Post on 26-May-2020
MATHEMATICAL LOGIC*
R.A. Bull
(received 1 June, 1971)
The purpose of this talk is to give a sketch-map of mathematical
logic, with highlights on some points which have amused me in my recent
reading. It is very much an amateur performance. The flow chart in
the abstract for the talk (see page 66) may help you to follow where I
am going.
l. Philosophical studies
There are two opposing tendencies in mathematical logic: asking
logical questions about mathematics; and giving mathematical answers in
logic. Of course this split is an illuminating exaggeration and the
counter-examples are important. For example, to answer a logical ques
tion about mathematics one must often formalise the reasoning involved
in that mathematics; and some technical results in mathematics are of
great logical significance.
The basic form of logical questions about mathematics is: 'What
am I doing when I do this piece of mathematics?' If this question is
really hurting, then one needs some kind of consolation; and there seem
to be two kinds of consolation available, that offered by the philos
ophers, and that offered by the mathematicians. This split is another
illuminating exaggeration.
One kind of answer to the question, 'What am I doing when I do
this piece of mathematics?' is the reductionist answer, 'You are not
doing mathematics, you are doing X '- where X is something nice and
* Invited address delivered at the sixth New Zealand Mathematics
Colloquium, held at Wellington, 17-19 May, 1971.
Math. Chronicle 2(1972), 17-27.
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unproblematic. This is an unfortunate answer, because in fact mathe
matics is not X, it is mathematics. The earlier years of this century
were notable for three large-scale reductionist answers to the basic
question. Russell said that mathematics is really a branch of formal
logic. Brouwer said that mathematics ought to be the science of mental
constructions (intuitionism). Thus one cannot say 'either a is true or
a is false', until one has constructed a proof for one of these assert
ions. Hilbert hoped that mathematics was the science of manipulating
symbols according to formal rules. Thus even though the content of
mathematics was problematic, the results could be obtained by finitary
methods. All these answers are obviously silly. This is yet another
illuminating exaggeration: if X is a good approximation to mathematics
then a study of X will throw light on mathematics; and while X is not
all of existing mathematics, it can be a very interesting new branch of
mathematics, as these three X's have turned out to be.
But what turned mathematical logicians from reductionist answers
to the basic question, was GOdel's proof that if X was adequate for the
theory of the natural numbers under addition and multiplication, then X
was not nice and unproblematic. This meant that substituting X for
mathematics could not be the knock-out answer which was hoped for then.
To give a later, more general variant of GOdel's result: Let arithmetic
be the theory with the natural numbers and variables over them, addit
ion and multiplication, equality, logical operations including quanti
fication. A satisfactory formalisation of arithmetic would include
numerals n and a derivability predicate |— with the properties:
f- is effectively defined;
for each statement P in arithmetic, there is a formula a with
Pin) true iff f-a(n).
There is no formal theory with these properties.
GOdel's work has been crucial to the history of mathematical
logic, and I should give three of his main results (in rather stretched
variants). These are classic instances of technical results which are
of philosophical significance. He showed that any theory formalised as
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a predicate calculus is model-complete with respect to suitable models;
so that adding predicate constants and axioms to the predicate calculus,
the formulas which are derivable are those which are valid in some class
of models. He showed that any formalisation of arithmetic is not model-
complete with respect to the intended model alone; so that any formal
mathematical theory of interest involves us in non-standard models. He
constructed a non-standard model for basic set theory plus the axiom of
choice plus the continuum hypothesis, within basic set theory, showing
that the axiom of choice and the continuum hypothesis are consistent
relative to basic set theory.
It seems to me that there is still room for a philosophical
discussion of mathematics which is not reductionist, but which helps to
remove the problematic feeling about topics such as the mode of exist
ence of mathematical entities. Wittgenstein, who gave helpful discus
sions of meaning in ordinary language, left rough notes on the nature
of mathematics; but these are generally considered to be unsatisfactory.
Otherwise very little work of this 'therapeutic' kind has been done for
mathematics. I want discussions of 'ordinary' mathematics rather than
of formalised mathematics; I envisage much of the discussion being
itself mathematical. The only work which fits my needs are the very
brilliant analyses of Kreisel - except that they soon get out of my
depth!
