SOME NOTIONS. AND METHODS ON THE BORDERLINE OF
ALGEBRA AND METAMATHEMATICS
ALFRED
TARSKI
The content of this paper is an outline of the general theory of arithmetical
classes and a discussion of some of its applications.1 Roughly speaking, an
arithmetical class is any set of algebraic systems whose definition involves no
set-theoretical terms; thus, e.g., the set of all groups and that of all lattices are
arithmetical classes, while the set of all simple groups and that of all denumerable
lattices are not. The tendency to distinguish between those constructions and
derivations which involve set-theoretical notions from those which do not
involve them is undoubtedly a pronounced trend of modern algebraic research.
The theory of arithmetical classes provides this trend with a theoretical framework and exhibits its reach and limitations.
The fact that we shall concern ourselves with those algebraic notions which
involve no set-theoretical elements by no means implies that we shall avoid
set-theoretical apparatus in studying these notions. On the contrary, it will be
seen that set-theoretical constructions and methods play an essential part in
the development of the general theory of arithmetical classes.
The notion of an arithmetical class is of a metamathematical origin ; whether
or not a set of algebraic systems is an arithmetical class depends upon the form
in which its definition can be expressed. However, it has proved to be possible
to characterize this notion in purely mathematical terms and to discuss it by
means of normal mathematical methods. The theory of arithmetical classes
has thus become a mathematical theory in the usual sense of this term, and
in fact it can be regarded as a chapter of universal algebra.
The theory concerns arbitrary algebraic systems formed by a nonempty set
A, some operations O 0 , Oi, • • • under which A is closed, some relations Ä 0 ,
Ri, • • • between elements of A, and possibly some distinguished elements
Co, Ci, • • • of A.2 In the present paper we are interested exclusively in systems
with finitely many operations, relations, and distinguished elements (though the
set A of all elements of the system may be infinite) ; this restriction, however, is
not essential for the major part of our discussion. To simplify the discussion,
we formulate definitions and theorems exclusively for systems SI = <A, +>
1
Some fundamental facts concerning the theory of arithmetical classes have been stated
in [11]. The ideas which have been systematically developed in this theory can already be
found in earlier papers of the author; see [13, second part, pp. 298 ff.] and [14]. The numbers
in brackets refer to the bibliography at the end of the paper.
2
As is known, we could restrict ourselves without loss of generality to systems formed
by a set and certain relations between its elements.
705
706 ^
ALFRED TARSKI
formed by a set A and one binary operation + . Such systems will be referred to
for brevity as algebras, and the set of all algebras will be denoted by Œ.3,4
A property of algebras is called arithmetical if it is expressed by a sentence of
the arithmetic of algebras, i.e., of that part of the general theory of algebras which
can be formalized within elementary logic (the lower predicate calculus). In
such a sentence no set-theoretical terms occur; the symbol + is the only nonlogical constant; and all variables are assumed to range over elements of an algebra (and not, e.g., over sets of, or relations between, such elements). Thus,
e.g., the sentence for all x and y, x + y = y + x is an arithmetical sentence, and
the property expressed by this sentence—the commutativity of an alge'bra—
is ah arithmetical property. Instead of arithmetical properties we shall speak
of arithmetical classes. By an-arithmetical class we understand the set of all
algebras which have a certain arithmetical property in common, i.e., in which
a' certain arithmetical sentence holds. Thus the set of all commutative algebras
iä an arithmetical class.
To obtain a precise and purely mathematical definition of an arithmetical
class we imitate the metamathematical definition of a sentence in a formalized
arithmetic of algebras. Sentences are particular cases of sentential functions; in
fact, a sentence is a sentential function without free variables. Sentential functions are defined recursively in terms of simplest, so-called atomic sentential
functions, which are described explicitly; a sentential function in general is obtained from atomic sentential functions with the help of certain operations which
consist'in combining expressions by means of sentential connectives (or, and,
not, etc.) and quantifiers (for every x, for some x). It is convenient to assume that
the only variables used in the formalized arithmetic are the symbols x, x1,
x", • • • , x(h\ • • • , and the only atomic sentential functions are the expressions
xw = x{l) and xik) + x{l) = xim). (Thus, the expression x' + x" = x" + x1 is
not regarded as a sentential function; it is replaced by the expression for some
x, x' + x" = x and x" + xf = x.)
In a given algebra % = <A, -\->, every sentential function <£ determines the
set of all those finite sequences, with terms in A, which satisfy <_>. The number of
terms in these sequences varies dependent on the number of free variables in $.
For instance, the sentential function x + #' = x" determines the set of all threetermed sequences (ordered triples) <y0 , yi, y2> such that yo ,yi , y2 are members
* 3 To avoid any appearance of antinomial constructions we can agree to restrict ourselves
to algebras <A, +> in which A is a subset of a certain infinite set U fixed in advance.
With this restriction, the set d and all other sets involved in the discussion become
legitimate, mathematical entities whose existence can be derived from axioms of set theory.
