Foundational Studies, Selected Works Vol I

675 Pages·1979·12.15 MB·english

by  Andrzej Mostowski

STUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 93
Editors
J. BARWISE, Madison
D. KAPLAN, Los Angeles
H. J. KEISLER, Madison
P. SUPPES, Stanford
A. S. TROELSTRA, Amsterdam
Advisory EditorialBoard
K. L. DE BOUVERE, Santa Clara
H. HERMES, Freiburg i. Br.
J. HINTIKKA, Helsinki
J. C. SHEPHERDSON, Bristol
E. P. SPECKER, Zurich
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM. NEW YORK. OXFORD
ANDRZEJ MOSTOWSKI
FOUNDATIONAL STUDIES
SELECTED WORKS
VOLUME I
Editorial Committee
Kazimierz Kuratowski (Chairman), Wiktor Marek, Leszek Pacholski,
Helena Rasiowa, Czeslaw Ryll-Nardzewski, Pawel Zbierski (Secretary)
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM. NEW YORK. OXFORD
PWN-POLISH SCIENTIFIC PUBLISHERS
WARSZAWA
©
Copyright 1979 by
PWN-POUSH SCIENTIFIC PUBLISHERS
Warszawa
All Right Reserved. No part ofthis publication may bereproduced,stored ina retrieval
system,oftransmittedinany form or byany means,electronic,mechanical,photocopying.
recording or otherwise without the prior permission from the publishers
Publishers:
PWN - Polish Scientific Publishers- Warszawa
North-Holland Publishing Company- Amsterdam. New York. Oxford
Sole distributor for the U.S.A. and Canada:
ELSEVIER NORTH-HOLLAND, INC.
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New York, N. Y. 10017
Library of Congress Cataloging in Publication Data
Mostowski, Andrzej,
Foundational studies.
(Studies in logic and the foundations of mathematics; v. 93)
Some of the papers translated from Polish, French or German.
"AbibliographyofworksofAndrzei Mostowski (compiled by W. Marek)":p.
1. Logic, Symbolic and mathematical- Collected works.
2. Mostowski, Andrzej--Bibliography. I. Kuratowski,
Kazimierz. II. Title. III. Series.
QA9.M7554 511'.3 77-18025
ISBN 0-444-85102-X (v, I)
ISBN 0-444-85103-8 (v. 2)
Printed in Poland
Editorial Note
The present selection of A. Mostowski's work comprises within one
edition the most important ofhis papers, written during nearly forty years
of scholarly activity and scattered so far over various publications. We
hope that it will gain the approval ofall readers who concern themselves
with the foundations of mathematics.
In selecting the papers, the Editorial Committee aimed at including,
on the one hand, the most important of A. Mostowski's contributions to
mathematics and, on the other hand, those papers which have preserved
their topical interest and are the most frequently quoted ones.
xrx-xuv,
In the five introductory articles (pp. vol. f) which discuss
the main trends in A. Mostowski's research, the numbers ofbibliographical
reference marks refer to the full bibliography of A. Mostowski's works
included in volume one.
Most of A. Mostowski's papers, which were originally published in
English, have been reproduced photographically without any changes.
In those papers, a double pagination occurs: at the outer corners the
running pagination of the volume, and at the inner corner, in square
brackets, the reference to the complete bibliography and the pagination
of the original version. It is intended to facilitate the use of the inter-
national reference marks within each paper which refer to the original
page number.
Only a few ofthe papers included in this edition (namely those marked
in the Bibliography with the numbers [1], [2], [3], [5], [6], [7], [51], [79])
have been translated from Polish, French or German, so at the inner
corners in those papers the references to the complete bibliography are
given only.