2. Mathematical studies
Another kind of answer to 'What am I doing when I do this piece
of mathematics?' is to continue the mathematical analysis of the intui
tive foundations of that piece of mathematics. In an ordinary mathe
matical theory we have formal mathematics down to a certain level, and
below that we rely on our understanding of the terms. This level tends
to get shifted down. Much work has been done on the mathematical analy
sis of mathematical concepts; the importance of mathematical analysis of
mathematical proofs is now being realised. The 19th century was notable♦
for its mathematical clarification of the concepts of the calculus. In
this century, the main targets of mathematical clarification have been
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the concepts of sets, of computable functions, and of models of a for
mal mathematical theory. These are very large subjects; I shall only
comment on some details.
The set theorists have arrived at a much better intended inter
pretation for the concept of a set. Earlier in the century, the inten
ded interpretation was that a set was the domain of a predicate, that
is a class, with some suitable restriction to avoid paradoxes. Four
different restrictions led to the four main set theories: Russell res
tricted the formulas which could be called predicates: Quine restric
ted the predicates from which classes could be formed; Zermelo et al
restricted sets to being the intersections of classes and given sets;
von Neumann et al restricted sets to being the classes which were memb
ers of classes. The new method is a construction of the collection of
sets, with stages typed by ordinals. Writing E(a) for the collection
of sets available at the a th stage, we take:
£(0 ) = {<*>}
Z(a + 1) = {1(a)}
E(a) = U E($), for limit ordinal a.$<a
With this intended interpretation the theories ZF and NGB are true,
together with the axiom of choice. The ordinal a which types the col
lection Z(a) of all sets of a set theory is characteristic; for example
if a Zermelo-type set theory is E(a) then the corresponding von-Neumann-
type set theory is E(a + 1). Much work is being done on E(a) for very
large ordinals a, but the question of the continuum hypothesis remains
open.
Take a mathematical theory; form the set of sentences which are
valid in this theory; form the class of models in which this set of
sentences is valid. Model theory considers this closure of a given
mathematical theory. I shall illustrate the extent to which models can
differ from the intended model, and the extent to which non-standard
models are relevant to the intended model. If a theory has a model
with cardinality H for any infinite X? , then it has a model with
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cardinality V . This is easy to show, but the axiom of choice is
equivalent to the converse, namely: If a theory has a model with
cardinality q then it has a model with cardinality X , for each
infinite X . So if a theory has a model with any infinite cardinality
then it has models with every infinite, cardinality. This shows that a
formal set theory has models which are very non-standard as regards card
inality. However non-standard models can be easier to handle than stan
dard ones. Abraham Robinson has shown that analysis has non-standard
models with members h such that 0 < h < x, for each standard positive
real number x. The derivative f'(a) of / at a is precisely the standard
part of the fraction
f(a + h) - f(a) for such non-standard h.
h
It can be shown that the results which hold for the standard members
of this model are precisely the results which hold in the standard
model.
It is in the analysis of mathematical proofs that intuitionist
mathematics comes into its own, because its intended interpretation is
in terms of proofs rather than in terms of sets, as for classical math
ematics. The real numbers of intuitionist mathematics are so weak that
there are reals x and y with none of x < y3 x = y3 x > y. However,
proofs of arithmetic and large fragments of classical analysis can be
translated into proofs in intuitionist analysis. Further, distinctions
made in intuitionist mathematics, but lost in the intended interpreta
tion of classical mathematics, can be used to get a better analysis of
proofs than can be obtained in classical mathematics. See the works
of Kreisel, and the references given there.
3. Formal logics
I now turn from the very large subject of foundations of mathe
matics to the smaller subject of the mathematics of formal systems of
logic. The central system is the classical first-order predicate cal
culus. Here we have propositions which are true or false, individuals
from some domain, predicates (that is, mappings from individuals to
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propositions), and the logical operations, A for 'and', ~ for 'not',
Vx for 'for all x', v for 'or', -* for 'implies', 3x for 'for some x'.
It is in terms of this calculus that mathematical theories are usually
formalised. Given a mathematical theory we have the set of formulas
which are valid when that theory is used as a model; and given a set
of formulas we have the class of models in which all the formulas are
valid. A mathematical theory can be studied by setting up an applied
predicate calculus with a derivability predicate |— in terms of an eff
ectively defined set of axioms and finitary derivation rules, so chosen
that the derivable formulas in the predicate calculus approximate to the
true statements in the mathematical theory. The applied predicate cal
culus itself can be studied formally where the original theory cannot,
because j- is defined effectively while the original theory is based on
intuitive concepts.