4
(H is what is called in [2] a species of algebraic systems ; it can be replaced in the whole
discussion by any other species. Every set of algebraic systems considered in the theory of
arithmetical classes is assumed to be a species or a subset of a species, i.e., to consist of
similar algebraic systems.
6
The method applied here was developed in [14] ; for its geometric interpretation see [4]. A
closely related method was* used in [12] to define the notion of truth and other semantical
notions.
BORDERLINE ÖF ALGEBRA AND METAMATHEMATICS
707
of A (yo, yx, 2/2 G A) and 2/0 + 2/1 = 2/2. To simplify the construction we replace
finite sequences by infinite ones. The set of all natural numbers is denoted by
ca; and the set of all infinite sequences y = <y0, 2/1 , • • • > whose terms are in A,
i.e., the set of all functions on w to A, is denoted by Au. The atomic sentential
functions x{k) = xil) and xik) + xil) =* x{m) determine the sets Ik,\ and Skii,m of
all those sequences y G Aw for which yk = yi and yk + yt = ym , respectively.
There are operations on subsets X of A" (X £ Au) corresponding to those metamathematical operations which are used in constructing sentential functions.
These are, first, the familiar operations of set addition X u Y, set multiplication
X n Y, and complementation X *= A" — X. (The symbol X is used in the present
paper in those cases in which it appears clear from the context with respect to
which set the complement of X is taken.) Secondly, we have two sequences of
operations: the operations Vo, Vi , • • • of outer cylindrification and the operations A0 , Ai , • • • of inner cylindrification. Given any X £_; A", the set VkX (or
AkX) consists by definition of all sequences y G A" such that re G Xfor some (or
for every) x G Au which differs from y at most in its /cth term. The reasons for
the choice of the term cylindrification become clear if we think of Au as an
abstract infinite-dimensional analytic space over A. We now define the family
of all arithmetical sets X ΠAu (i.e., all sets determined by arithmetical sentential functions) as the smallest family which contains the sets Ik,i and Skli,m
and is closed under the operations u, n, ~, V*, and Ak (k, l, m = 0, 1, • • • ; the
operations n and Ak could be omitted in this definition).
If we consider a sentential function $ as referring to, not a particular algebra
31 = <A, +>, but the totality Q, of such algebras, then what is determined by <3>
is not a set X £ Aw, but a function $ (in the mathematical sense) whose domain
is Ct (in symbols, D($) = Ct) and which correlates a set .1(310 £ A" with every
algebra 31 = <A, + > ; ^(SI) consists, of course, of all those sequences y G Aa
which satisfy $. A function 57 determined in this way by some arithmetical
sentential function $ is called an arithmetical function. We arrive at a mathematical definition of this notion by means of a construction entirely analogous
to that which was outlined above for the notion of an arithmetical set of sequences in a particular algebra. The construction will be described here in a
formal way.
DEFINITION 1. (i) By F we denote the set of all functions JF such that D($) = Ct,
and ff(Sr) £ Aa for every 3t = <A, +> G Ct.
(ii) $k,i and &k,i,mfor k, l, m = 0, 1, • • • are the functions defined by the formulas:
D(àk,i) = F>($kìl,m) = a
and, for every 31 = <A, +>,
4 M ( 9 0 = {x I ^ G A", xk =
x
x
xi],
u
SjM.mCÄ) = i I G A , xh + Xl = xm}.
(iii) The functions âkti and &k,i,m for k, l, m = 0, 1, • • • are called
FUNCTIONS; the set of all elementary functions is denoted by EF.
ELEMENTARY
708
ALFRED TARSKI
A symbol of t h e form {x \ • • •} denotes t h e set of all elements x which satisfy
the formulas t o t h e right of t h e stroke |.
D E F I N I T I O N 2. Let ff, g G F and let k = 0, 1, • • • .
(i) The UNION ff u g is the function 5C defined by the conditions: Z)(3C) = Ct,
and 3C(3t) = ff(3t) u g(3T) for every 21 G a .
(ii) Analogously we define, in terms of operations on sets of sequences, the INTERSECTION ff n g, the COMPLEMENT ff, as well as the U N I O N U (3C* | i G I)
INTERSECTION D(3Ci | i G / ) of an arbitrary system of functions
with elements i of a set I.
and
the
3C* G F correlated
(iii) The OUTER CYLINDRIFICATION (or EXISTENTIAL QUANTIFICATION) V^ff is the
function 3C defined by the conditions: D(3C) = Ct, and, for every 2Ï = <A, + > ,
3C(3I) is the set of all sequences y G Aa such that x G ff(3l) for some sequence
x G Au which differs from y at most^ in its kth term. Similarly we define the
I N N E R CYLINDRIFICATION (or UNIVERSAL QUANTIFICATION) by changing
u
for some
sequence" to "for every sequence".
T h u s t h e symbols previously introduced for operations on sets of sequences
will also be used t o denote t h e corresponding operations on functions; it will
always b e clear from t h e context which meaning of t h e symbols is intended.
