Expressibilityin First-order Logic

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```					6     Expressibility in First-order Logic
6.1    First and Higher-order logic
Clearly, ﬁrst-order logic has greater expressive power than propositional
logic. Many arguments and concepts which can be expressed with the ap-
paratus of predicates, relations and quantiﬁers cannot be expressed without
these, using only truth-functional sentential connectives. First-order logic
with identity has even greater expressive power.
Consider a series of domains: a domain of individuals, a domain of prop-
erties of those individuals (we call these properties of order 1), a domain of
properties of properties of individuals (we call these properties of order 2),
and so on. A logic is said to be of zero order if it contains no quantiﬁers. It
is of ﬁrst order if it contains quantiﬁers ranging only over individuals. It is of
second order if it contains quantiﬁers ranging over properties of individuals
(and doubtless, individuals too). It is of nth order if it contains quantiﬁers
ranging over properties of orders less than n. ω-order logic (or ﬁnite type
theory) contains quantiﬁers ranging over properties of every order. One can
even consider type theories allowing transﬁnite orders. A wﬀ of nth-order
logic which is not a wﬀ of any logic of order < n is called an nth-order wﬀ.
The wﬀ
(∀x)(∀y)(∀z)(Rxy ∧ Ryz ⊃ Rxz)
is a typical wﬀ of ﬁrst-order logic. It expresses the assertion that R is a
transitive binary relation. But
(∀P )(P (0) ∧ (∀x)(P (x) ⊃ P (x )) ⊃ (∀x)P x)
(where x is the successor of x) is a wﬀ of second-order logic. It is second-
order because it quantiﬁes over properties. It expresses Peano’s ﬁfth postu-
late, his induction axiom. Peano set out to characterise arithmetic by a set
of axioms or postulates. Some of these axioms may easily be expressed by
ﬁrst-order wﬀs with identity: e.g.,
(∀x)¬(x = 0),
i.e., 0 is not a successor. But the principle of induction cannot be so ex-
pressed. What Peano wanted to say was that every number has every prop-
erty of 0 which belongs to the successor of every number which has it. But
as we have seen, this is a second-order wﬀ, quantifying over properties of
individuals. We might try to express it by a ﬁrst-order schema:
α(0/x) ∧ (∀x)(α ⊃ α(x /x)) ⊃ (∀x)α,
that is, an inﬁnite set of wﬀs, one for every ﬁrst-order wﬀ α. However, we
will ﬁnd that this does not fully capture the principle of induction. Induc-
tion is not ﬁrst-order expressible. We need to quantify over all ﬁrst-order
properties, and not all of these are expressible in ﬁrst-order wﬀs, α. For
there are uncountably many ﬁrst-order properties, but only countably many
wﬀs.
So even at the level of ﬁrst- versus second-order logic, interesting ques-

50
Deﬁnition 6.1        • A property f is expressible in a logic L if (and only
if ) there is a set X of wﬀs of L every model of which has the property
f.

• f is expressible by ‘φ’ (‘φ’ being a predicate of the language, f and
‘φ’ being n-place) if there is a set of wﬀs containing ‘φ’ in all of whose
models ‘φ’ is interpreted by f .

There are six important features of ﬁrst-order logic which are not shared by
second and higher-order logic. Several of these features entail restrictions
or limitations on the expressive power of any logic which has them, so they
are often referred to as limitative results:

