The Foundations of Mathematics in Poland After World War II

U

is an element of the filter

A of individuals, the group

G, and the filter

of a class which is a model of set theory.

Thus the choice of the U leads to a construction

Now appropriate choice of the above

parameters yields "pathological" models in which normally the axiom of choice is invalid although some of its consequences may hold.

Numerous papers by Mostowski

and other mathematicians using this method provided deeper insight into the role of the axiom of choice in mathematics.

Among them one is compelled to mention

the investigations of Mostowski on the finite axioms of choice [13].

An interest-

ing feature of these investigations was an application of the methods of finite groups to the investigation of interconnections between the axioms of choice for families of n-element sets (for various n). Investigations in the foundations of set theory were conducted at that time mostly by Mostowski.

In evaluating the developments in the early fifties, one must

mention two of his results.

Seeking a semantic proof of the incompleteness of

Peano arithmetic [15], Mostowski proves the following theorem (and also finds another undecidable statement) which nowadays is generally called "Mostowski's contraction lemma" - one of the basic tools for modern investigations in the foundations of set theory. This theorem states the following. 2 where E C A , with the following properties: 1°



is extensional, i.e., (x) (y) [(z) (zEx

2°



is well founded, i.e., there is no infinite E-descending sequence

Then there is a transitive set

B

such that

~

zEy)

Given a structure

x = y]

~

~
r

E

I

B> (where E is the

membership relation). This lemma is a generalization of the following fact:

Every well-ordering is

similar to some von Neumann ordinal. It is no exaggeration to say that every paper investigating transitive models of set theory uses t.hi.s lemma.

It's simply not stated anymore.

Another basic result due to Mostowski [16], a result with far-reaching consequences, is a collection of facts on seemingly paradoxical properties of the Godel-Bernays theory of classes.

Mostowski proves that in that theory the full axiom of induc-

tion (in the form of a schema: formulas

p), is not provable.

ones in which formula

~

~(O)

& (n) (P(n)

P(n+l)) =

(n)~(n)

, for all

The unprovable instances are impredicative, i.e.

contains quantifiers ranging over all classes.

FOUNDATIONS OF MATHEMATICS IN POLAND

131

A consequence of this surprising phenomenon was the formulation of the so-called Kelley-Morse theory of classes (sometimes called the impredicative theory of classes) which was intensively investigated much later in Warsaw.

The last-

modern system of set theory is based on the following intuitions, which originated with Cantor and subsequently von Neumann. possible kinds:

The set-theoretical objects are of two

"small" ones, which are able to belong to another object - these

are sets - and "large" ones, which do not belong to any object - these are proper classes.

Such an approach eliminates semantical paradoxes (in this scheme

Burali-Forti paradox just states that the ordinals do not constitute a set while Cantor's paradox means that the class of all sets is not a set).

These purposes

(elimination of paradoxes) are achieved by the Godel-Bernays theory of classes but a distinguishing feature of the Kelley-Morse theory of classes ([23]) is a convincing formalization of Frege's principle of the existence of extensions of properties.

This is the following scheme of class existence: (EZ) (x) (x E Z = X is a set & ¢(X»

where the only limitation on

¢

is that

Z

,

does not appear in it.

This schema

simply states that each property of sets determines a class (let us notice that Mostowski in the aforenamed paper shows that the scheme of class existence is unprovable in Godel-Bernays theory of classes).

Note that in

¢

some quantifiers

may range over all classes; thus, in a sense, this scheme is "self-referring".

On the other hand, the consistency of the Kelley-Morse theory of classes is provable in

ZFC + "there exists an inaccessible cardinal" (this seems to be a

fairly weak set theory in times when measurable and strongly compact cardinals seem to appeal to a lot of people).