D E F I N I T I O N 3 . (i) d and 01 are functions defined by the conditions: D(d) =
.DCll) = Ct, 3(31) = A and 01(31) = A" for every 31 = <A, + > .
(ii) Given ff, g G F we say that ff is INCLUDED à g, î £ g, if ff(3t) ç ; g(3ï)
for^every 3t-G-Ct.
—
A denotes as usual t h e e m p t y set.
All t h e theorems will be stated in this paper without proof; most of t h e m can
easily be derived from t h e definitions of t h e notions involved.
T H E O R E M 4. (i) The set F together with the operations u, n, and ~ forms a complete atomistic Boolean algebra.
(ii) In this algebra d and *U are respectively the zero and the unit elements; C is
the inclusion relation; and U and fi are the join and the meet operations on arbitrary
tf systems of elements.
T H E O R E M 5. For any ff, g G F and k, I = 0, 1, • • • we have:
(i) Vkd = 3 and Afc0l = 01.
(ii) V f c ï C F ^ ï C VAff; A*ff G F and Affi £ ff.
(iii) V*(ff n VAS) = VAff n V Ä g, Afc(ff u A*g) = AAff u A Ä g.
(iv) VÄff = AÄff and Affi =
Vffi.
(v) VÄVz-T = ViVffi and AjbAiff =
AiAffi.
T H E O R E M 6. For any ff G F and k,l,m
(i) **.! G F .
= 0,l,
• • • we have:
BORDERLINE OF ALGEBRA AND METAMATHEMATICS
709
(ii) éhtk = 0L.
(iii) / / k ?* m and l 9± m, then ôkti = Vm(âktm n tfj,w).
(iv) i"/ k ^ I, then Vk(dkìi n ff) n Vk(ók,i n ff) = %.
It is known that all the equations which involve the operation symbols U, fi,
~,Vk , Ak, the special function symbols d, OL, âkiï , and some variables ff, g, • • • ,
and which are identically satisfied when the variables ff, g, • • • range over arbitrary functions in F, can be derived in a purely formal way from the identities
stated (explicitly or implicitly) in Theorems 4-6. 6 As examples the following
equations may be mentioned (where k and I range over arbitrary natural numbers) :
VfcV*ff = Vffi.
V*(ffu g) = Vffiu V*g.
VfcAjff n ViAffi = It.
Vjb(tf*,i n ff n g) = Vjfe(tf*.i n ff) n Vk(ëk,i n g)
for k ^ I.
Various identities are known which, in addition to the operations defined in
Definition 2 and the functions d, OL, êk,i, involve also the function Sfc.z.m ; for
instance:
S M , ™ n &k,i,n n äm,n — %•
THEOREM 7. For every function ff G F the following two conditions are equivalent:
(i) Affi = ff (or, what amounts to the same, Vffi = ff) for every natural number
k;
(ii) there is no algebra 3Ï = <A, +> for which ff(3t) ^ A and ff(3l) ^ Au.
DEFINITION 8. The set of the ARITHMETICAL FUNCTIONS, in symbols AF, is the
intersection of all sets X C F which include EF and are closed under the operations
u, "", and Vkfor k = 0, 1, • • • .
THEOREM 9. (i) EF
C
AF.
(ii) The set AF is closed under the operations u, n,
k = 0, 1, • • • .
—
, as well as Vk and Ak for
THEOREM 10. The set AF is denumerable (i.e., of the power bio).
THEOREM 11. In order that ff G AF it is necessary and sufficient that ff be representable in the form
ff = O A - - - On(g)
6
This statement is based on some unpublished work by Professor L. H. Chin, Mr. F. B.
Thompson, and the author.
710
ALFRED TARSKI
where each of the operations Oi(i = 1, 2, • - • , n) coincides with one of the operations Vo > Vi, • • • and A0 , Ai , • • • , and where g is a finite union of finite intersectims^ of elementary functions and their complements.
This is^ the so-called theorem on canonical representation.
\
THEOREM 12. For every ff G AF there exist only finitely many natural numbers
kfor which\ffi ^ $ (or Vffi 5* ff).
The set of all natural numbers k for which A*ff ^ ff can be referred to as the
dimension index of the function ff.
Much deeper than all the preceding is the compactness theorem for arithmetical
functions:
THEOREM 13. / / K (Z AF and fl(ff | ff G K) = d, then there is a finite set L C K
for which f1(ff | ff G L) = d.
A mathematical proof of Theorem 13 is rather involved. On the other hand,
this theorem easily reduces to a metamathematical result which is familiar from
the literature, in fact to GödePs completeness theorem for elementary logic.7
We proceed to defining the notion of an arithmetical class.
DEFINITION 14. (i) For every ff G F we put
eje(ff) = {311 3t = <A, +> G a, ff(3l) = A"}.
(ii) A set S C a is called an ARITHMETICAL CLASS if S = (3<£(ff) for some ff G
AF. The family of all arithmetical classes is denoted by AC,
15. (i) For every ff G F and k = 0, 1, • • • we have C«£(ff) = e£(Affi).