• Compactness

• Completeness

• Semi-decidability

• the L¨wenheim property
o

• the Tarski property

• the Lindstr¨m property.
o

6.2   Compactness
First-order logic is compact; second-order logic is not. Compactness is prop-
erly expressed using the notion of a model. Recall that a set of wﬀs X has
a model if there is some interpretation under which every member of X is
true. A logic is compact if, for every set X of wﬀs of the language of that
logic, X has a model iﬀ every ﬁnite subset of X has a model.
This is clearly trivial if X is a ﬁnite set of wﬀs. It becomes interesting
when we consider inﬁnite sets of wﬀs. It is also trivial from left to right. If
a set of wﬀs has a model, then every ﬁnite subset of it has the same model.
But from right to left it is of deep signiﬁcance. It can reveal the existence
of unexpected models. We can show this by a famous example.
Example 1
Arithmetic has a standard model. It is the set of natural numbers, which
(after §4) we can call ω, ordered by <, on which operations of addition,
multiplication and exponentiation are deﬁned. Let X consist of the set of
all wﬀs of ﬁrst-order logic which are true in ω. That is, take a language
of ﬁrst-order logic with identity, with an individual constant 0, a one-place
successor operation, , three two-place functions, add, mult and exp, and
a two-place relation <; we can formulate a large (clearly inﬁnite) set of
wﬀs interpreted in ω, where the constant 0 is mapped to zero, the successor
and arithmetical operations (functions) are interpreted by the corresponding
operations on ω, and < is interpreted by the order on ω. Let the term ‘n’
be shorthand for 0 ... (with n s), the nth successor of 0. X consists of all
such wﬀs which are true on this interpretation.

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We now apply Compactness. First, we augment our language by a new
constant, d. Consider the sets of wﬀs
X0 = X
X1 = X0 ∪ {d = 0}
X2 = X1 ∪ {d = 1}
...
Xn+1 = Xn ∪ {d = n}
...
Xω =         Xn
n<ω
In other words, we augment X by an inﬁnite set of wﬀs of the form d = n.
Consider any ﬁnite subset Y of Xω . Only ﬁnitely many wﬀs of the form
d = n will be in Y . Hence, for some k, no wﬀ of the form d = n for n ≥ k
is in Y . So ω provides a model for Y , where d is interpreted as k. Hence by
Compactness, Xω has a model. But ω cannot be a model for Xω , for there
is no member of ω to interpret d. For if we let n, say, model d, then the wﬀ
d = n will be false under the interpretation; yet d = n is a member of Xω ,
so then not every member of Xω will be true under this interpretation.
So Xω has a model, but that model is not ω. Moreover, X is a subset of
Xω , so this new model is also a model of X. So X has a model other than
ω. But X consists of all ﬁrst-order truths about arithmetic. So the set of
all ﬁrst-order truths about arithmetic does not characterise ω categorically.
Deﬁnition 6.2 A set of wﬀs is categorical if all its models are isomorphic,
that is, have the same structure, i.e., are essentially the same bar renaming
of elements.
We will show that this new model of X is not isomorphic to ω. So X is not
categorical.
2

ω is called the standard model of arithmetic. We now see that there
is also (at least one) non-standard model of arithmetic. (In fact, there are
uncountably many.) What does this non-standard model look like? Well, it
must have elements corresponding to 0, 1, 2 and so on. So it starts like ω:
0 1 2
.   .   .
ω

However, it must also contain some object, not in ω, which interprets d.
Let’s call it d too. Now X contains all (ﬁrst-order) truths of arithmetic.
Among these is
(∀x)(x = 0 ⊃ x > 0).
So d > 0. But d = n for any n < ω. So d is, in some sense, “inﬁnite”, i.e.
comes after all the ﬁnite numbers:
0 1                                                d
. . .                                              .
ω

Moreover, among X we ﬁnd
(∀x)(∃y)y = x

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and
(∀x)(x = 0 ⊃ (∃y)x = y ).
In other words, d has both a successor and an immediate predecessor, and
those in turn have successors and predecessors. So d belongs to a doubly
inﬁnite array (in fact, a copy of the integers) following ω:
0 1 2                                      d−1   d d+1
.   .   .                                    .   .    .
ω

We also have
(∀x)(∃y)y = 2x
and
(∀x)(∃y)(x = 2y ∨ x = 2y + 1).
So there are elements 2d > d and d (or d−1 ) < d. Suppose 2d = d + n, for
2      2
some ﬁnite n. Then d = n. That is, if 2d were ﬁnitely accessible from d,
d would itself be ﬁnite. So 2d is not in ω nor in the doubly inﬁnite array
containing d. That is, there are yet more sections of “inﬁnite” numbers:
.
.

d
2
.   .    .
.
.
0 1 2                                      d−1   d    d+1
.   .   .                                    .   .    .
ω
.
.
2d
.   .    .