We are grateful to all publishers of the original papers here included
for their consent to the reproduction of those papers, which has, enabled
us to prepare this edition of A. Mostowski's work and present it to the
readers. In particular the following permissions to reprint Mostowski's
papers were given: Thirty years offoundational studies, Lectures on the
development of mathematical logic and the study of the foundations of
xvrr
mathematics in 1930-1964, Acta Philosophica Fennica (1965),
pp. 1-180; Models ofset theory, Lectures delivered in Varenna, Septemher
VIII EDlTORJALNOTE
1968, C.I.M.E., Italy 1968, pp. 67-179; The classical and os-complete
arithmetic, Reprinted with permission of the publisher American Mathe-
matical Society from Journal of Symbolic Logic, copyright © 1958,
Volume 23, No.2, pp. 188-206; An exposition offorcing, in: Algebra and
logic, Lecture Notes in Mathematics 450, Springer-Verlag, Berlin-Heidel-
©
berg-New York, by Springer-Verlag, Berlin-Heidelberg 1975,pp. 220-
282; Observations concerning elementary extensions of co-models I, re-
printed with permission of the publisher American Mathematical Society
from Proceedings of Symposia in Pure Mathematics, copyright © 1974,
Volume 25, pp. 349-355; On extenddbility ofmodels ofZF set theory to
the models ofKelley-Morse theory ofclasses, in: Logic Conference, Kiel
1974, Lecture Notes in Mathematics 499, © by Springer-Verlag, Berlin-
Heidelberg 1975, pp. 460-542.
Andrzej Mostowski (1913-1975)
Professor Andrzej Mostowski was born in LWDw on November lst,
1913. He studied mathematics in Warsaw in the years 1931-1936. On re-
ceiving his M.A. degree he went to Vienna, where he studied under Kurt
Godel, and then to Zurich. In 1938Mostowski defended hisdoctoral disser-
tation, written under the supervision of Alfred Tarski and devoted to the
interrelationship of various definitions of the notion of infinite set. He then
took up a post at the Meteorological Institute in Warsaw. During the Nazi
occupation of Poland he worked as an accountant at a tile factory. In the
years 1942-1944 he taught at the Underground University of Warsaw,
where he was unofficially appointed a "docent". After the war Professor
Mostowski was for a short time on the staff of the JagiellonianUniversity
in Cracow, where-s-in 1945~hepresented his "habilitation" thesis, devoted
to the axioms of choice for finite sets. In 1946he returned to Warsaw and
was made assistant professor at the University of Warsaw. He became
associate professor in 1947and full professor in 1951.For the next sixteen
years he occupied firstthe ChairofAlgebraand then the Chairofthe Foun-
dations of Mathematics at the University of Warsaw, and in 1952he was
Dean of the Faculty of Mathematics and Physics. In the years 1948-1968
he worked also at the Institute ofMathematics of the Polish Academy of
Sciences, acting as head of the Division of the Foundations of Mathema-
tics.
In 1956 Professor Mostowski was elected an associate memberand in
1963 a full member ofthe Polish Academy of Sciences. He received Po-
lish State Prizes in 1953and 1966 and the Jurzykowski Foundation Prize
in 1972. In 1973'he was elected a member of the Finnish Academy of
Sciences. He was on the editorial boards ofseveral learned journals, inclu-
ding the Fundamenta Mathematicae, the Journal of Symbolic Logic, and
the Annals of Mathematical Logic; he was also a co-editorofthe Seriesfor
Mathematics, Physicsand Astronomyofthe Bulletin ofthe Polish Academy
of Sciences. His long association with the North-Holland Publishing Com-
pany helped to raise the "Studies in Logic and the Foundations of Mathe-
matics" series to its present-day importance.
The academic year 1948-1949 Mostowski spent at the Institute for
Advanced Study at Princeton, and in 1969-1970 he was a fellow of the
x ANDRZEJ MOSTOWSKI (1913-1975)
All Souls College at Oxford. He took part in numerous congresses and
conferences all over the world.
In 1972 Professor Mostowski was elected President of the Section of
Logic, Methodology and Philosophy of Scienceof the International Union
for the History and Philosophy of Science. He died in Vancouver RC.
on August 22nd, 1975.
A Bibliography of works of Andrzej Mostowski
(compiledby W. Marek) (*)
[1] Abzdhlbare Boolesche Kiirper und ihre Anwendung aufdie allgemeine
Metamathematik, Fund. Math. 29 (1937), pp. 34-53.