(ii) For every ff G AF there exists a g G AF such that 6£(ff) = <3£(g), and
Afc g = g for every natural number k.
THEOREM
A function ff G F which satisfies one of the two equivalent conditions 7(i)
and 7(ii) (i.e., whose dimension index is empty) is called a simple function. In
view of Theorem 15(ii) we can equivalently transform Definition 14(h) by assuming that ff is not an arbitrary but a simple arithmetic function. The content
of Definition 14(h) thus transformed is in perfect agreement with the intuitive
notion of an arithmetical class. Tn fact, while arbitrary arithmetical functions
correspond to arbitrary sentential functions, simple arithmetical functions correspond to sentences, i.e., sentential functions without free variables; ff being
a simple function correlated with an arithmetical - sentence <3>, C£(ff) is clearly
the set of all those algebras 31 G Πin which $ holds. To give an example, consider the set S of all commutative algebras. Let
ff = AoAiV2(So,i,2 n §1,0,2).
7
For a proof of Gödel's theorem see, e.g., [3].
BORDERLINE OF ALGEBRA AND METAMATHEMATICS
711
As is easily seen, ff is a simple arithmetical function and e«£(ff) = S; hence S is
an arithmetical class.
THEOREM 16. (i) AC is a field of subsets of G,; in other words, the system
<AC, u, n, ~~> (where X = (1 — Xfor every X G AC) is a Boolean algebra.
(ii) AC is denumerable.
THEOREM 17. J / K C A C and V\(X | X G K) = A, then there is a finite family
L Ç K such that fi(9C | X G L) = A.
This theorem—the compactness theorem for arithmetical classes—is clearly a
corollary of Theorem 13. As an improvement of Theorem 17 we obtain
THEOREM 18. If K C AC and f](X | X G K) G AC, then there is a finite family
L £ K such that ft(X \ X G L) = Ci(X \ X G K ) /
An especially important particular case of Theorems 17 and 18 is given in
the following theorem.
THEOREM 19. If Sn G AC, Sn+i £ S„ and Sn+i ^ Sn for every n G co, then
fì(Sn I n G to) ?± A and, ?ftore generally, f)(&n \ n G co) $ AC.
In view of the fact that the family AC is closed under complementation, we
can derive from Theorems 17-19 their duals in which D and A are replaced by
U and a. Theorems 18 and 19 are frequently applied to show that various sets
of algebras are not arithmetical classes. As an application of this kind we give
THEOREM 20. (i) A set S C a of finite algebras is in AC if and only if, together
with every algebra 31, S contains all algebra® 33 isomorphic to 31 (3Ï == 93), and there
is a number n such that every algebra 31 G S has at most n elements.
(ii) hi particular, the set of all finite algebras and hence also the set of all infinite
algebras 31 G Giare not in AC.
When saying that an algebra 31 = <A, +> is finite, or has n elements, etc.,
we have of course in mind that the set A is finite, or has n elements, etc.
We are often led to consider families which are more comprehensive than
AC; in fact, the families AC«, , AC« , AC_.fi , AC«, , etc. (K being a family of sets,
Kc and K s are respectively the families of all denumerable unions and all denumerable intersections of sets in K.) Among these families, AC« is especially
important from the viewpoint of the intuitions underlying our whole discussion.
A set S of algebras is in AC if it can be characterized axiomatically by means of
a single arithmetical sentence or—what amounts to the same—by means of a
finite system of arithmetical sentences; it is in A CO if it can be characterized
axiomatically by means of any finite or infinite system of arithmetical sentences.
712
ALFRED TARSKI
For instance, the set of all infinite algebras is easily seen to be in AC3 (though,
by Theorem 20(ii), not in AC), and the set of all finite algebras is in AC, . Theorem 17 remains valid if AC is replaced by AC5 . The following important result
essentially due to G. Birkhoff should be mentioned here : If S is a set of algebras
which (i) together with any algebra 31 contains every subalgebra of 31 and every
algebra isomorphic to 31, and (ii) together with any algebras 3t. , i G J, contains
their cardinal (direct) product, then S G AC§ .8
DEFINITION 21. Two algebras St, 93 G ß are said to be ARITHMETICALLY EQUIVALENT, in symbols 31 = 93, if every set S G AC contains either both of these algebras or neither of them.
Obviously, = is an equivalence relation on the set Œ.
THEOREM 22. For any algebras 31, 93 G Œ,
(i) the formula 31 ^ 93 implies 31 = 93;
(ii) if 31 is finite, then the two formulas are equivalent.
THEOREM 23. For every algebra 31 G Ct of an infinite power a and for every
infinite cardinal ß there is an algebra 93 G G, of the power ß such that 3Ï = 93.
/ / in addition ß ^ a, then such an algebra 93 can be found among the subalgebras
of%.
