proceedings of the American Mathematical
Society 5, 323–8.
Conrad, P., and J. Dauns, 1969, ‘An Embedding Theorem for Lattice-ordered Fields’,Pacific
Journal of Mathematics 30, 385–97.
Ehrlich, P. (ed.): 1994, Real Numbers, Generalizations of the Reals, and Theories of Continua Kluwer Academic Publishers, Dordrecht, Holland.
Ehrlich, P.: 1994a, ‘General Introduction’, in (Ehrlich 1994, pp. vii–xxxii).
Ehrlich, P.: 1995, ‘Hahn’s Über die nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them’, in Jaakko
Hintikka (ed.), From Dedekind to Gödel, Essays on the Development of the Foundations
of Mathematics, Kluwer Academic Publishers, Dordrecht, Holland.
Ehrlich, P.: (Forthcoming), ‘Dedekind Cuts of Archimedean Complete Ordered Abelian
Groups’, Algebra Universalis.
Eisenhart, L. P.: 1939, Coordinate Geometry, Ginn and Company, Boston.
Elman, R. and T. Y. Lam: 1972, ‘Quadratic Forms over Formally Real Fields and Pythagorean Fields’, American Journal of Mathematics 94, 1155–94.
Eves, H.: 1965, A Survey of Geometry, Volume Two. Allyn and Bacon, Inc., Boston.
Fuchs, L.: 1963, Partially Ordered Algebraic Systems. Pergamon Press.
Hahn, H.: 1907, ‘Über die nichtarchimedischen Grössensysteme’, Sitzungsberichte
der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch – Naturwissenschaftliche Klasse 116 (Abteilung IIa), 601–655.
Henkin, L., P. Suppes, and A. Tarski (eds.): 1959, The Axiomatic Method, North-Holland
Publishing Company, Amsterdam.
Hilbert, D.: 1899, Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des
Gauss-Weber Denkals in Göttingen, Teubner, Leipzig.
Hilbert, D.: 1900a, ‘Über den Zahlbegriff’, Jahresbericht der Deutschen Mathematiker –
Vereinigung, 8, 180–4. Reprinted in (Hilbert 1930, pp. 241–246).
FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES
75
Hilbert, D.: 1900b, Les Principes Fondamentaux de la géometrie. Gautier-Villars, Paris.
French translation by L. Laugel of an expanded version of (Hilbert 1899). Reprinted in
Annales Scientifiques de L’école Normal Supérieure 17 (1900), 103–209.
Hilbert, D.: 1902, Foundations of Geometry. English Translation by E. Townsend of an
expanded version of (Hilbert 1899).
Hilbert, D.: 1903, Second Edition of (Hilbert 1899).
Hilbert, D.: 1904a, ‘Über die Grundlagen der Logik und Arithmetik’, in Verhandlungen des
Dritten Internationalen Mathematiker – Kongresses in Heidelberg vom 8. bis 13. August
1904. Teubner, Leipzig, 1905.
Hilbert, D.: 1904b, ‘On the Foundations of Logic and Arithmetic’, English translation by
Beverly Woodward of (Hilbert 1904a) in J. van Heijenoort (ed.), From Frege To Gödel:
A Sourcebook in Mathematical Logic, 1879–1931 (Second Printing), Harvard University
Press, Cambridge, Massachusetts, 1971.
Hilbert, D.: 1909, Third Edition of (Hilbert 1899).
Hilbert, D.: 1930, Seventh Edition of (Hilbert 1899).
Hilbert, D.: 1956, Eighth Edition with supplements by Paul Bernays of (Hilbert 1899),
Teubner, Stuttgart.
Hilbert, D.: 1971, Foundations of Geometry. Open Court, LaSalle, Illinois, English translation by Leo Unger of Paul Bernays’ Revised and Enlarged Tenth Edition of (Hilbert
1899).
Hilbert, D. and P. Bernays: 1934, Grundlagen der Mathematik I. Springer Verlag, Berlin.
Holland C.: 1963, ‘Extensions of Ordered Groups and Sequence Completions’, Transactions of the American Mathematical Society 107, 71–82.
