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A General Setting for Dedekind's Axiomatization of the Positive Integers

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A GENERAL SETTING FOR DEDEKIND’S AXIOMATIZATIONOF THE POSITIVE INTEGERS
GEORGE WEAVER
Abstract.
A Dedekind algebra is an ordered pair (
B
,
h
) where
B
is a non-empty set and
h
is a similarity transformation on B. Among the Dedekindalgebras is the sequence of the positive integers. From a contemporary per-spective, Dedekind established that the second-order theory of the sequenceof the positive integers is categorical and ﬁnitely axiomatizable. The purposehere is to show that this seemingly isolated result is a consequence of moregeneral results in the model theory of second-order languages. Each Dedekindalgebra can be decomposed into a family of disjoint, countable subalgebrascalled the conﬁgurations of the algebra. There are
ℵ
0
isomorphism types of conﬁgurations. Each Dedekind algebra is associated with a cardinal valuedfunction on
ω
called its conﬁguration signature. The conﬁguration signaturecounts the number of conﬁgurations in each isomorphism type that occurs inthe decomposition of the algebra. Two Dedekind algebras are isomorphic iﬀ their conﬁguration signatures are identical. The second-order theory of anycountably inﬁnite Dedekind algebra is categorical, and there are countablyinﬁnite Dedekind algebras whose second-order theories are not ﬁnitely axiom-atizable. It is shown that there is a condition on conﬁguration signatures nec-essary and suﬃcient for the second-order theory of a Dedekind algebra to beﬁnitely axiomatizable. It follows that the second-order theory of the sequenceof the positive integers is categorical and ﬁnitely axiomatizable.
1.
Introduction
In a letter to Hans Keferstein (see Dedekind [1890]) Dedekind posed
the semantic completeness problem
for the theory of the sequence of the positive integers:What are the mutually independent fundamental properties of thesequence
N
, that is, those properties that are not derivable fromeach other and from which all others follow.Dedekind oﬀered
the ﬁnitary categoricity solution
to this problem: he exhibited aﬁnite set of properties of the sequence of the positive integers and claimed(1) that no one of them was derivable from the others; and(2) that all models of these properties are isomorphic.From a modern perspective, the reasoning from the second claim to semantic com-pleteness is weak (see Dedekind [1887]
§
134, pages 95-96). In essence, Dedekindargued that isomorphisms preserve the satisfaction of properties. Therefore, giventhe second claim, any property satisﬁed by the sequence of positive integers is sat-isﬁed by all models of the ﬁnite set Dedekind presented. Hence, every propertysatisﬁed by the sequence of the positive integers is a consequence of this set. With
Date
: 5-21-09.1991
Mathematics Subject Classiﬁcation.
03B15, 03C85.
Key words and phrases.
second-order logic, categoricity, Dedekind, positive integers.
1
2 GEORGE WEAVER
some modiﬁcation in terminology, this general line of reasoning was repeated inthe early decades of the 20
th
century (Corcoran [1980], Scanlan [1991]). However,these inferences are not evident. There is no explanation of the conditions underwhich a property is satisﬁed or not satisﬁed. The speciﬁcation of the collection of properties satisﬁed by the sequence, while suggestive, is imprecise. For example,consider the following from Dedekind [1887]: among the properties of the sequenceof positive integers isany theorem in which we leave entirely out of consideration thespecial character of the elements
n
and discuss only such notionsas arise from the arrangement
φ
.This imprecision concerning the properties not only weakens the reasoning but italso supports the contention that the semantic completeness problem does not ad-mit of a mathematical solution. For example, Kershner and Wilcox [1950] (pp.361-362) maintained that the claim to the eﬀect that isomorphic systems satisfythe same properties is ”too broad and vague to be amenable of rigorous proof”. Fur-ther, the inference from this claim to semantic completeness appears to presupposesomething like Tarski’s
no counter interpretation
deﬁnition of logical consequence.Providing a mathematical solution to the semantic completeness problem in-volves ﬁnding a precise way to specify the collection of properties (as well as spellingout the conditions under which these properties are satisﬁed or not satisﬁed). Theproperties that Dedekind exhibited are each expressible by a sentence in a second-order language. Call the set of second-order sentences that express the properties
Dedekind’s axioms