Investigations of the Kelley-Morse theory

of classes have been conducted in Warsaw in recent years by Mostowski and his collaborators [12], and it seems that Kelley-Morse theory of classes is-recentlyquite popular among topologists and category theorists. Among other results connected with foundations of mathematics one must mention those of Kuratowski [9], who found interesting consequences of GOdel's axiom of constructibility in descriptive set theory. In the late fifties and the beginning of the sixties, the foundations of set theory was in a sort of standstill. Cohen's breakthrough.

Thus the explosion of results following

His method of forcing led to numerous independence proofs

in set theory without individuals.

In Poland investigations of the subject were

conducted by Mostowski and a group of his younger collaborators.

A survey of

investigations on forcing is Mostowski's monograph "Constructible sets with applications".

The presentation is based on the following topological inter-

pretation (due to Ryll-Nardzewski and-independently-to Takeuti). the method is based on the following construction:

Let

Roughly speaking

M be a countable,

132

W. MAREK

transitive model of

ZFC

set theory.

We try to make it "thicker" by adding new

elements but in such a way that it will not become "longer" i.e. no new ordinal is added. from all

M a

Let and

a

by closure of the set

M U {a}

It turns out that the set of

dual in Cantor set topology for each axiom such that

generic over

M[a] M.

q,

The model

M[a)

under some operations.

are suitable for our purposes namely we need

a model of ZFC.

a

a f M

be a set of natural numbers,

a

a

a

such that

such that

M[a]

of set theory.

satisfies all axioms of ZFC.

The set

a

M[a)

F

arises But not

q,

is again is resi-

In this way we find is called Cohen

Using topologies other than Cantor set topology and considering

other topological spaces leads to models in which the axiom of constructibility, the continuum hypothesis or other interesting statements do not hold. Another major idea in the foundations of mathematics emerges in the beginning of the sixties, the so-called axiom of determinacy of Mycielski and Steinhaus. G (where Xc P(w». X In this way a 0-1 sequence a

Let

us imagine the following game

Players I and II choose a

number 0 or 1 in turn.

is formed (i.e. a subset of

w).

The subset thus constructed will eventually be in

agree that the player I wins if see that the game X = {a : 0 E a} sequence.

a EX.

X

or outside of it.

Otherwise player II wins.

We

It is easy to

G is determined for some X - at least, for instance if x then clearly I wins, just by putting 1 at the oth place of the

Similarly whenever

X

is denumerable, II has a winning strategy:

consecutively omitting elements of

X

(these results are far from optimal -

recently A. Martin proved that for every Borel set statement:

For every

X ~~w)

, G

X, G is determined). The x is determined (i.e. for each xc P(w) either

X I or II possesses a winning strategy in

G is called the axiom of determinacy. x) This statement implies (Mycielski [27], [28]) that all sets of reals are Lebesguemeasurable, thus contradicting the axiom of choice.

But on the other hand, it

implies some weaker forms of the axiom of choice (i.e. for families of denumerable sets of reals).

For these and other reasons (it was proved by Solovay that the

axiom of determinacy has very great metamathematical strength) the determinacy is being investigated very intensively. Now we shall discuss model theory, which is the newest and most intensively studied branch of foundational research. Basically the theory of models was initiated by Tarski.

In Poland model-theoretic

research was started by Mostowski in his study on direct products.

An important

paper by Mostowski [17] was later generalized by Feferman and vaught.

Their in-

vestigations were later considerably extended by a group of mathematicians in Wroclaw centred around Ryll-Nardzewski. the following problem. U({Ai}iEI)

A typical example of their interests is

Given an operation

U(·)

on families of structures (i.e.

is a structure), to find in what way the theory of the structure

U({A.}. )depends on the properties of I and of the family J. J.EI . additional parameters may be involved).

{Th (Al.')J.'EI}

(some

FOUNDATIONS OF MATHEMATICS IN POLAND

133

Throughout the fifties Rasiowa, Sikorski and others were developing an algebraic approach to the foundations of mathematics.