The concept of validity requires some set theory, but the concept
of derivability is effective and finitary. So although model-theoretic
methods are usually stronger, proof-theoretic methods can be more inter
esting, and I shall talk about them on their own first.
The theory of derivable formulas )- a is difficult to analyse
directly; instead derivable sequents o ,,...,01̂ (— are stud
ied. The semantics for sequents is a ,...,0^ f= 3 .»•••.> 3g iff
(^A.-.Aop * (s^.-.vep.
Most of the derivation rules for sequents introduce logical operators
into sequents; for example, the rules for A - introduction are:
r , a , 3 f ~ A r |— A , a r (— A , 3 «
T , a A 3 } - A r ( - A , o t A 3
Note that in these rules each formula of an antecedent sequent is a
subformula in the consequent sequent. A derivation which has this sub-
formula property is called a direct derivation.
Gentzen showed that there is a derivation of |— a in the ordinary
predicate calculus iff there is a direct derivation of |— a in the seq
uent calculus. Typing the sequents in a direct derivation with ordinals,
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he obtained a finitary consistency proof for predicate calculus rela
tive to the existence of suitable ordinals. This argument has been
extended to give finitary consistency proofs for formal arithmetic and
formal analysis relative to the existence of suitable ordinals.
A full formalisation of many statements requires quantifiers
over predicates, as well as over individuals; for example, the induction
principle refers to all properties of the integers. Adding such quanti
fiers to the first-order predicate calculus gives the second-order pre
dicate calculus. Stronger mathematical theories can make use of logi
cal objects of higher type again, although these can be avoided by using
sets instead. The simple theory of types due to Church can handle pre
dicates and functions of all finite types, in a very elegant manner.
We have types 0,1, and 018 where a and 8 are types. A formula of type 0
is a proposition; a formula of type 1 is an individual; and a formula
of type a8 is a function from objects of type 3 to objects of type a.
The apparatus of simple type theory consists of: variables x , where
a is a type, etc; B ̂ (of type a), the value of the function
with argument B \ x ^ A^ (of type aB) the function determined by the
formula by taking
k n A : y a -* B 3 8 a y 3 a
where B is obtained from A by substituting y Q for x ;a a ° 8 8
quantifiers CT0(0a) (given a predicate forming propositions from
objects of type a, n , .A is the proposition 'for all x , A of x J o(oa) oa r r a oa ais true1); a choice function l r a finite number of suitable axioms
J a(oa)and derivation rules. This calculus is sufficient for all known mathe
matics.
I now return to model-theoretic considerations. To study the
relation between a mathematical theory and its formalisations it is
necessary to study the class of models with respect to which model
completeness holds for the formal system, that is, the definitions of
(= for which (- a iff f= a. The relations between these three systems
are liable to be complicated.
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Applied first-order predicate calculi are known to be model-
complete with respect to some models. However, if the intended model
is characterised by higher-order properties then completeness will
involve non-standard models, as with arithmetic. Henkin has proved
model-completeness for higher-order predicate calculi, including simple
type theory. However, for higher-order predicate calculi this complet
eness involves secondary models, where ranges over a subset of all
objects of type a on the domain. So here again model-completeness will
involve non-standard models, as with arithmetic. Even in arithmetic,
model-completeness must involve models which are non-standard in some
way. To summarise: for many theories both first-order and higher-
order predicate calculi permit non-standard models; the higher-order
calculi have greater logical elegance; but first-order predicate calc
ulus is easier to use for most practical purposes.
Appendix A. In tu itio n is m
To see that intuitionism is reasonable, remember that its inten
ded interpretation of a statement a is 'a is provable1, rather than
'a is (set-theoretically) true'. Set theory makes the truth of
a v ~ a trivial, but if one is interested in proofs then it is not
trivial, as intuitionism observes. Kripke has given a semantics for
intuitionism, in terms of a mathematician passing through a tree of
situations with increasing information at each node. He knows what
information is available at future trees, but not which path he will
actually take through the tree.