The-mathematical proof of this theorem is related to that of Theorem 13 and
is rather involved. On the other hand, Theorem 23 can easily be recognized as
a mathematical translation of a familiar metamathematical result—in fact, of
an extension of the Löwenheim-Skolem theorem.9
By Theorems 22 (ii) and 23, the equivalence of the formulas 3Ï = 93 and
3Ï = 93 f or every algebra 93 is a characteristic property of finite algebras 31.
The following theorem is a rather special consequence of Theorem 13; as will
be seen later, it has many interesting applications.
THEOREM 24. If St G a, and ifffTOG AF and ffw+i(3I) C ffn(3l) ^ A for every
natural n, then there is an algebra 93 G @> such that 31 = 93 and fl(ffn(93) | n G co)
DEFINITION 25. A sets Πa is called an ARITHMETICAL TYPE if S = {931 31 = 93}
for some algebra 31 G Œ. The family of all arithmetical types is denoted by AT.
Thus, arithmetical types are partition sets under the equivalence relation = .
In metamatheiriatical terminology, a set of algebras is an arithmetical type if
it is axiomatically characterized by means of a complete and consistent system
of arithmetical sentences.10
8
Cf. [2].
This extension of the Löwenheim-Skolem theorem follows from the results in [5] ; see
also [3] and [11].
10
The term "complete" is used here in the sense of [13, second part, p. 283].
9
BORDERLINE OF ALGEBRA AND METAMATHEMATICS
713
THEOREM 26. (i) S G AT if and only if S = f ì ( 9 C | 3 l G 9CG AC) for some
3Ï G a.
(ii) AT Aas £Ae power 2N°.
By Theorem 26(i), every arithmetical type is in ACÔ . By Theorems 20(i) and
22(ii), the arithmetical types of finite algebras are in AC; this property, however, is not characteristic for finite algebras.
THEOREM 27. (i) If I is a prime ideal in the Boolean algebra <AC, u, n, ~> of
Theorem 16 and S ~ fì(9C | X G AC - I), then S G AT.
(ii) If S G AT,tfftenöftere is exactly one prime ideal I m <AC, u, n, ~> for which
S = fì(9C| 9C G AC - I), in facti = {9C | 9C G AC, S n 9C = A}.
Theorem 27(i) is a consequence of Theorem 17.
DEFINITION 28. A set S C ß is said fo be ARITHMETICALLY CLOSED if, together
with every algebra 31, it contains all algebras 93 such that 31 = 93. ÎT/ie family
of all arithmetically closed sets is denoted by ACL.
29. (i) S G ACL # and only if & = V(X\
(ii) ACL Aas the power 22«0.
THEOREM
X £ K) for some K G AT.
THEOREM 30. ACL is a complete field of subsets of Ö., and hence <ACL, u, n, ~~>
is a complete and atomistic Boolean algebra. AC is a subfield of ACL, and AT is
the set of all atoms of ACL.
From Theorem 30 it follows that not only AC but also all the families ACff ,
ACs , ACcrg , ACsa , • • • are included in ACL; by Theorem 29(h), ACL is much
more comprehensive than all these families. On the other hand, simple examples
of sets of algebras are known which are not in ACL. E.g., by Theorem 23, the
isomorphism type of an infinite algebra 31 (i.e., the set of all algebras 93 with
31 == 93) as well as the set of all algebras of any given infinite power a are not
arithmetically closed.
By Theorem 30, the Boolean algebra 31 = <AC, u, n, ~> is a subalgebra of
the Boolean algebra 93 = <ACL, u, n, ~>; by Theorem 27, 33 proves to be isomorphic to the Boolean algebra formed by all the sets of prime ideals in 31.
Hence, by means of familiar results from Stone's representation theory for
Boolean algebras, the theory of arithmetical classes acquires a topological interpretation. 11 To obtain this interpretation directly, we proceed as follows.
With every set X C Πwe correlate the set G(9C) defined by the formula
e(9c) = n c y | X C ^ G AC).
Using exclusively Theorem 16(i), we easily show that the set d of all algebras
11
For Stone's representation theory and the notions involved in the following discussion
see, e.g., [1].
714
ALFRED TARSKI
<A, +> is a topological space with 6 as closure operation. Since, however, {31} 9e
6({3Ï}) for any 31 G GL ({31} denoting as usually the set containing 3Ï as the only
element), it is not a topological space in the narrower sense (i.e., not a ï\-space).