The algebraic methods are based on

the application of the theory of Boolean algebras (or other distributive lattices - in the case of nonclassical logics) to an algebra of formulas of the language. A fundamental achievement of this method is an algebraic proof of completeness theorem, based on the so-called "Rasiowa-Sikorski lemma" [32]. algebra

B be given and let

assume moreover that

a. a

{a,

a

!1

JEI

i

that there exists an ultrafilter i.e. if

,L~ Eco

~J

ij

u

'EI ,J i

Let a Boolean

B

be a "matrix tl of elements of

exists for all

iECil

in the algebra

B which preserves the inf's

{aij}jEI, ~ U then also

a, E U a,

(for all

Then the lemma states

iECil).

~

The usefulness of algebraic methods in investigations of classical logic and subsequent extensions of these methods so they become useful also for the study of nonclassical logics started a broad current of research of a group of mathematicians (mainly in Warsaw) centred around Rasiowa and Sikorski. A surprising contribution to algebraic logic was presented in the late sixties by Scott and Solovay in their work on Boolean-valued models of set theory.

It

turns out that the construction (described above) of a set of natural numbers generic over

M may be viewed as an application of Rasiowa-Sikorski lemma.

The

investigations of the so-called Martin's axiom, one of the most tempting alternatives for the continuum hypothesis, are also connected with Rasiowa-sikorski result.

The investigations of Rasiowa and Sikorski were summed up in their mono-

graph "The Mathematics of the Metamathematics" (and later in Rasiowa's "An Algebraic Approach to Nonclassical Logic") grouping and classifying the algebraic methods and the results proved using these methods. Another important branch of model theory is Ehrenfeucht's results on elementary equivalence and the characterization of this notion by the methods of game theory (so called Ehrenfeucht's games [2]). been applied in abstract model theory.

The methods originated there have recently Other important results of Ehrenfeucht's

was so-called "omitting types" theorem, which provides an essential means of constructing models. At approximately the same time (the late fifties), basic results on categoricity in power were obtained by ~s and Ryll-Nardzewski.

We say that the theory

categorical if - up to isomorphism - it possesses just one model.

T

is

For instance,

Peano arithmetic with a second order (nonelementary) induction axiom is categorical.

However no elementary theory with infinite models is categorical, so for

elementary theories we consider "categoricity in power". categorical in power

k

if any two models of

T

of power

A theory k

T

is

are isomorphic.

For instance, the theory of dense linear orderings without endpoints is categori-

W. MAREK

134 cal in power \~o

Los [10] gave the following sufficient condition for an

0

elementary theory to be complete (this is so called the If a denumerable theory

T

power, then it is complete. denumerable theory

T

~os-Vaught

criterion):

with no finite models is categorical in any infinite ~os

formulated the following conjecture:

If a

is categorical in some nondenumerable power, then it is

categorical in all nondenumerable powers.

This conjecture, proved in 1962 by

Morley, initiated a line of research which recently has been very actively pursued. Ryll-Nardzewski found very important criterion for categoricity in power '\'0' [34]. Let

~l

'

be formulas with free variables among

~2

is equivalent over

T

with

~2

T ~ ~l ~ ~2'

iff

xl, ••• ,x

We say that n' Ryll-Nardzewski theorem

states that a complete theory without finite models is categorical in power iff for all

n, the number of equivalence classes of formulas with variables

~l

o

is finite. It is impossible to overestimate an importance of the notion of ultraproduct, introduced by Zos, [11].

Let

{Ai}iEI

the sake of simplicity we assume U be an ultrafilter in lence relation f

~

g iff f

~U

and

I.

Ai

In the product

by stipulating g

be a family of similar structures (for
R> i

where

i~I Ai

R ~ Ai x Ai) and let i we introduce an equiva-

f ~U g ~ {i : f(i) = g(i)} E U

are equal "almost everywhere").

In

(i.e.

Ai 1"11

TI

we

iEI

introduce a relation

R as follows:

R(f/~U ' g/~U) ~ {L : R

(f(i) , g(i»} E U • i The fundamental result on ultraproducts, proved by Los, is the following: A sentence

~

holds in

{L : Ai ~ ~} E U. products

0

TI

iEI

A.I~U ~

iff it is true "almost everywhere"

i.e.