.
.

And so it goes on. In fact, the model consists of an initial section like the
natural numbers, followed by a collection of copies of the integers, indeed,
as many copies of the integers as there are rationals (for between d and 2d
there is 3d and so on). Recall that ω ∗ is (the order-type of) the mirror image
2
of ω. So the integers have order-type ω ∗ + ω. The order-type of the ratio-
nals is η. Hence the order-type of this non-standard model of arithmetic is
ω + (ω ∗ + ω)η. (This order-type is not well-ordered—it is not an ordinal.)
This model, we will see later, is not the only non-standard model of
arithmetic. But in fact it can be shown that any countable non-standard
model of arithmetic has this order-type. (Any countable collection of count-
able sets is countable; the above model augments ω by a countable collection
(of the order-type of the rationals) of countable sets (of order-type of the
integers). So it is countable.)
This non-standard model of arithmetic contains elements which are in
a certain sense inﬁnite. This is because the notion ‘ﬁnite’ is not ﬁrst-order
expressible. There is no wﬀ of ﬁrst-order logic, or any (even inﬁnite) set of
ﬁrst-order wﬀs all of whose models are ﬁnite. We can see this from the next
example:

53
Example 2
If a set X of wﬀs of ﬁrst-order logic has arbitrarily large ﬁnite models,
it has an inﬁnite model. The method of proof is as in Example 1. This time
we add inﬁnitely many new constants {dn : n < ω} to our language, and
augment X successively:
Y0 = X
Y1 = Y0 ∪ {d0 = d1 }
Y2 = Y1 ∪ {d0 = d2 , d1 = d2 }
...
Yn+1 = Yn ∪ {d0 = dn+1 , d1 = dn+1 , . . . , dn = dn+1 }
...
Yω =         Yn
n<ω

In other words, Yω extends X by adding all wﬀs of the form di = dj , where
i = j.
The reasoning is as before. Take any ﬁnite subset Z of Yω . Then Z is a
subset of Yk for some k. Recall that X has arbitrarily large ﬁnite models.
Take a ﬁnite model of X with k+1 elements. Assign d0 , d1 , . . . , dk to diﬀerent
elements of X, and assign di , i > k, arbitrarily (so each such di is assigned
to the same element as some dj , j ≤ k). Then each of ‘di = dj ’, if i, j ≤ k, is
true (although many of di = dj , i > k or j > k, are false). Hence this model
is also a model of Yk , and so of Z. That is, each ﬁnite subset of Yω has a
model. So by Compactness, Yω has a model. However, this model cannot
be ﬁnite, for Yω contains every wﬀ of the form di = dj , i = j. That is, Yω
has an inﬁnite model. But X is a subset of Yω , and so X has an inﬁnite
model.
So no wﬀ, or set of wﬀs, of ﬁrst-order logic, has models all of which are
ﬁnite, though of arbitrary ﬁnite size. We must be clear what this means.
First, note that there certainly are wﬀs (and sets of wﬀs) all of whose models
are ﬁnite. For example, all models of the wﬀ

(∃x)(∀y)x = y

have exactly one member. All models of

(∃x)(∃y)(x = y ∧ (∀z)(x = z ∨ y = z))

have exactly two members; and so on. So it is quite clearly possible within
ﬁrst-order logic (with identity) to capture the notions 1, 2, 3 and so on, i.e.,
particular ﬁnite sizes. What is not possible is to capture the notion ‘ﬁnite’,
that is, an arbitrary ﬁnite size.
Secondly, note (contra Forbes, Modern Logic p. 289) that one can capture
the notion ‘inﬁnite’. The set of wﬀs

{(∀x)¬Rxx, (∀x)(∃y)Rxy, (∀x)(∀y)(∀z)(Rxy ∧ Ryz ⊃ Rxz)}

has only inﬁnite models. No ﬁnite model could make all these wﬀs true.