On the other hand, Theorem 16(i) also implies that C({8t}) = 6({®}) or
C({3t}) n <3({93}) = A for arbitrary 31, 93 G d. Hence we can transform this
space into a topological space in the narrower sense by "identifying" two points
31, 93 G GL in case (3({3t}) = C({93}). The new space can be referred to as the
arithmetical space over GL. The definition of closure in the new space remains
unchanged. As is easily seen, the formula <3({3I}) = <B({93}) is equivalent to
3Ï = 93; hence the points of the new space, i.e., the sets X of the form X =
<3({3Ï}) for some 31 G GL, are simply arithmetical types, and the power of the
space is 2**° (see Theorem 26). The point sets of the space are arbitrary (not sets,
but) unions of points; by Theorem 27 (i), they coincide with ^arithmetically
closed sets. ACs and AC., are respectively the families of all closed and all open
sets; by the formula AC = ACs n AC,, which is a consequence of Theorem 17,
arithmetical classes coincide with those point sets which are both closed and
open. The families AC5 , AC.,, ACs, , AC^ , • • • are what are called Borelian
classes of point sets. The arithmetical space over GL is easily seen to be totally
disconnected; by Theorem 16(h) it is separable and by Theorem 17 it is bicompact. (Thus, the bicompactness of the space discussed is a consequence of the
completeness of elementary logic.) The theory of totally disconnected, separable ,-and bicompact spaces is a well developed branch of topology; and various
results obtained in this theory can be automatically carried over to the theory
of arithmetical classes.12 In this way we arrive, e.g., at the conclusion that every
arithmetical class is a union of finitely many or denumerably many or 2K0 arithmetical types.
Within the framework of the ideas outlined in the present paper various special studies are carried on. In order to obtain stronger and deeper results we
usually concentrate in these studies upon algebraic systems of some special
kind, e.g., upon Abelian groups, arbitrary groups, or—turning now to algebraic
systems with more than one operation—upon rings, fields, lattices, Boolean
algebras, etc. To apply the results of our general discussion to such special sets
of algebraic systems, we first subject all the notions involved to a process of
relativization. Thus, V being the set of all algebraic systems in which we are
interested, we introduce the notions of an arithmetical function, arithmetical
class, etc., relative to V, in symbols AF(eü), AC(eü), etc. In case V is a subset of
GL, the definitions of relativized notions are obtained in the following way: by
modifying Definition l(i), we agree to denote by F(eü) the set of all functions
9F such that D($) = V, and ff(3I) C A" for every 31 = <A, +> G V; in all the
subsequent definitions we replace F by F(eü). It turns out, in particular, that
arithmetic classes relative to 13 are simply intersections of *ü with arithmetical
12
For related applications of topology to metamathematics see [6].
BORDERLINE OF ALGEBRA AND METAMATHEMATICS
715
classes in the absolute sense, i.e., relative to d;13 and if V is itself an arithmetical
class in the absolute sense, then arithmetical classes relative to *U coincide with
those arithmetical classes in the absolute sense which are subsets of V. Most
theorems stated in this paper automatically extend to relativized notions. In a
few important cases, however, some restrictive assumptions concerning "U are
necessary. For instance, Theorems 17-19 do not apply to arithmetical classes
relative to an arbitrary set "U, but they prove to hold under the assumption that
V is itself a member of AC or, more generally, of AC$ (in the absolute sense).
Whenever a set V of algebras is studied from the viewpoint of the theory of
arithmetical classes, the main problem is that of giving an exhaustive description of all arithmetical classes relative to V. The solution of the problem usually
consists in (i) singling out certain special arithmetical classes referred to as basic
classes; (ii) showing that the family of all arithmetical classes (relative to V)
coincides with the field of subsets of V generated by the basic classes—or, in
other words, that every arithmetical class can be represented as a finite union
of finite intersections of basic classes and their complements (to V) ; (iii) establishing a criterion which permits us to decide in each particular case whether or
not two representations of the type just mentioned yield the same arithmetical
class. As a rule, such a description of arithmetical classes is preceded by and
derived from an analogous description of arithmetical functions. In obtaining
the latter description the crucial point consists in showing that, 93 being the
Boolean algebra generated by basic functions, the operation V* (or A^) performed on an arbitrary function in 93 yields a new function in 93. Hence the
method applied is referred to as the method of eliminating quantifiers.1*
A detailed description of arithmetical classes has many important consequences. It gives us a clear insight into the structure of arithmetical classes and,
speaking more precisely, permits us to determine the isomorphism type of the
Boolean algebra formed by these classes. It enables us to decide in various special cases whether or not a given subset of V is an arithmetical class. It leads to
a description of all arithmetical types (relative to V) and provides us with a
criterion for arithmetical equivalence of any two algebras in "Ü.
So far, an exhaustive description of arithmetical classes has been obtained for
a few special sets of algebras; the most important among them are the sets of
all Abelian groups, all algebraically closed fields, all Boolean algebras, and all
well ordered systems (i.e., systems formed by a set A and a binary relation R
which establishes a well-ordering in A).15 The results are especially simple in
the case of the set V of all algebraically closed fields. In fact, p being 0 or a
prime number, let Qp be the set of all algebraically closed fields with characteristic p. All these sets Qp , with the exception of C0 , are easily seen to be in AC( e ü).
They are chosen as basic classes, and it is shown that every set in AC(eü) can
13
Or, more generally, to the species which includes V as a subset; cf. footnote 4.
For information concerning this method see [10, pp. 15, 50].
15
See [7], [9], and [11].
14
716
ALFRED TARSKI
o
be represented in a unique way either as a union of finitely many sets <2P (p ^ 0)
or as an intersection of finitely many complements of these sets. In consequence,
the family AT(13) proves to consist of all the basic classes <3P and, in addition,
of the set (30 ; two algebraically closed fields prove to be arithmetically equivalent if and only if they have the same characteristic.