{2] 0 niezaleznoicidefinicji skonczonosciwsystemie logiki (On the inde-
pendence ofdefinitionsoffinitenessin asystem of logic), Dodatek
doRocznikaPolskiegoTowarzystwaMatematycznego,vol.11(1938),
pp.1-54.
[3] Uber gewisseuniverselle Relationen; Ann. Soc.Pol. Math. 17(1938),
pp. 117-118.
[4] Uber den Begriffeiner endlichen Menge, Sprawozdania Towarzystwa
Naukowego Warszawskiego (Cl.III) 31 (1938), pp. 13-20.
[5] (with A. Lindenbaum) llber Unabhdngigkeit des Auswahlaxioms und
einiger seiner Folgerungen, Sprawozdania Towarzystwa Naukowego
Warszawskiego(Cl.III) 31 (1938), pp. 27-32.
{6] (with A. Tarski) Boolesche Ringe mit geordneter Basis, Fund. Math.
32 (1939), pp. 69-86.
[7] UberdieUndbhangigkeitdes WohlordnungssatzesvonOrdnungsprinzip,
Fund. Math. 32 (1939),pp. 201-252.
[8] Bemerkungen zum Begriff der inhaltlichen Widerspruchsfreiheit,
J. Symbolic Logic 4 (1939), pp. 113-114.
[9] Remarquessur fanote de M. Sierpinski "Un theoreme sur lesfamilles
d'ensembles et ses applications", Fund. Math. 33 (1945), pp. 7-8.
[10] AXiom ofchoiceforfinite sets, Fund. Math. 33 (1945), pp. 137-168.
[11] 0 zdaniach nierozstrzygalnych w sformalizowanych systemach mate-
matyki (On sentencesundecidable in formalized systems of mathe-
matics), Kwartalnik Filozoficzny 16 (1946), pp. 223-277.
[12]* Zarys teorii Galois (An outline of Galois theory), supplement to
Zasadyalgebry wyzszej (Foundations ofhigher algebra) by W. Sier-
piriski,Monografie Matematyczne 11,Warszawa 1946,pp. 371-428.
(*) Unfortunately, dealingwith the bibliographyofa scholarasprolificasA. Mos-
towskione cannot besure that all thepapershave beenfound and introduced. Stillwe
hope that all the major papers have been included.
Wenoticethat thebooksareadditionallymarked withanasterisk.
XII ABIBLIOGRAPHYOF WORKS OF A"IDRZEJ MOSTOWSKI
[13] On definable sets ofpositive integers, Fund. Math. 34 (1947), pp.
81-112.
[14] On absolute properties of relations, J. Symbolic Logic 12 (1947),
pp. 33-42.
[15] On the principle ofdependent choices, Fund. Math. 35 (1948), pp.
127-130.
[16] Proofs ofnon-deducibility in intuitionisticfunctional calculus, J. Sym-
bolic Logic 13 (1948), pp. 204-207.
[17] Onasetofintegersnotdefinablebymeans ofone-quantifierpredicates,
Ann. Soc. Pol. Math. 21 (1948), pp. 114-119.
[18] Un theoreme sur les nombres cos2rck/n, Colloq. Math. l/ (1948),
pp. 195-196.
[19]* Logika matematyczna, Kurs uniwersytecki (Mathematical logic,
University course), Monografie Matematyczne 18, Warszawa-Wro-
claw 1948, pp. VIII + 338.
[20] An undecidable arithmetical statement, Fund. Math. 36 (1949), pp.
143-164.
[21] Sur l'interpretation geometrique et topologique des notions logiques,
Actes du XvemeCongres International de Philosophie (Amsterdam,
11-18 aout 1948), Amsterdam 1949,pp. 610-617.
(22] (with A.Tarski)Arithmeticalclassesand typesofwellorderedsystems,
Bull. Amer. Math. Soc. 55 (1949),p. 65.
[23] A general theorem concerning the decision problem, Bull. Amer.
Math. Soc. 55 (1949), pp. 65-66.