54
Thirdly, although we have cast these considerations in terms of the car-
dinality of the whole model, one could make the same points about the
extension within the model of a particular predicate. No set of wﬀs can
ensure that, e.g., the extension of the predicate ‘P x’ is ﬁnite, though of
arbitrary ﬁnite size. We can ensure that it is a singleton, by the wﬀ

(∃x)(P x ∧ (∀y)(P y ⊃ x = y)),

or that it has 2, 3 or any particular ﬁnite number of members, or that it is
inﬁnite—by similarly relativising the above sets of wﬀs to P x. But no set
of wﬀs can ensure that it is only ﬁnite, and so, we say, the notion ‘ﬁnite’ is
not ﬁrst-order expressible.
But ‘ﬁnite’ is second-order expressible. Recall the notion of ‘well-ordered’
from §4. This also is not ﬁrst-order expressible. But it is second-order
expressible, by the wﬀ

(∀P )((∃x)P x ⊃ (∃x)(P x ∧ (∀y)(Ryx ⊃ ¬P y)),

where R expresses a (transitive and anti-symmetric) ordering relation. That
is, any non-empty collection (deﬁned by) P has a least member under the
ordering. Note the quantiﬁcation over all properties of individuals. Then
a predicate P has a ﬁnite extension if every ordering relation on P is well-
ordered.

6.3   Completeness
We have not said why ﬁrst-order logic is compact—we have only remarked
on the consequences for expressibility. The usual proof of compactness infer
it as a corollary of completeness (or rather, of the Existence Lemma), and
takes the following form: suppose X has no model. Then (as in Lemma 5.10)
X is inconsistent (in some logic L), that is, X L ⊥. But proofs are ﬁnite.
So for some ﬁnite X ⊆ X, X L ⊥. So for some ﬁnite X ⊆ X, X has
no model. Contraposing, if every ﬁnite subset of X has a model, X has a
model.
The Henkin proof of the Existence Lemma made crucial use of Lin-
denbaum’s Lemma. Although it is not usual to make explicit mention of
this, Lindenbaum’s Lemma invokes an implicit appeal to a result (or princi-
ple) known as Zorn’s Lemma, itself a version or equivalent of the Axiom of
Choice. This notorious principle lay hidden in many mathematical proofs
until its presence was recognised only at the start of the twentieth century.

Postulate 6.1 (The Axiom of Choice) Given any collection of non-empty
sets, there is a choice function which “chooses” an element from each set.

This innocuous-sounding principle is amazingly strong and indeed, inde-
pendent of the other axioms of set theory, as ﬁnally shown by Paul Cohen
around 1960. It has many equivalents, one of which is Zorn’s Lemma:

Postulate 6.2 (Zorn’s Lemma) If every linearly ordered subset (i.e., chain)
of a set has an upper bound in the set, then the set has a maximal element.