Kaplansky, I.: 1942, ‘Maximal Fields with Valuations’, Duke Mathematical Journal 9,
303–21.
Krull, W.: 1932, ‘Allgemeine Bewertungstheorie’, Journal für die Reine and Angewandt
Mathematik 167, 160–96.
Lam, T. Y.: 1980, ‘The Theory of Ordered Fields’, in B. McDonald (ed.), Ring Theory and
Algebra III: Proceedings of the Third Oklahoma Conference, Marcel Dekker, Inc., New
York.
Lam, T. Y.: 1983, Orderings, Valuations and Quadratic Forms, Regional Conference Series
in Mathematics #52. American Mathematical Society, Providence, R.I.
Martin, G.: 1975, The Foundations of Geometry and the Non-Euclidean Plane. SpringerVerlag, New York.
Pejas, W.: 1961, ‘Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie’,
Mathematische Annalen 143, 212–35.
Prestel, A.: 1974, ‘Euklidsche Geometrie ohne das Axiom von Pasch’, Abhandlungen aus
dem Mathematischen Seminar der Univeristät Hamburg 41, 82–109.
Prestel, A.: 1975, Lectures on Formally Real Fields. (Monografias de Mathemática 22).
Instituto de Mathemática Pura e Aplicada. Rio de Janeiro. Reprinted (with corrections)
in 1984. Lecture Notes in Mathematics #1093. Springer-Verlag, Berlin.
Prestel, A. and M. Ziegler: 1975, ‘Erblich euklidische Körper’, Journal für die Reine und
Angewandt Mathematik 274/275, 196–205.
Priess-Crampe, S.: 1973, ‘Zum Hahnschen Einbettungssatz für Angeordnete Körper’,
Archiv der Mathematik 24, 607–14.
Priess-Crampe, S.: 1983, Angeordnete Strukturen, Gruppen, Körper, projektive Ebenen.
Springer-Verlag, Berlin.
Priess-Crampe, S. and R. von Chossy: 1975, ‘Ordungsverträgliche Bewertungen eines
angeordneten Körpers’, Archiv der Mathematik 26, 373–87.
76
PHILIP EHRLICH
Rayner, F. J.: 1976, ‘Ordered Fields’, in Seminaire de Théorie des Nombres 1975–76.
(Univ. Bordeaux I. Talence). Exp. No. 1, pp. 1–8, Lab. Théorie des Nombres, Centre Nat.
Recherche Sci., Talence.
Ribenboim, P.: 1990, ‘Algebraic Extensions of Fields of Generalized Power Series’, La
Société royale du Canada. Comptes Rendus Mathématique de l’Académie des Sciences
(Mathematical Reports) 12(5), 209–12.
Ribenboim, P.: 1992, ‘Fields: Algebraically Closed and Others’, Manuscripta Mathematica
75, 115–50.
Schwabhäuser, W.: 1956, ‘Über Die Vollständigkeit Der Elementaren Euklidischen Geometrie’, Zeitschrift für mathematische Logik and Grundlagen der mathematik 2, 137–65.
Schwabhäuser, W., W. Szmielew, and A. Tarski: 1983, Metamathematische Methoden in
der Geometrie, Springer-Verlag, Berlin.
Szczerba, L.: 1986, ‘Tarski and Geometry’, The Journal of Symbolic Logic 51, 907–12.
Szmielew, W.: 1983, From Affine to Euclidean Geometry, D. Reidel Publishing Company,
Dordrecht, Holland.
Tarski, A.: 1948/1951, A Decision Method for Elementary Algebra and Geometry, The Rand
Corporation, Santa Monica; Second Edition, University of California Press, Berkeley and
Los Angeles.
Tarski, A.: 1959, ‘What is Elementary Geometry?’, in (Henkin, Suppes, and Tarski 1959,
pp. 16–29).
Viola, T.: 1960, ‘Sul postulato della completezza lineare dell’ Hilbert’, Annali Di Matematica Pura Ed Applicata 49, 367–73.
Department of Philosophy
Ohio University
Athens, OH 45701
USA