. Satisfaction of formulas in second-order languages is easilydeﬁned and it is a simple matter to show that satisfaction is preserved under iso-morphisms. Hence, isomorphism implies equivalence in second-order languages.Therefore, every second-order sentence true of the sequence of the positive integersis a consequence of the axioms. Thus, it seems natural to reformulate Dedekind’sresult within the model theory of second-order languages: there is a ﬁnite set of second-order sentences such that each member of the set is true of the sequence of the positive integers and all models of the set are isomorphic. Hence, the second-order theory of the sequence of the positive integers is ﬁnitely axiomatizable andcategorical. It is arguable that this interpretation of Dedekind’s work might benarrower than he intended because there are properties in his sense that are not ex-pressible in a second-order language. For example, there are passages in Dedekind[1887] in which it appears that some of these properties involve inﬁnite sequencesof quantiﬁers (pp. 46-52). However, because Dedekind’s axioms are categorical,broader interpretations are available. Since satisfaction of formulas in
ω
-order lan-guages (including those that include sentences with inﬁnite sequences of quantiﬁers)is preserved under isomorphisms, the
ω
-order theory of the sequence the positiveintegers is also ﬁnitely axiomatizable. More generally, if
L
is a language that in-cludes the second-order language and satisfaction for formulas in
L
is preservedunder isomorphisms, then the theory of the sequence of the positive integers in
L
is also ﬁnitely axiomatizable.The second-order interpretation of Dedekind is used below. The purpose of thispaper is to place this interpretation of Dedekind in a more general context. Aclass of algebras (the Dedekind algebras) that includes the sequence of the positive
A GENERAL SETTING FOR DEDEKIND’S AXIOMATIZATION OF THE POSITIVE INTEGERS3
integers is introduced. A
Dedekind algebra
1
is an ordered pair
B
=(
B
,
h
B
) where
B
is a nonempty set (
the domain of
B
)
and
h
B
is an injection (or
similarity transformation
) on
B
. In essence, these algebras are the focus of the second, third,fourth and ﬁfth sections of
The Nature and Meaning of the Numbers
. Among themis the sequence of the positive integers. This paper departs from Dedekind in threerespects: (1) in formulating the discussion in terms of second-order languages andtheir model theory; (2) in investigating conditions both necessary and suﬃcientfor the second-order theory of a countably inﬁnite Dedekind algebra to be ﬁnitelyaxiomatizable and categorical, rather than investigating the second-order theoryof a particular Dedekind algebra (viz. the sequence of the positive integers); and(3) in focusing on conﬁgurations (the smallest set containing an object and closedunder both
h
A
and its inverse), rather than chains (kette).The paper is divided into eleven sections. While the primary focus of the paperis countably inﬁnite Dedekind algebras, many of the results presented for the count-able case are consequences of results that hold for all inﬁnite Dedekind algebras.Sections two and three present some topics from universal algebra. Sectiontwo introduces the notion of the conﬁguration of an element in the domain of a Dedekind algebra and classiﬁes the types of these conﬁgurations. It is shownthat the conﬁgurations in a Dedekind algebra form a partition of the domain of thealgebra and that there are
ω
-many types of conﬁgurations: progressions, bigressionsand, for each
n
,
n
-loops. Section three introduces the notion of the conﬁgurationsignature of a Dedekind algebra and establishes that the conﬁguration signaturecodes the structure of the algebra. The conﬁguration signature of the Dedekindalgebra
A
is a cardinal valued function deﬁned on
ω
that counts the number of conﬁgurations in
A
of the various types: the value at 0 is the number of progressions,the value at 1 is the number of bigressions and the value at
n
+ 1 is the numberof n-loops . For the purposes of this preliminary discussion, think of the value of this function at
n
as the number of conﬁgurations of type
n
. It is shown that twoDedekind algebras are isomorphic iﬀ their conﬁguration signatures are identical.The intuition behind the next three sections is that the second-order theory of inﬁnite Dedekind algebra
A
is categorical, if its conﬁguration signature is
pointwise describable
in the sense that for each
n
, the value of the
A
’s conﬁguration at
n
is describable by a second-order formula. There are two basic components of theintended description. Section four introduces some fundamental facts about thesecond-order theories of Dedekind algebras. It is shown that for each
n
there is asecond-order formula
φ
n
(
S
) that is satisﬁed by a set
B
in
A
iﬀ
B
contains exactlyone object from each conﬁguration of type
n
in
A
. Hence, if
B
satisﬁes
φ
n
(
S
) in
A
,then the cardinality of
B
is the value of
A
’s conﬁguration signature at
n
. Sectionﬁve reviews the notion of a second-order charaterizable cardinal (Garland [1967])and establishes that if the cardinal
β
is second-order characterizable, then there isa formula,
β
(
S
), that is satisﬁed by all and only sets of cardinality
β
. In sectionsix it is shown that the second-order theory of
A
is categorical if, for each
n
, thevalue of
A
’s conﬁguration signature at
n
is zero or a second-order characterizablecardinal.