55
The use of Zorn’s Lemma in Lindenbaum’s Lemma is to assert the existence
of the maximal set which we need in order to construct the model.3
We also implicitly invoked the Axiom of Choice in §4.6 when we claimed
that the cardinals were initial ordinals. The ordinals are intrinsically or-
dered; the cardinals are not. The Axiom of Choice is equivalent to the
Well-Ordering Principle, that every set can be well ordered, from which it
follows that the cardinals (as the sizes of those sets) can be identiﬁed with
certain ordinals.
it come about that compactness fails for second-order logic? For it seems
to depend only on the ﬁniteness of proof, and are not second-order proofs
ﬁnite too? But another assumption was in play in the above proof, namely,
that every consistent set has a model. This is essentially the completeness
theorem. Suppose that every consistent set has a model (where consistency
is deﬁned, as in Deﬁnition 5.1, relative to some system of proof, L), and
suppose X |= A, i.e., A is a logical consequence of X. Then X {¬A}
has no model, so X {¬A} L ⊥, whence X L A. That is, every logical
consequence of X may be derived from X in L, i.e., L is complete. This was
the argument in Theorem 5.3.
Second-order logic is incomplete. There is no proof procedure which can
capture second-order logical consequence. Second-order logical consequence
is essentially unaxiomatisable.
In 1950, Leon Henkin published an article with the strange title “Com-
pleteness in the theory of types”; strange, because if second-order logic is
incomplete, so much the more so is type theory (for type theory is eﬀectively
ω-order logic). However, Henkin’s result did not contest the incompleteness
of higher-order logic. What Henkin observed was that one can deﬁne a dif-
ferent notion of higher-order consequence, a weaker notion that essentially
reduces it to a ﬁrst-order theory, and so regains completeness. The idea is
this: in second-order logic, there are quantiﬁers ranging over all properties
of individuals; in third-order, over all properties of properties of individuals;
and so on. Henkin introduced the idea of general models, models which al-
low domains of interpretation in which the quantiﬁers range over arbitrary
collections (subsets) of properties, not necessarily over all. What Henkin
called “standard models”, where all properties must be considered, are then
a special case of what he called “general models”, where only some arbitrary
subset need be taken in the model. Henkin-validity is truth in all general
models. General models increase the chance of ﬁnding a counterexample to
a sequent, and so fewer sequents turn out Henkin-valid. Indeed, so many
fewer sequents are valid in the general sense that the proof procedures turn
out to be complete.
What Henkin’s observation amounted to was to construe higher-order
logic as a many-sorted ﬁrst-order theory, where the higher-order variables
are construed as set variables. But thereby one loses expressibility, for ﬁrst-
order set theory is subject to the limitative constraints we have listed, and
ultimately to Skolem’s paradox, which we consider later.
3
See, e.g., J. Zalabardo, Introduction to the Theory of Logic, Lemma 6.110.

56
6.4      Semi-decidability
First-order logic is not decidable. That is, there is no eﬀective or mechanical
procedure or test, guaranteed to terminate in a ﬁnite time with a decision
as to whether a given sequent or wﬀ is valid. There is such a test for
propositional logic, namely, truth-tables, but there is none for ﬁrst-order
logic.
Nonetheless, there is a positive test for validity in ﬁrst-order logic. That
is, there is an eﬀective procedure which, if given a valid sequent, will ter-
minate ﬁnitely in conﬁrmation that it is valid. The problem, of course, is
that, given an invalid sequent, it may not terminate. There is no negative
test for invalidity. We say that ﬁrst-order logic is semi-decidable.

Proposition 6.1 (Church’s Thesis) The informal notion of eﬀective proce-
dure should be identiﬁed with the formal notion of a recursive function.

Deﬁnition 6.3      • A recursive function is one deﬁnable from the initial
functions (successor, constant and identity) by composition, recursion
and minimization.

• A set if recursive if its characteristic function is a recursive function.

• A set if recursively enumerable (r.e.) if it is the range of a recur-
sive function.

• A logic is decidable if its consequence relation is recursive; it is semi-
decidable if its consequence relation is r.e.

It follows that a set is recursive iﬀ it and its complement are r.e.
One can see the semi-decidability of ﬁrst-order logic (that is, that conse-
quence in ﬁrst-order logic is recursively enumerable) by considering semantic
tableaux or trees, and their use in proving the completeness of ﬁrst-order
logic. There are two main methods used for proving completeness, that is,
showing that every consistent set has a model: the Henkin method, which we
looked at in §5 above, and proceeds by Lindenbaum’s Lemma, extending a
simply consistent set to a maximally consistent (in general, well-rounded4 )
o
one; and what is often called the G¨del method, which proceeds by con-
structing a tree or tableau on a given wﬀ or sequent. This wﬀ or sequent
lies at the root of the tree, and above (or more usually, below) it are placed
subformulae or sequents of subformulae. Initial segments of this tree, tracing
out a path from the root to a particular wﬀ or sequent, are called branches.
Wﬀs or sequents above/below which no further wﬀ or sequent may be placed,
that is, where the tree terminates, are called leaves.
In the case of propositional logic, the complexity of the wﬀ or sequent is
reduced at each stage of construction. Consequently, the tree is always ﬁnite
for propositional logic. Two cases arise: how they are described depends
somewhat on whether it is a tree or tableau, and whether of wﬀs or sequents.
But essentially, if the root wﬀ or sequent is valid, the tree terminates in
4
See J. Zalabardo, Introduction to the Theory of Logic, §5.4.

