The first-order theory of the class
$\mathcal {WO}$
of well-ordered sets
$\langle L,<\rangle $
was developed by Tarski and Mostowski, and an in-depth analysis was finally published by Doner, Mostowski, and Tarski [Reference Doner, Mostowski, Tarski, Macintyre, Pacholski and Paris1]: among other results, they provided an explicit axiomatization for the theory, and proved it decidable. Their main technical tool is a syntactic elimination of quantifiers, which however takes some work to establish, as various somewhat nontrivial properties of Cantor normal forms are definable in the theory after all. Alternatively, by way of hammering nails with a nuke, the decidability of
$\operatorname {\mathrm {Th}}(\mathcal {WO})$
follows from an interpretation of the MSO theory of countable linear orders in the MSO theory of two successors (S2S), which is decidable by a well-known difficult result of Rabin [Reference Rabin5]. Our goal is to point out that basic properties of
$\operatorname {\mathrm {Th}}(\mathcal {WO})$
can be proved easily using ideas from Läuchli and Leonard’s [Reference Läuchli and Leonard4] proof of the decidability of the theory of linear orders. A similar technique was also used by Shelah [Reference Shelah7].
Let
$\mathcal L_<$
denote the set of sentences in the language
$\{<\}$
. The theory of (strict) linear orders is denoted
$\mathsf {LO}$
; the
$\mathcal L_<$
-theory
$\mathsf {TI}$
extends
$\mathsf {LO}$
with the transfinite induction schema
$$\begin{align*}\forall x\:\bigl(\forall y\:(y<x\to\varphi(y))\to\varphi(x)\bigr)\to\forall x\:\varphi(x)\end{align*}$$
for all formulas
$\varphi $
(that may in principle include other free variables as parameters, though the parameter-free version is sufficient for our purposes). We will generally denote a linearly ordered set
$\langle L,<\rangle $
as just L. Given linearly ordered sets I and
$L_i$
for
$i\in I$
, let
$\sum _{i\in I}L_i$
denote the ordered sum with domain
$\{\langle i,x\rangle :i\in I,x\in L_i\}$
and lexicographic order
$$\begin{align*}\langle i,x\rangle<\langle j,y\rangle\iff i<j\text{ or }(i=j\text{ and }x<y).\end{align*}$$
Put
$L\cdot I=\sum _{i\in I}L$
. We write
$L\equiv _kL'$
if
$L\vDash \varphi \iff L'\vDash \varphi $
for all
${\varphi \in \mathcal L_<}$
of quantifier rank
$\operatorname {\mathrm {rk}}(\varphi )\le k$
. It follows from the basic theory of Ehrenfeucht–Fraïssé games (see [Reference Hodges3, Reference Läuchli and Leonard4]) that for each
$k<\omega $
,
$\equiv _k$
has only finitely many equivalence classes as there are only finitely many formulas of rank
$\le k$
up to logical equivalence, and that
$\equiv _k$
preserves sums and products.
Lemma 1. If
$L_i\equiv _kL^{\prime }_i$
for each
$i\in I$
, then
$\sum _{i\in I}L_i\equiv _k\sum _{i\in I}L^{\prime }_i$
. If
$I\equiv _kI'$
, then
$L\cdot I\equiv _kL\cdot I'$
.
Proof Sketch
Duplicator can win the game
$\mathrm {EF}_k\bigl (\sum _{i\in I}L_i,\sum _{i\in I}L^{\prime }_i\bigr )$
by playing auxiliary games
$\mathrm {EF}_k(L_i,L^{\prime }_i)$
for each
$i\in I$
on the side. When Spoiler plays an element in the ith summand, Duplicator simulates it in
$\mathrm {EF}_k(L_i,L^{\prime }_i)$
, finds a response using a fixed winning strategy (given by the assumption
$L_i\equiv _kL^{\prime }_i$
), and plays the corresponding element in the main game. The second assertion of the lemma is similar.
We come to the main theorem. It was originally proved in [Reference Doner, Mostowski, Tarski, Macintyre, Pacholski and Paris1, Theorem 31, Corollaries 30, 32] by tedious quantifier elimination. (The equivalence of (i) and (ii) also follows from Ehrenfeucht [Reference Ehrenfeucht2], who proved by induction on k that ordinals congruent modulo
$\omega ^k$
are
$\equiv _k$
-equivalent.) Instead, we give a short argument inspired by the proof of [Reference Läuchli and Leonard4, Theorem 2] that needs almost no ordinal arithmetic and no explicit EF game strategies.
Theorem 2. The following are equivalent for all
$\varphi \in \mathcal L_<$
:
-
(i)
$\mathcal {WO}\vDash \varphi $
. -
(ii)
$\alpha \vDash \varphi $
for all
$\alpha <\omega ^\omega $
. -
(iii)
$\mathsf {TI}\vdash \varphi $
.