The intuitionist theory of sequents is as for the classical the
ory, except that the right hand side must have at most one formula.
In classical semantics a v 3 is ~(~a A ~3), a -* 3 is ~(a A ~ 3 ) , 3a;a is
~ V x ~ a ; if r 1 |- A 1 is obtained from r (— A by making these replacements
then r (— A in classical logic iff r f (— A ' in intuitionist logic.
Godel used an extension of this argument to embed classical arithmetic
in intuitionist arithmetic.
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Appendix B. Non-classical logic
In classical logic it is assumed that each proposition is either
true or false, but for many purposes this assumption does not worki
Kripke and Hintikka have given semantics for a number of logical
operators, such as 'it is necessarily the case that a', 'it ought to be
the case that a', 'A knows it is the case that a1, M believes it is the
case that a'. This is done by taking a set K of possible worlds, and
a relation R on K, with the intended interpretation that ctRb means
'world b is compatible with world a', under an appropriate criterion
of compatibility, for example, under necessary truth, or under a moral
code, or under A's knowledge, or under 4's beliefs. The formal logical
operator o is then defined by taking Da to be true in world a iff a is
true in every world which is compatible with world a, that is,
V(uaLtd) = T iff Vb(dRb + 7(ct,B) = T).
Unlike almost all formal mathematical theories, these logics with
out quantifiers are decidable. A good method of proving decidability
through semantics is the finite model property, due to Harrop. He show
ed that if a calculus is model-complete with respect to each member of a
set of finite models, then the calculus is decidable. Using this result,
proving decidability becomes a matter of algebraic manipulation of the
models of a logic.
When quantifiers are introduced in non-classical logics, con
siderable logical and technical difficulties arise. This can be expres
sed in terms of the Kripke semantics, by saying that the different pos
sible worlds may have different languages, making it difficult to work
out 7(a,a) when a contains both □ and V. Several mathematical logicians
are working on these problems at present.
Appendix C. Henkin's method
Proofs of model-completeness can be obtained by defining a model
which satisfies a in terms of the direct proof of |— a, giving a fini-
tary proof of model-completeness. A more elegant method due to Henkin
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takes the set of derivable formulas of the given calculus and uses its
maximal consistent extensions.
First, some notation:
A set of formulas r is inconsistent iff I— ~(a A...A a ) for some1 r
{a ,...,a } c r.1 r -
r is consistent iff it is not inconsistent, r |— a ('a is derivable
from r 1) iff r U {'mj} is inconsistent; that is r [— a iff
f- (a A •. Aa ) a for some {a , a } c r.1 i r j* ' v —~ -completeness, T (— ~a iff not r |— a.
V-completeness, r |— Vxa(x) iff r (— a(t) for each variable t.
r is a theory iff r is consistent and r |— a implies a € r. (Thus for
a theory T we have: ~ -completeness, ~a € r iff not a € T;
V-completeness, Vxa(x) € r iff a it) € r for each variable t.)
Henkin's method of constructing a model satisfying given 6 with
{6) consistent now runs: there is a ~ -complete, V-complete theory A
with 3 € A; define 7(a) = T iff a € A, then 7 is an evaluation and
7(3) = T. The trouble with Henkin's construction is that the V-complete
theory obtained has a bigger language than the given set of formulas.
The following result, due to Thomason, obtains a V-complete theory in the
same language as the given set of formulas. I would be prepared to call
it the fundamental theorem of formal logic, at any rate for the formal
logics with countable languages. Given consistent, V-complete r there
is a ~ -complete, V-complete theory A with r £ A. The proof can be obta
ined through the following lemmas:
1. r (- a + 3 iff r U {a} (- 3.
2. If r is V-complete then r U {a} is V-complete.
3. Given V-complete r, for each formula a(x) we have
r |— 3xa(x) -»• a it), for some variable t.
4. Enumerate the formulas with x free as a (x), a (x), a (x),... .1 2 3
Given consistent, V-complete r there is a theory r f with r £ r'
and 3xa^(#) € r 1, for some variable t^, for i = 1,2,3, ..
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5. Given a theory P with 3xa. (x) ■+ a. (t.) € r', for i - 1,2,3,...is 1s 1s
there is a ~ -complete, V-complete theory A with T'£ A .
University of Canterbury
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