57
such a way that it is, or can easily be converted to, a proof; whereas if
the root wﬀ or sequent is invalid, the tree terminates in such a way that
a counterexample (an interpretation invalidating the wﬀ or sequent) may
be read oﬀ. Completeness and decidability of propositional logic follow
immediately.
In ﬁrst order logic, however, a third case is possible. For the complex-
ity of the wﬀ or sequent is not always reducible at each stage. It never
increases, but for certain quantiﬁed wﬀs it is not decreased either. Con-
sequently, the construction of the tree may not terminate. As before, if
it does terminate, then, if valid, a proof is forthcoming, and if invalid, a
counterexample. The crucial fact is that the third case only arises if the wﬀ
or sequent is invalid. The non-terminating tree delivers a counterexample.
(Once again, this requires Zorn’s Lemma.) Hence, if the sequent is invalid,
there is a counterexample; and if it is valid, there is a proof. So the calculus
is complete. Moreover, if it is valid, the procedure of constructing the tree
terminates, ensuring the establishment, in a ﬁnite time, of that fact. So
ﬁrst-order logic is semi-decidable.
Semi-decidability goes the way of completeness in higher-order logic.
Clearly, given that higher-order logic is incomplete, no terminating proce-
dure for valid sequents can be constructed. There is no positive or negative
test for validity. Validity in higher-order logic is not r.e.

6.5        o
The L¨wenheim property
We have seen that although ‘inﬁnite’ can be expressed in ﬁrst-order logic,
‘ﬁnite’ cannot. We might then wonder whether ‘uncountably inﬁnite’ can
o
be so expressed. The L¨wenheim property possessed by ﬁrst-order logic
shows that it cannot be: that is, that no set of wﬀs of ﬁrst-order logic has
only uncountable models; similarly, that no set of wﬀs can so characterise a
predicate P x that its extension in every model is uncountable.
o
L¨wenheim’s result of 1915 in which this was shown was later extended
and given a more rigorous proof by Skolem in 1919 and 1922. Hence the
result recording this feature of ﬁrst-order logic is commonly known as the
o
(Downward) L¨wenheim-Skolem theorem. It states that every set of wﬀs of
ﬁrst-order logic which has a model (for short, is satisﬁable) has a countable
model. This is a corollary of what we proved in § 5 above: the model we
deﬁned had as its domain the countable set of terms (variables) of the lan-
guage. Certain extra subtleties can be added: if the language lacks identity,
then every satisﬁable set of wﬀs has a countably inﬁnite model. The ad-
dition of identity can force every model to be ﬁnite (as we have seen), so,
with identity present, we have that any satisﬁable set of wﬀs has a (ﬁnite or
inﬁnite) countable model.
It is presupposed here that the language itself is countable, that is, that
there are at most countably many constants, variables and predicates, and
so that there are countably many wﬀs. Uncountable (ﬁrst and higher-order)
languages can be constructed, usually by the addition of uncountably many
(individual) constants. A language with κ non-logical symbols is said to have
o
cardinality κ. The (downward) L¨wenheim-Skolem theorem for a ﬁrst-order