1
The use of the term ‘Dedekind Algebra’ in the above sense was introduced in Weaver [1996].The author is not aware of an earlier use of this term in this or any other sense. This term isused in Potter [2004] (page 90) to refer to members of the isomorphism type of the sequence of the positive integers.
4 GEORGE WEAVER
Since all ﬁnite and non-zero cardinals and the cardinal
ℵ
0
are second-order chara-terizable, the second-order theory of every countably inﬁnite Dedekind algebra iscategorical. Since there are 2
ℵ
0
diﬀerent conﬁguration signatures of countably inﬁ-nite Dedekind algebras, not all of their theories are ﬁnitely axiomatizable. Sectionseven provides a condition that separates those countably inﬁnite Dedekind alge-bras whose second-order theories are ﬁnitely axiomatizable (and categorical) fromthose whose theories are merely categorical. In fact, something stronger is estab-lished: there is a condition that separates those inﬁnite Dedekind algebras whosesecond-order theories are ﬁnitely axiomatizable and categorical from those whosetheories are not. Consider the countable case and focus on the conﬁguration signa-tures of these algebras, rather than the algebras themselves. For each
n
, the valueof the signature at
n
is a describable cardinal. Some of the signatures are morecomplex than others. For example, some of these functions will only take ﬁnitelymany diﬀerent values. Of these, some are constant valued functions, while othersare not. Roughly speaking, the second-order theory of such a Dedekind algebra iscategorical and ﬁnitely axiomatizable provided that the action of its conﬁgurationsignature is
globally describable
in the sense that there is a second-order formula
φ
(
S,S
′
) such that the sets
B
,
B
′
satisfy
φ
(
S,S
′
) in the Dedekind algebra iﬀ theconﬁguration signature of the algebra at the cardinality of
B
is the cardinality of
B
′
. When the algebra is inﬁnite, the notion of a describable conﬁguration is aslightly more complex.In section eight it is shown that if
A
is any inﬁnite Dedekind algebra whosesecond-order theory is ﬁnitely axiomatizable, then that theory is categorical. It fol-lows that the describability condition mentioned in the above paragraph separatesthose inﬁnite Dedekind algebras whose second-order theories are ﬁnitely axiomati-zable from those whose theories are not.Section nine brieﬂy discusses Dedekind’s concern to produce independent setsof axioms and it is observed that the sets of axioms exhibited in section seven areindependent.Section ten provides an application of Dedekind algebras in set theory: proofsof both the Cantor-Schr¨oder-Bernstein Theorem and Banach’s Theorem. Finally,section eleven outlines how the notion of the conﬁguration of an object can beextended to a class of algebras that includes the Dedekind algebras. Conﬁgurationsignatures can be associated with each member of this class and it can be shownthat the conﬁguration signatures of a member of the class codes the structure of themember. However, the number of types of conﬁgurations is uncountable. Hencesome results from section four fail for this extended class.2.
Configurations
Following Burris and Sankappanavar [1980],
B
=(
B
,
h
B
) is a
mono-unary algebra
where
B
is a non-empty set (
the domain of
B
) and
h
B
is a unary function on
B
. A
Dedekind algebra
is a mono-unary algebra in which the unary function is an injectionon the domain. Since the proper class of Dedekind algebras is a universal Horn class(quasi-variety), it is closed under subalgebras, direct, reduced and ultraproducts,subdirect products, directed unions and direct limits (see Wechler [1992]). However,the class of Dedekind algebras is not closed under homomorphic images.Given a Dedekind algebra
B
,
P
B
is the binary relation on
B
that correspondsto
h
B
. When (
a
,
b
)
∈
P
B
,
a
is said to be
the parent of
b
(in
B
)
. Notice that
a
is
A GENERAL SETTING FOR DEDEKIND’S AXIOMATIZATION OF THE POSITIVE INTEGERS5
the parent of
b
in
B
iﬀ
h
−
1
B
(
b
)=
a
. Since
h
B
is an injection, parents are unique.
b
is a ﬁxed point of
h
B
(i.e.
h
B
(
b
)=
b
) iﬀ
b
is its own parent in
B
. Members of
B
not in
h
B
[
B
] (the image of
B
under
h
B
) have no parents in
B
. Let
A
B
be thetransitive closure of
P
B
(i.e. the smallest transitive binary relation on
B
extending
P
B
). When (
a
,
b
)
∈
A
B
, we say that
a
is an ancestor of
b
(in
B
)
.
a
is an ancestorof
b
iﬀ there is
n
≥
1 such that
h
n
B
(
a
)=
b
.