The above theorem is the key to applications of ultra-

It is difficult to describe the numbp.r of papers devoted to this

construction and its applications. books on ultraproducts.

It is enough to say that there are entire

Important results by Tarski, Scott, Keisler and Shelah

(chronologically) established the ultraproduct construction as a basic tool of model theory. Another 'important result and research technique introduced at the same time is the method of "indiscernibles" of Ehrenfeucht and Mostowski.

Their paper [3J on

models with automorphisms contains a method applied later in numerous studies in model theory (in particular in

Silve~'s

papers on large cardinal numbers).

shown by Ehrenfeucht and Mostowski, for any complete theory

As

T without finite

models there is a model which has a large number of automorphisms (its automorphism group contains given group).

As it often happens the most important result

for applications turned out to be the main lemma on which the theorem is based.

FOUNDATIONS OF MATHEMATICS IN POLAND

135

It states that for a theory as above and an arbitrary linear ordering < A there is a model

M

of

T

sequences (in the sense of

and an imbedding '"

and~)

~

: A

~

of element of

M

r

'"

>

such that increasing of the same length

~(A)

have exactly the same properties. Recently "abstract" model theory has been studied in many places.

In particular,

the study of generalized quantifiers was initiated by Mostowski [19] who considered the generalized quantifier "There exists infinitely many

x

such that ••• " •

Another important result by Mostowski which motivated abstract model theory was his paper [22], where he proves that Craig's and Beth's theorems fail in some extensions of first order logic.

The important results of Mostowski on quantifiers

(also [25]) were the inspiration for the investigations of Fuhrken and Vaught and, later, of Keisler and Krivine and McAloon. The field of Recursion Theory is represented by research on hierarchies of sets and functions, decidability and metamathematical investigations on second order arithmetic (analysis).

Of basic importance was the early work of Mostowski [14]

on the arithmetical (Kleene-Mostowski) hierarchy.

In the late forties and the

fifties Mostowski stUdied the incompleteness of Peano arithmetic.

Apart from

obtaining important extensions of GOdel results, Mostowski wrote a monograph "Sentences undecidable in formalized arithmetic" which is one of the most widely studied monographs of the subject. An important result of the forties was the proof by Szmielew [36], of the decidability of abelian groups. Later research in recursion theory was conducted by Grzegorczyk both on computable functionals and on the hierarchy of recursive functions [6], [7].

Grzegorczyk

also did some important work on decidability. Research in the metamathematics of analysis (on the borderline between recursion theory and set theory) were initiated by a paper of Grzegorczyk, Mostowski and Ryll-Nardzewski [8].

Formalized analysis or the theory dealing with properties

on integers and reals, like every first order theory, has numerous models.

Among

them one discerns models with a faithful interpretation of the natural numbers, these are the so called W-models.

The study of these models by the authors named

above, in COmbination with later deep results by Kreisel, Baiwise, Kripke, Platek, Sacks and others, led to the development of a completely new field; generalized recursion theory.

A slightly different line of research initiated by Mostowski

in [20], was investigation of so called B-models of analysis, which were intensively stUdied by a group of mathematicians around Mostowski in Warsaw [26], [1],

[25].

When we sum up the activity in the Foundations of Mathematics in Poland after W.W. II, it is quite clear that the individual prevailing over the development

W. MAREK

136

of this domain of mathematics in Poland was Andrzej Mostowski.

His efforts led

to the establishment and education of a large group of mathematicians dealing with foundations and the mathematization of the subject.

The lines of research

initiated by Mostowski were and are actively developed in foundational studies. Throughout the post-war period, foundational research was connected with the intensive investigations in computer science conducted by Ehrenfeucht and Pawlak and later by Blikle and Mazurkiewicz.