58
language of cardinality κ says that every satisﬁable set of wﬀs has a model
of cardinality ≤ κ. Let us say that a logic has the L¨wenheim property if
o
o
it satisﬁes the downward L¨wenheim-Skolem theorem. Second-order logic
o
lacks the L¨wenheim property.
That ‘uncountable’ cannot be expressed (in a countable ﬁrst-order lan-
guage) follows immediately. For the theorem clearly states that no set of
wﬀs has only uncountable models. This fact gives rise to what is known as
o
Cantor’s theorem (indeed, K¨nig’s theorem) can be proved in ﬁrst-order
ZF, Zermelo-Fraenkel set theory. This is a theory of sets based on ﬁrst-order
logic, with a vocabulary of (at least) variables, logical constants, identity and
‘∈’, for set membership. Axioms are laid down to characterise ‘∈’. Cantor’s
theorem (Theorem 4.2) states that no set is equinumerous with its power
set. In particular, ω (whose existence is asserted in one of the axioms) is
strictly smaller than its power set. So a theorem of (ﬁrst-order) ZF asserts
the existence of an uncountable set.
In the standard model of ZF, the power set of ω, P(ω), is indeed un-
countable. The standard model is taken to be the real universe of sets. But
o
by the L¨wenheim-Skolem theorem, any set of wﬀs (such as ZF) with a
model has a countable model. So there is a model of ZF in which the power
set of ω is countable.
This is thought to be paradoxical. Skolem believed it showed that there
are no really uncountable sets, but that countability is relative to a theory.
However, the obvious explanation would seem to be that the notion ‘un-
countable’ can no more be expressed in ﬁrst-order terms than can ‘ﬁnite’.
The reason the countable model of ZF makes the axioms and theorems of
ZF true—in particular, Cantor’s theorem—is that there is so little in it (and
in its “power set of ω”) that it does not contain the function which in fact
enumerates its power set of ω. There is such a function; that is why it is
countable. But that function is not in the model, so the model “thinks” the
set is uncountable, and so makes Cantor’s theorem true.

6.6   The Tarski property
Not only is ‘uncountable’ not expressible in ﬁrst-order terms; neither is
o
‘countable’. This is shown by the so-called Upward L¨wenheim-Skolem the-
orem. Actually, this is a strange name for it, since it was undreamed of by
o
L¨wenheim, and (since, as we’ve seen, Skolem thought no sets were really
uncountable) dismissed as nonsense by Skolem. It was, it seems, ﬁrst stated
by Tarski. Let us say that a logic has the Tarski property if it satisﬁes
o
the upward L¨wenheim-Skolem theorem, which says that any set of wﬀs
on a language of cardinality κ which has an inﬁnite model, has a model
of every inﬁnite cardinality ≥ κ. First-order logic has the Tarski property;
second-order logic lacks it.
It follows immediately that ‘countable’ is not expressible in ﬁrst-order
logic, since by the theorem no set of wﬀs of ﬁrst-order logic has only count-
able models. Again, as with ‘ﬁnite’, we need to put this precisely. Clearly,
there are sets of wﬀs with, for example, only models of cardinality 1, and so

59
on for every particular ﬁnite cardinality. So there are sets of wﬀs with mod-
els of a particular countable cardinality. But ‘countable’ itself should allow
models of any countable cardinality. But any set of wﬀs with a countably
inﬁnite model has uncountable models. Hence ‘countable’ is not ﬁrst-order
expressible.
Recall the set X of all ﬁrst-order wﬀs true in ω. We saw above that
this set has non-isomorphic countable models, the standard model, ω, and
non-standard ones of order-type ω + (ω ∗ + ω)η. The Upward L¨wenheim-
o
Skolem theorem shows that it also has uncountable models. Recall the
notion ‘categorical’ from § 4.

Deﬁnition 6.4 A set of wﬀs is κ-categorical, for κ a (ﬁnite or inﬁnite)
cardinal, if all its models of cardinality κ are isomorphic.

Because ﬁrst-order logic has the Tarski property, the only categorical sets of
wﬀs are ones all of whose models are ﬁnite—and by Compactness, no bigger
than some ﬁxed ﬁnite cardinality. Nonetheless, there are ℵ0 -categorical sets
of ﬁrst-order wﬀs. But the set of ﬁrst-order truths of arithmetic is not one
of them.
‘Countable’ and ‘uncountable’ are second-order expressible, for second-
o
order logic lacks both the L¨wenheim and the Tarski properties. In fact, the
set of second-order wﬀs true in ω is categorical.

6.7              o
The Lindstr¨m property
o
The Lindstr¨m property is not a new property, diﬀerent from those we have
o
met before, but a combination of them. Lindstr¨m proved that there was a
certain combination of properties of ﬁrst-order theories such that any wﬀ of
a logic possessing this combination of properties has exactly the same models
as some ﬁrst order wﬀ. He was therefore led to claim that this combination
o
of properties, the Lindstr¨m property, characterises ﬁrst-order logic. Any
logic with this property, or set of properties, is essentially ﬁrst-order (or
elementary).
The combination is this:

o
Deﬁnition 6.5 A logic has the Lindstr¨m property if it

• has the L¨wenheim property and
o

• either has the Tarski property or is compact.