A
B
(
b
) is
the set of ancestors of
b
(in
B
)
.When
B
is the set of positive integers and
h
B
is the successor function, then
A
B
is the strictly less than relation and for
b
∈
B
,
A
B
(
b
) is the strict initial segmentof
b
.Since
A
B
is transitive, if
a
is an ancestor of
b
, then
A
B
(
a
)
⊆
A
B
(
b
). If
a
hasancestors and
a
=
b
and
A
B
(
a
)
⊆
A
B
(
b
), then
a
is an ancestor of
b
; and if
a
is anancestor of
b
and
A
B
(
a
) is a proper subset of
A
B
(
b
), then
a
is an ancestor of
b
.Finally, since
h
B
is 1-1, if
a
and
b
are diﬀerent ancestors of
c
, then either
a
is anancestor of
b
or
b
is an ancestor of
a
.Given the Dedekind algebra
B
,
A
⊆
B
and
b
∈
B
,
A
is
the conﬁguration of
b
(in
B
) iﬀ
A
is the smallest set such that
b
∈
A
,
A
B
(
b
)
⊆
A
and
A
is closed under
h
B
. Let Σ be the collection of subsets of
B
that include
{
b
}∪
A
B
(
b
) and are closedunder
h
B
.
B
∈
Σ. And, the intersection of Σ is in Σ. Hence, every member of
B
has a unique conﬁguration in
B
. The set
A
is a conﬁguration in
B
provided
A
isthe conﬁguration of some element in
B
. Each conﬁguration is a chain (or Kette)in the sense of Dedekind [1888], but not all chains are conﬁgurations. It is shownbelow that the conﬁgurations in
B
partition
B
.
Lemma 2.1:
Assume that
B
is a Dedekind algebra, that
a
and
b
∈
B
and that
a
is the ancestor of
b
in
B
. Then,
a
and
b
have the same conﬁguration in
B
.
Proof:
Assume that
a
is an ancestor of
b
in
B
. Let
A
′
be the conﬁguration of
a
in
B
and
A
be the conﬁguration of
b
in
B
. Both
A
and
A
′
are closed under
h
B
.It suﬃces to establish the following:(1)
b
∈
A
′
and
A
B
(
b
)
⊆
A
′
; and(2)
a
∈
A
and
A
B
(
a
)
⊆
A
.It follows that
A
⊆
A
′
and
A
′
⊆
A
.By assumption,
A
B
(
a
)
⊆
A
B
(
b
) and
a
∈
A
B
(
b
). Since
A
B
(
b
)
⊆
A
, condition (2)above is satisﬁed. Also by assumption, there is
n
≥
1 such that
h
n
B
(
a
)=
b
.
a
∈
A
′
and
A
′
is closed under
h
B
. Thus,
b
∈
A
′
. Let
c
be any ancestor of
b
in
B
. Either
a
=
c
,
a
is an ancestor of
c
or
c
is an ancestor of
a
. If
c
=
a
or
c
is an ancestor of
a
,then
c
∈
A
′
. Suppose that
a
is an ancestor of
c
. There is
n
≥
1 such that
h
n
B
(
a
)=
c
.
a
∈
A
′
and
A
′
is closed under
h
B
. Hence,
c
∈
A
′
; and (1) above is established.Suppose that
A
is the conﬁguration of
a
in
B
and that
b
∈
A
. Then, either
b
=
a
or
a
is an ancestor of
b
or
b
is an ancestor of
a
. Hence, by Lemma 2.1,
A
is also theconﬁguration of
b
in
B
. The following is immediate.
Lemma 2.2:
Assume that
B
is a Dedekind algebra and that
A
and
A
′
areconﬁgurations in
B
. Then
A
=
A
′
iﬀ
A
∩
A
′
is non-empty.There is another charaterization of the conﬁgurations in
B
.
Lemma 2.3:
Assume that
B
is a Dedekind algebra,
b
∈
B
and
A
⊆
B
. Then,the following are equivalent:

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