The method of eliminating quantifiers which is used in deriving results of this
type is of metamathematical origin; it has been frequently applied to establish
a decision procedure for various theories formalized within elementary logic.
In fact, whenever the decision problem for the arithmetic of algebraic systems
of a set 13 has been solved in an affirmative way, a detailed description of arithmetical classes relative to V has also been obtained. If, conversely, the solution
of the decision problem for the arithmetic of V is known to be negative, the task
of exhaustively describing arithmetical classes relative to V appears to be hopeless; this applies, e.g., to the set of all algebras <A, +>, or all groups, or all fields,
or all lattices.16
However, even in those cases in which the structure of the family of arithmetical classes is not known, various special problems involving this family can
be successfully discussed. We can ask, e.g., the question whether a given special
set S of algebraic systems is an arithmetical class. If an arithmetical definition
of S is known, it obviously implies an affirmative answer to this question. If, on
the other hand, no such definition of S is available, the conjecture that S is not
an_arithmetical class_arises in a natural way; often, however, a_confirmation of
this conjecture presents considerable difficulties. Sometimes an application of
Theorem 18, 19, or 24 leads to the desirable result. Let, for instance, S be the
set of all fields of characteristic 0, and Sw be the set of all fields whose characteristic is not a prime number ^n. Clearly, all the sets STO for n = 0, 1, 2, • • •
are in AC, and
S = f1(Sn | n G w).
Therefore S is in ACfi and hence also in ACL; on the other hand, Theorem 19
implies that S is not in AC. The same holds, e.g., for the set of all algebraically
closed fields or all groups without elements of finite order > 1 ; see also Theorem
20. On the other hand, let 3 be the set of all groups without elements of infinite
order. From Theorem 24 we easily conclude that, for every group ® G 3 in
which the orders of elements are not bounded above, there is an arithmetically
equivalent group § which is not in 3. Hence 3 is not in ACL and a fortiori not
in AC. In an entirely analogous way we can show by means of Theorem 24 that
the set of all well ordered systems (or, more generally, of all simply ordered systems without densely ordered subsystems) and the set of all ordered rings with
Archimedean order are neither in ACL nor in AC. A different argument leads to
the conclusion that the set of all directly indecomposable groups is not in ACL
or AC. Various problems in this domain remain open. Thus, it is known that
16
See [8] and [15].
BORDERLINE OF ALGEBRA AND METAMATHEMATICS
717
the set of all simple groups (or all simple algebras) is not in AC ; it is not known,
however, whether this set is in ACL.17
When new notions are introduced in mathematics, the question of their usefulness and applicability is often raised. Mathematicians want to know whether
the discussion of the new notions leads to interesting results whose significance
is not restricted to the intrinsic development of the theory of these notions. We
believe that the theory of arithmetical classes has good chances to pass the test
of applicability. To support this statement we want to discuss some applications
of this theory which may be of general interest to mathematicians and especially to algebraists; in some of these applications the notions of the theory itself
are not involved at all.
Theorem 13 and its consequences (Theorems 17, 19, 24) provide us with a
rather general method of constructing (or, at least, proving the existence of)
algebraic systems with some properties prescribed in advance. Consider, for
instance, the problem of the existence of non-Archimedean ordered fields. Let
a ' denote the set of all algebraic systems 31 = <A, +, -, < , 0, 1> with two binary operations + and •, a binary relation < , and two distinguished elements
0 and 1; all the notions of the theory of arithmetical classes are understood to
refer to GLf instead of to Œ. For any given natural number n, let $n be the function in F defined as follows: for 31 = <A, + , -, < , 0, 1>, ^(31) is the set of all
sequences x G A1* such that Xi = 0, xk+i = xk + 1 for k = 1, 2, • • • , n, and
xn+i < xQ . Taking for 31 the field of all real numbers, we see that the hypothesis
of Theorem 24 is satisfied. Hence there is an algebraic system 93 such that
31 = 93 and fl(9^(93) | n G w) ^ A; a moment's reflection shows that 93 is a real
closed field with a non-Archimedean order. Of course, as is well known, such
fields can be constructed by means of direct, specific methods; still it seems interesting that their existence is an almost immediate consequence of a general
theorem from the theory of arithmetical classes. Moreover, by analyzing the
argument just outlined, we notice that it leads to a stronger result, which—
neither in its general form nor in various particular cases—seems to be easily
obtainable by means of direct construction methods: For every ordered ring 9Î
there exists an arithmetically equivalent ring © with a non-Archimedean order.
(Hence, as was mentioned above, the set of all rings with Archimedean order is
not in ACL.) The following theorem obtained in an entirely analogous way may
also be of interest : For every integral domain 35 which is not a field there exists
an arithmetically equivalent integral domain © containing two elements x and
y such that x is not a unity element, y ^ 0, and xn divides y for every natural
number n. We should like to mention some further results in the same general
17
Among the results mentioned in the last paragraph, those concerning various sets of
groups have been communicated to the author by Dr. W. Szmielew; they were originally
obtained as consequences of her results in [9], without the help of Theorems 18, 19, or 24.