Proof Clearly, (iii)
${}\to {}$
(i)
${}\to {}$
(ii). For (ii)
${}\to {}$
(iii), if
$\mathsf {TI}\nvdash \varphi $
, let L be a countable model of
$\mathsf {TI}+\neg \varphi $
, and
$k=\operatorname {\mathrm {rk}}(\varphi )$
; it suffices to show that there exists
$\alpha <\omega ^\omega $
such that
$L\equiv _k\alpha $
. Put
$$\begin{align*}S=\left\{c\in L:\forall a,b\in L\:\bigl(a<b\le c\to\exists\alpha<\omega^\omega\:[a,b)\equiv_k\alpha\bigr)\right\}.\end{align*}$$
While the definition speaks of half-open intervals
$[a,b)$
, the conclusion also holds for
$[a,b]$
: if
$[a,b)\equiv _k\alpha $
, then
$[a,b]\equiv _k\alpha +1$
by Lemma 1. Clearly, S is an initial segment of L, and
$0\in S$
, where
$0=\min (L)$
(which exists by
$L\vDash \mathsf {TI}$
).
Claim 2.1. S is definable in L.
Proof Since there are only finitely many formulas of rank
$\le k$
up to equivalence, we can form
$\theta _k=\bigwedge \{\theta \in \mathcal L_<:\operatorname {\mathrm {rk}}(\theta )\le k,\forall \alpha <\omega ^\omega \,\alpha \vDash \theta \}$
. Then for any linear order
$L'$
,
$L'\equiv _k\alpha $
for some
$\alpha <\omega ^\omega $
iff
$L'\vDash \theta _k$
. In particular,
$c\in S$
iff
$L\vDash \forall x,y\,\bigl (x<y\le c\to \theta _k^{[x,y)}\bigr )$
, where
$\theta _k^{[x,y)}$
denotes
$\theta _k$
with quantifiers relativized to
$[x,y)$
.
First, assume that S has a largest element, say m. If
$S=L$
, then
${L=[0,m]\equiv _k\alpha} $
for some
$\alpha <\omega ^\omega $
, and we are done. Otherwise, we will derive a contradiction by showing that the successor of m (which exists by
$\mathsf {TI}$
), denoted c, belongs to S. Indeed, if
$a<b\le c$
, then either
$b\le m$
, in which case
$[a,b)\equiv _k\alpha $
for some
$\alpha <\omega ^\omega $
as
$m\in S$
, or
$b=c$
, in which case
$[a,b)=[a,m]\equiv _k\alpha $
for some
$\alpha <\omega ^\omega $
as well.
Consequently, we may assume that S has no largest element. Put
$S_{\ge a}=\{{b\in S}:{b\ge a}\}$
.
Claim 2.2. For every
$a\in S$
, there is
$\alpha <\omega ^\omega $
such that
$S_{\ge a}\equiv _k\alpha $
.
Proof We use the idea of [Reference Läuchli and Leonard4, Lemma 8]. Let
$a<a_0<a_1<a_2<\cdots $
be a cofinal sequence in S. For each
$n<m<\omega $
, let
$t(\{n,m\})=\min \{{\alpha <\omega ^\omega }:[a_n,a_m)\equiv _k\alpha \}$
. Since
$\equiv _k$
has only finitely many equivalence classes, t is a coloring of pairs of natural numbers by finitely many colors; by Ramsey’s theorem, there is
$\beta <\omega ^\omega $
and an infinite
$H\subseteq \omega $
such that
$t(\{n,m\})=\beta $
for all
$n,m\in H$
,
$n\ne m$
. Let
$\{b_n:n<\omega \}$
be the increasing enumeration of
$\{a_n:n\in H\}$
, and
$\alpha <\omega ^\omega $
be such that
$[a,b_0)\equiv _k\alpha $
. Then,
$S_{\ge a}=[a,b_0)+\sum _{n<\omega }[b_n,b_{n+1})\equiv _k\alpha +\beta \cdot \omega <\omega ^\omega $
by Lemma 1.
Now, if
$S=L$
, then
$L=S_{\ge 0}\equiv _k\alpha $
for some
$\alpha <\omega ^\omega $
by Claim 2.2. Otherwise, there exists
$c=\min (L\smallsetminus S)$
by Claim 2.1 and
$L\vDash \mathsf {TI}$
. We again derive a contradiction by showing
$c\in S$
: if
$a<b\le c$
, then either
$b<c$
and
$[a,b)\equiv _k\alpha $
for some
$\alpha <\omega ^\omega $
as
$b\in S$
, or
$b=c$
and
$[a,b)=S_{\ge a}\equiv _k\alpha $
for some
$\alpha <\omega ^\omega $
by Claim 2.2.
We have so far not actually used any results of Läuchli and Leonard [Reference Läuchli and Leonard4], only their methods. But we will do so now: in order to prove the decidability of
$\operatorname {\mathrm {Th}}(\mathcal {WO})$
, we need the following lemma.