As we have seen, second-order logic, and logics of higher-order, lack these
o
properties. Lindstr¨m’s interest in this combination of properties is revealed
in:

o
Theorem 6.1 (Lindstr¨m’s Theorem) Any set of wﬀs of a logic with the
o
Lindstr¨m property has the same models as some set of ﬁrst-order wﬀs.

o
Thus the Lindstr¨m property seems to capture something about the expres-
sive power (what models it characterises) of a logic.

60
If ﬁrst-order logic is so impoverished in expressive power, while even
second-order logic is so much stronger, why have we, and do others, concen-
trated so exclusively on the weaker theory? The answer lies in that phrase,
“so much stronger”. Second-order logic is “too hot to handle”. Consider, for
example, the independence results of § 4: GCH is not decided by ZF. ZF
is a ﬁrst-order theory. Consider its extension to ZF2 , a second-order theory
replacing ZF’s ﬁrst-order axiom schemata by second-order axioms quanti-
fying over all properties. Adding the denial of the existence of certain large
cardinals (really large cardinals), ZF2 is categorical. In particular, it decides
CH. But ZF2 is so strong that we have no way of knowing which way it de-
cides it. Too little is known about second-order set theory, or second-order
logic. (But see Stewart Shapiro’s Foundations without Foundationalism.)
What ﬁrst-order logic is poor in, is distinguishing one inﬁnite cardi-
nal from another. This is ironic, given modern logic’s origins in Frege’s,
Dedekind’s, Peano’s and Russell’s work. However, for them logic was not
restricted to ﬁrst-order. That restriction came in during the 1930s and later
o
in response to G¨del’s incompleteness theorem and other results. But for
other purposes, the expressive power of ﬁrst-order logic is impressive. Com-
pactness, completeness and semi-decidability are advantageous from this
perspective, making the logic tractable. Even the downward L¨wenheim-o
Skolem theorem has its merits, in supplying manageable countable models
for analytical treatment. Once we are aware of its limitations, ﬁrst-order
logic is an excellent theory.
J. Zalabardo, Introduction to the Theory of Logic, ch. 6 §§11-13, ch. 7.
G. Hunter, Metalogic §§52-53.
A. Fraenkel, Y. Bar-Hillel and A. Levy, Foundations of Set Theory, ch. V.
S. Shapiro, Foundations without Foundationalism, chs. 4-5.
P. Benacerraf, ‘Skolem and the sceptic’, Aristotelian Society Supp. vol. 59
(1985), pp. 85-115.
o
J. Dawson, ‘The compactness of ﬁrst-order logic from G¨del to Lindstr¨m’,    o
History and Philosophy of Logic 14, 1993, pp. 15-37.
Matti Eklund, ‘On how logic became ﬁrst-order’, Nordic Journal of Philo-
sophical Logic 1, 1996, pp. 147-67. [See http://www.hf.uio.no/iﬁkk/ﬁlosoﬁ/njpl/]
Exercise 6.1 Show that a predicate P has a ﬁnite extension if every order-
ing relation on P is well-ordered. [You may appeal to the Axiom of Choice.]

o
Exercise 6.2 Show using Compactness and the Downward L¨wenheim-Skolem
theorem that ‘most As are Bs’ is not expressible in ﬁrst-order logic. (Hint:
suppose the ﬁrst-order wﬀ φ(A, B) expressed ‘most As are Bs’. Use the
usual Compactness argument on the result of adding successively to φ(A, B)
the ﬁrst-order wﬀs ‘at least n As are B’ and ‘at least n As are not B’, for
o
every n; then use L¨wenheim-Skolem to get a countable model.)

Exercise 6.3 Let Lω1 ω be an extension of ﬁrst-order logic which allows
countably inﬁnitely long conjunctions and disjunctions. Show that ‘ﬁnite’
can be expressed in Lω1 ω .

61

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