[24] (with A. Tarski) Undecidability in the arithmetic of integers and in
the theory ofrings, 1. Symbolic Logic 14 (1949),p. 76.
[25] La vieet l'oeuvre de S. Dickstein, Prace Matematyczno-Fizyczne 47
(1949), pp. VII-XII.
[26] Kilka refleksji na temat zadafi logicznych z "Matematyki" (Some
remarks concerning problems on logic in "Maternatyka"), Mate-
matyka 3 (3) (1950), pp. 6-11.
[27] Some impredicative definitions In the axiomatic set-theory, Fund.
Math. 37 (1951), pp. 111-124. Correction: ibid. 38 (1952), p. 238.
[28] On the rulesofproofin thepurefunctional calculus ofthefirst order,
J. Symbolic Logic 16 (1951), pp. 107-111.
[29] A classification of logical systems, Studia Philosophica 4 (1951),
pp. 237-274.
(30] (with K. Kuratowski) Sur un probleme de fa theorie des groupes et
a
son rapport la topologie, Colloq. Math. 2 (1951), pp. 212-215.
ABIBLIOGRAPHY OF WORKSOF ANDRZEJ MOSTOWSKI XIII
[31] GroupsconnectedwithBooleanalgebras (Partial solution oftheProb-
lem P92), Colloq. Math. 2 (1951),pp. 216-219.
[32] Onmodelsofaxiomaticsystems, Fund. Math. 39(1952),pp. 133-158.
[33] Ondirectproductsoftheories,J. Symbolic Logic 17(1952),pp. 1-31.
[34]* (with K. Kuratowski) Teoria mnogosci (Set theory), Monografie
Maternatyczne 27, PWN-Polish Scientific Publishers, Warszawa-
Wroclaw 1952, pp. IX + 311.
(35]* Sentences undecidable Informalized arithmetic, An exposition ofthe
theory of Kurt Giidel, Studies in Logic and the Foundations of
Mathematics, North-Holland Publishing Co., Amsterdam 1952,
pp. VIII + 117.
(36] On asystem ofaxioms whichhasnorecursivelyenumerablearithmetic
model, Fund. Math. 40 (1953),pp. 56-61.
[37] A lemma concerning recursive functions and its applications, Bull.
Acad, Pol. Sci. (el. III) 1 (1953), pp. 277-280 (Russian version:
pp. 275-279).
[38] 0 tzw. konstruktywnych prawach IV matematyce (On constructive
principles in mathematics), Mysl Filozoficzna 1 (7) (1953), pp.
230-241.
[39]*(with M. Stark) Algebra wytsza, cz. I (Higher algebra, part I),
Biblioteka Matematyczna 1, PWN-Polish Scientific Publishers,
Warszawa 1953, pp. VI + 308.
[401 (with H. Rasiowa) 0 geometrycznej interpretacji wyraten logicznych
(A geometrical interpretation of logical formulae), Studia Logica
1 (1953), pp. 254-273.
[41J* (with R. M. Robinson and A. Tarski) Undecidability and essential
undecidability in arithmetic, part II of Undecidable theories, by
A. Tarski in collaboration with A. Mostowski and R. M. Ro-
binson, Studies in Logic and the Foundations of Mathematics,
North-Holland Publishing Co., Amsterdam 1953, pp. IX+98.
[421* (with M: Stark) Algebra wytsza, cz. II (Higher algebra, part11),
Biblioteka Maternatyczna 3, PWN-Polish Scientific Publishers,
Warszawa 1954, pp. VII+173.
[43]* (with M. Stark) Algebra wytsza, cz. III (Higher algebra, part III),
Biblioteka Matematyczna 4, PWN-Polish Scientific Publishers,
1st ed., Warszawa 1954, 3rd ed., Warszawa 1967, pp. VII+268.
[44] PodstawymatematykinaVIIIZjeidzie Matematykow Polskich (Foun-
dations of mathematics at the VIIlth Congress ofPolish Mathema-
ticians), Mysl FiIozoficzna 2 (12) (1954),pp. 328-330.