Lemma 3. The relation
$\{\langle \alpha ,\varphi \rangle \in \omega ^\omega \times \mathcal L_<:\alpha \vDash \varphi \}$
is recursively enumerable (whence decidable).
Here, we assume
$\alpha <\omega ^\omega $
is represented by a finite string describing its Cantor normal form (CNF) in a natural way. Lemma 3 is a special case of [Reference Läuchli and Leonard4, Theorem 1]: more generally, Läuchli and Leonard prove uniform decidability of linear order types described by “terms” using a constant
$1$
, a binary function
$+$
, unary functions
$x\cdot \omega $
and
$x\cdot \omega ^*$
, and a certain variable-arity shuffle operation. It is easy to see that the CNF of an ordinal
$\alpha <\omega ^\omega $
can be transformed to such a term using
$1$
,
$+$
, and
$x\cdot \omega $
, hence Lemma 3 follows.
We include a proof of Lemma 3 to make the paper more self-contained. It turns out that for well orders, it is more convenient to consider terms using
$1$
,
$+$
, and
$\omega \cdot x$
rather than
$x\cdot \omega $
: then we can directly axiomatize the theory by a finite sentence without expanding the language with extra predicates as in [Reference Läuchli and Leonard4]. This argument can be found, e.g., in Rosenstein [Reference Rosenstein6].
Proof of Lemma 3
Given
$\alpha <\omega ^\omega $
, we can compute an
$\mathcal L_<$
-sentence
$T_\alpha $
such that
$\operatorname {\mathrm {Th}}(\alpha ,<)=\mathsf {LO}+T_\alpha $
by induction on
$\alpha $
as follows. We can take
$\forall x,y\,x=y$
for
$T_1$
. It is well known that
$T_\omega $
can be defined by axioms postulating that a least element exists, every element has a successor, and every non-minimal element has a predecessor. If
$T_\alpha $
and
$T_\beta $
are already constructed, let
$T_{\alpha +\beta }$
be
$\exists x\,\bigl (T_\alpha ^{<x}\land T_\beta ^{\ge x}\bigr )$
, where
$T_\alpha ^{<x}$
denotes the sentence
$T_\alpha $
with all quantifiers relativized to
$(-\infty ,x)=\{y:y<x\}$
, and similarly for
$T_\beta ^{\ge x}$
: in particular, any
$L\vDash \mathsf {LO}+T_{\alpha +\beta }$
contains an element
$a\in L$
such that
$(-\infty ,a)\equiv \alpha $
and
$[a,\infty )\equiv \beta $
, which implies
$L\equiv \alpha +\beta $
by Lemma 1.
Finally, we consider
$\omega \cdot \alpha $
for a limit
$\alpha $
. Let
$\lambda (x)$
denote the formula
$\forall y<x\,\exists z<x\,y<x$
, meaning “x is not a successor.” We define
$T_{\omega \cdot \alpha }$
as the conjunction of
$T_\alpha ^\lambda $
and axioms postulating that for each x,
$\max \{{y\le x}:\lambda (y)\}$
and
$\min \{{y>x}:\lambda (y)\}$
exist. Clearly,
$\omega \cdot \alpha \vDash T_{\omega \cdot \alpha }$
. Conversely, if
$L\vDash \mathsf {LO}+T_{\omega \cdot \alpha }$
, then
$L^\lambda :=\{x\in L:L\vDash \lambda (x)\}\vDash T_\alpha $
, thus
$L^\lambda \equiv \alpha $
, and
$L=\sum _{x\in L^\lambda }[x,x^+)$
, where
$x^+=\min \{{y>x}:y\in L^\lambda \}$
. It is easy to see that
$[x,x^+)\vDash T_\omega $
for each x, thus
$L\equiv \sum _{x\in L^\lambda }\omega =\omega \cdot L^\lambda \equiv \omega \cdot \alpha $
by Lemma 1.
The following consequence is Theorem 33 of [Reference Doner, Mostowski, Tarski, Macintyre, Pacholski and Paris1].
Theorem 4. The theory
$\operatorname {\mathrm {Th}}(\mathcal {WO})=\mathsf {TI}$
is decidable.
Proof
$\{\varphi \in \mathcal L_<:\mathsf {TI}\vdash \varphi \}$
is recursively enumerable as
$\mathsf {TI}$
is recursively axiomatized; by Theorem 2 and Lemma 3,
$\{{\varphi \in \mathcal L_<}:{\mathsf {TI}\nvdash \varphi }\}=\{{\varphi \in \mathcal L_<}:\exists {\alpha <\omega ^\omega }\,{\alpha \vDash \neg \varphi }\}$
is also recursively enumerable.
Acknowledgments
I am grateful to the anonymous reviewers for helpful suggestions, especially the simplification of the proof of Lemma 3. The main argument was developed while teaching a course on “Decidable theories” at the Charles University in Prague (https://users.math.cas.cz/~jerabek/teaching/decidable.html).
Funding
The author was supported by the Czech Academy of Sciences (RVO 67985840) and GA ČR project 23-04825S.
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