The remaining results are due to the author; for well ordered systems see [13, second part,
p. 301]. Added in proof: The answer to the question whether the set of all simple groups is
arithmetically closed has recently been found to be negative.
718
ALFRED TARSKI
direction whose proofs, based on Theorem 19, are even simpler: I. If S G AC
(or, more generally, S G ACô) and if, for every n, there is a group in § no element of which is of order ^n (except, of course, the unit), then there is a group
in S all elements of which are of infinite order. IL Dually, if S G AC (or, more
generally, S G ACff), and if S contains every group all elements of which are of
infinite order, then, for some natural number n, S contains every group no element of which is of order _grfc. I and I I remain valid if groups without elements
of order un are replaced by fields of characteristic ^n, and groups with all
elements of infinite order by fields of characteristic 0.18
Applications of a different type can be obtained from various special results
by which any two algebraic systems of a certain set are arithmetically equivalent. It has been shown, for instance, that any two real closed fields are arithmetically equivalent (and that, consequently, the set of all real closed fields is
an arithmetical type). Hence, if an arithmetical class S contains some real closed
field, e.g., the field of all real numbers, then it contains every real closed field
as a member; and, what is very important, the same still holds if S is not necessarily an arithmetical class but an arithmetically closed set. Let now, for every
natural number n, Sn be the set of all ordered fields 3t = <A, + , - , < , 0, 1>
satisfying the following condition: every polynomial in two variables over 3Ï of
degree n reaches a maximum on every rectangle, i.e., on every set of couples
<x, y> such that a ^ x ^ b and c ^ y _g d for some a, b, c, d G A, a < b, c < d.
Let S =- D (Sn-| n^ÇL-v); i.e*, S is the set of all ordered fields 3Lsuch__tbat _every
polynomial in two variables over 31 of an arbitrary degree reaches a maximum
on every rectangle. Clearly, all the sets Sw for n = 0, 1 • • • are in AC; hence,
S is in ACs and therefore in ACL. Obviously, the field of real numbers belongs to
S. Consequently, every real closed field belongs to S—a result which can also
be obtained directly, without the help of the theory of arithmetic classes, but
which is not quite trivial. In exactly the same way many different theorems which
have been established for the field of real numbers by essentially applying the
continuity of this field, and sometimes by using difficult topological methods,
automatically extend to all real closed fields.19
18
The method discussed in the last paragraph was mentioned by the author in his address
at the Princeton University Bicentennial Conference on the Problems of Mathematics, 1946;
some of its applications, e.g., to the problem of the existence of non-Archimedean ordered
fields, were explicitly pointed out. The problem of the existence of integral domains with two
elements x, y such that x is not a unity, y ^ 0, and xn divides y for n = 0,1, • • • was formulated by Professor R. M. Robinson and solved by the author with the help of the same
general method. Some other applications and an extension of the same method can be found
in the Princeton University doctoral dissertation of L. Henkin, The completeness of formal
systems, 1947. The question whether the method discussed provides an effective construction
of algebraic systems with prescribed properties or merely proves their existence is somewhat
involved and will not be discussed here. Added in proof: For a direct construction of integral
domains mentioned above see the recent paper of R. M. Robinson, Undecidable rings, Trans.
Amer. Math. Soc. vol. 70 (1951) pp. 149-150.
19
The arithmetical equivalence of all real closed fields was established in [10, p. 54].
The far-reaching applicability of this result to non-arithmetical theorems became clear
BORDERLINE
OF A L G E B R A A N D M E T A M A T H E M A T I C S
719
In connection with the last remarks the following observation seems to be
appropriate. The statement that any two real closed fields are arithmetically
equivalent is a mathematical translation of the metamathematical result by
which the arithmetic of real closed fields is a complete (and consistent) theory.10
This result implies as an immediate consequence that every arithmetical statement which holds in a particular real closedfield,e.g., in thefieldof real numbers,
automatically holds in any other real closed field. As we have seen above, the
consequence just mentioned extends to a comprehensive set of statements which
cannot be formulated in the arithmetic of real closed fields; in fact, to every
statement such that the set of all ordered fields in which this statement holds is
arithmetically closed, without being an arithmetical class. This extension is
immediate once the completeness theorem for real closed fields has been translated into the language of arithmetic classes; however, it could hardly be derived
in a purely metamathematical (syntactical) way from the completeness theorem
itself—unless we allow ourselves to apply some rather intricate semantical
notions and methods. At first sight the mathematical theory of arithmetical
classes seems to be merely a translation of the metamathematics of arithmetical
formalisms; actually this theory paves the way for constructions and derivations
which go far beyond purely metamathematical procedures.
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720
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UNIVERSITY OF CALIFORNIA,
BERKELEY, CALIF., IL S. A.