Inspirations from each side led to

interesting developments in both fields. Another important aspect of Foundations in Poland was an exchange of scientific thought with mathematicians from abroad. periods in Poland one lists:

Among logicians who spent longer

Addison, Benda, Hinman, Lopez-Escobar, Prikry,

Sayeki, Scott, Suzuki and many others. International activity of Polish logicians is also evident in the organization of international conferences and meetings.

Apart from the "Infinitistic methods"

symposium (Warsaw 1959) and an ASL meeting (Warsaw 1968) there was the Logical Semester at the Banach Center in Warsaw (Spring 1973) and more recently a series of three conferences on set theory and hierarchy theory (Sudety Mountains, 1974, 75, 76) was organized by Wroclaw group. The editorial activity of Polish authors - apart from the above mentioned monographs of Mostowski, Rasiowa and Sikorski - consisted in publishing handbooks and monographs including:

"Set Theory" by Kuratowski and Mostowski, "Mathematical

Logic" (in Polish) by Grzegorczyk. Foundational actiVity is conducted in Poland mainly in two centers:

Warsaw

and Wrocraw (although almost every University in Poland hires logicians and Logic and Foundations is a part of the courses required by teaching institutions). Recent research in Warsaw concerns chiefly algebraic logic, foundations of geometry, and foundations of set theory, whereas in

Wroc~aw

foundations of set

theory and model theory are the main subjects of interests. REFERENCES This bibliography does not exhaust important papers written in the post-war period in Poland.

It is just pertinent to the main lines of the article.

[1] Apt, K.R., Marek, W. (1974). Second order arithmetic and related topics Annals of Math. Logic ~, pp. 177-229. [2] Ehrenfeucht, A. (1961). An application of games to the completeness problem for formalized theories. Fund. Math. XLIX, pp. 129-141. [3] Ehrenfeucht, A., Mostowski, A. (1956). Models of axiomatic theories admitting automorphisms, Fund. Math. XLIII, pp. 50-68. [4]

Grzegorczyk, A. (1953).

[5] Grzegorczyk, A. (1955). 202.

Some classes of recursive functions, Diss. Math IV. Computable functionals, Fund. Math. XLII, pp. 168-

FOUNDATIONS OF MATHEMATICS IN POLAND

137

[6] Grzegorczyk, A. (1956). Some proofs of undecidability of arithmetic, Fund. Math. XLIII, pp. 166-177. [7] Grzegorczyk, A. (1974). Warszawa - Dordrecht.

Outline of Mathematical Logic, PWN-Reidel,

[8] Grzegorczyk, A., Mostowski, A., Cz. Ryll-Nardzewski (1958). The classical and w-complete arithmetic, Journal of Symb. Logic 22, pp. 188-206. [9] Kuratowski, K. (1948). Ensembles projectifs et ensembles singuliers, Fund. Math. XXXV, pp. 131-140. [10] ~s, J. (1954). On the categoricity in power of elementary deductive systems and some related problems, ColI. Math. III, pp. 58-62. [11] bas, J. (1955). Quelques remarques, theoremes et problemes sur la classes definissables d'algebres, In: Math. interpretation of formal systems, NorthHolland, Amsterdam, pp. 98-113. [12] Marek, W., 11ostowski, A. On extendability of the models of ZF set theory to the models of KM theory of classes, In: Springer Lecture Notes 499, pp. 460-522. [13] Mostowski, A. (1945). pp. 137-168.

Axiom of choice for finite sets, Fund. Math. XXXIII,

[14] Mostowski, A. (1947). XXXIV, pp. 81-112.

On definable sets of positive integers, Fund. Math.

[15] Mostowski, A. (1949). XXXVI, pp. 143-164.

An undecidable arithmetical statement, Fund. Math.

[16] Mostowski, A. (1951). Some impredicative definitions in the axiomatic set theory, Fund. Math. XXXVII, pp. 111-124. [17] Mostowski, A. (1952). Logic 17, pp. 1-31.

On direct product of theories, Journal of Symb.

[18] Mostowski, A. (1952). Sentences undecidable in formalized arithmetic, North -Holland, Amsterdam, 117 pp. [19] Mostowski, A. (1957). pp. 12-36.

On generalization of quantifiers, Fund. Math. XLIV,

[20J Mostowski, A. (1960). Formal system of analysis based on an infinitistic rule of proof, In: Infinitistic Methods, PWN - Pergamon Press, Warszawa - London. [21J Mostowski, A. (1965). Fennica12, 180 pp.

Thirty years of foundational studies, Acta Phil.

[22] Mostowski, A. (1968). Craig interpolation theorem in some extended systems of logic, In: Logic, Methodology and Phil. of Sciences III, North-Holland, Amsterdam, pp. 87-103. [23] Mostowski, A. (1967). Constructible sets with applications. North-Holland, Warszawa-Amsterdam, 269 pp. [24J Mostowski, A. pp. 220-282.

An exposition of forcing.

In:

PWN-

Springer Lecture Notes 450,

[25] Mostowski, A. (1975). Observations concerning elementary extensions of W-models I, Proceedings of MiS Symposia in Pure Mathematics XXV, pp. 349-355. [26] Mostowski, A., Suzuki, Y. (1969). Fund. Matho LXV, pp. 83-93. [27] Mycielski, J. (1964). pp. 205-224.

On w-models which are not S-models,

On the axiom of determinateness I, Fund. Math. LIII,

W. MAREK

138 [28] Mycielski, J. pp. 203-212.

(1966).

On the axiom of determinateness II, Fund. Math LIX,

[29] Mycielski, J., Cz. Ryll-Nardzewski, (1968). II, Fund. Math. LXI, pp. 271-281.

Equationally compact algebras

[30] Pacholski, L., Cz. Ryll-Nardzewski, (1970). products I, Fund. Math. LXVII, pp. 155-161.

On countably compact reduced

[31] Rasiowa, H. (1974). An algebraic approach to nonclassical logics. North-Holland, Warszawa-Ansterdam.

PWN-

[32] Rasiowa, H., Sikorski, R. (1950). A proof of the completeness theorem of Gadel, Fund. Math. XXXVII, pp. 193-200. [33] Rasiowa, H., Sikorski, R. (1963). PWN, Warszawa.

The mathematics of metamathematics,

[34] Ryll-Nardzewski, Cz. (1959). On the categoricity in power < ~' Bull. Acad. Pol. Sci. Sere Sci. Math. Astronom. Phys. VII, pp. 545 2 548. [35] Sierpinski, W. (1947). L'hypothese generalisee du continu et l'axiome du choix, Fund. Math. XXXIV, pp. 1-5. [36] Szmielew, W. (1955). XLI, pp. 203-271. [37] W)!glorz, B. (1966) • pp. 289-298. [38] WtglOrZ, B. pp. 9-93.

(1967).

Elementary properties of abelian groups, Fund. Math. Equationally compact algebras I, Fund. Matho LIX, Equationally compact algebras III, Fund. Math. LX,

POSTSCRIPT>I< Professor A. Tarski and Dr. S. Givant pointed to us certain inaccuracies and important omissions in our text. As this could lead to misunderstanding we notice most important of them: (1) Reporting the domains of foundational studies we omitted several branches of investigations such as:

al

Foundations of Geometry investigated in Warsaw and other places by Borsuk, Szmielew, Szczerba and others.

bl Universal Algebra as developed in Ryll-Nardzewski and others.

wroc~aw

by Marczewski, Mycielski,

(2) We generally omitted reference to parallel work of other investigators like in case of Ehrenfeucht games reference to parallel Fralsse work on elementary equivalence or in case of Ryll-Nardzewski result on ~o-categoricity independent work of Engeler and Svenonius. (3) We erronously stated that the Axiom of Determinataness implies the principle of dependent choices. As pointed by Professor J. Mycielski the axiom of determinateness implies only the existence of choice function for countable families of reals.

* Added on

March 20, 1977