On a notion of smallness for subsets of the Baire space
Author:
Alexander S. Kechris
Journal:
Trans. Amer. Math. Soc. 229 (1977), 191207
MSC:
Primary 04A15; Secondary 54H05
MathSciNet review:
0450070
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let us call a set of functions from into bounded if there is a countable sequence of functions such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of . We show that most of the usual definability results about the structure of countable subsets of have corresponding versions which hold about bounded subsets of . For example, we show that every bounded subset of has a ``bound'' and also that for any there are largest bounded and sets. We need here the axiom of projective determinacy if . In order to study the notion of boundedness a simple game is devised which plays here a role similar to that of the standard games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the  and  (or BanachMazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of whose special cases include countability, being of the first category and boundedness and for which one can generalize all the main results of the present paper.
 [Bu]
D. R. Busch, Some problems connected with the axiom of determinacy, Ph. D. Thesis, Rockefeller Univ., 1972.
 [D]
Morton
Davis, Infinite games of perfect information, Advances in game
theory, Princeton Univ. Press, Princeton, N.J., 1964,
pp. 85–101. MR 0170727
(30 #965)
 [Fe]
Jens
Erik Fenstad, The axiom of determinateness, Proceedings of the
Second Scandinavian Logic Symposium (Univ. Oslo, Oslo, 1970),
NorthHolland, Amsterdam, 1971, pp. 41–61. Studies in Logic and
the Foundations of Mathematics, Vol. 63. MR 0332479
(48 #10806)
 [Fr]
Harvey
M. Friedman, Borel sets and hyperdegrees, J. Symbolic Logic
38 (1973), 405–409. MR 0335248
(49 #30)
 [Fr]
, A basis theorem for L (circulated notes).
 [Gu]
D. Guaspari, Thin and wellordered analytical sets, Ph.D. Thesis, Univ. of Cambridge, 1972.
 [H]
Stephen
H. Hechler, On the existence of certain cofinal subsets of
^{𝜔}𝜔, Axiomatic set theory (Proc. Sympos. Pure Math.,
Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), Amer.
Math. Soc., Providence, R.I., 1974, pp. 155–173. MR 0360266
(50 #12716)
 [Ke]
A. S. Kechris, Lecture notes on descriptive set theory, M.I.T., Cambridge, Mass., 1973.
 [Ke]
Alexander
S. Kechris, The theory of countable analytical
sets, Trans. Amer. Math. Soc. 202 (1975), 259–297. MR 0419235
(54 #7259), http://dx.doi.org/10.1090/S00029947197504192357
 [Ke]
Alexander
S. Kechris, Measure and category in effective descriptive set
theory, Ann. Math. Logic 5 (1972/73), 337–384.
MR
0369072 (51 #5308)
 [Ku]
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
York, 1966. MR
0217751 (36 #840)
 [Ma]
D. A. Martin, determinacy implies determinacy, 1973 (circulated notes).
 [Ma]
, Countable sets, 1972 (circulated notes).
 [Mo]
Yiannis
N. Moschovakis, Descriptive set theory, Studies in Logic and
the Foundations of Mathematics, vol. 100, NorthHolland Publishing
Co., Amsterdam, 1980. MR 561709
(82e:03002)
 [Mo]
Yiannis
N. Moschovakis, Determinacy and prewellorderings of the
continuum, Mathematical Logic and Foundations of Set Theory (Proc.
Internat. Colloq., Jerusalem, 1968), NorthHolland, Amsterdam, 1970,
pp. 24–62. MR 0280362
(43 #6082)
 [Mo]
Y.
N. Moschovakis, Analytical definability in a playful universe,
Logic, methodology and philosophy of science, IV (Proc. Fourth Internat.
Congr., Bucharest, 1971), NorthHolland, Amsterdam, 1973,
pp. 77–85. Studies in Logic and Foundations of Math., Vol. 74.
MR
0540769 (58 #27486)
 [Mo]
Yiannis
N. Moschovakis, Uniformization in a playful
universe, Bull. Amer. Math. Soc. 77 (1971), 731–736. MR 0285390
(44 #2609), http://dx.doi.org/10.1090/S000299041971127891
 [My]
Jan
Mycielski, On the axiom of determinateness, Fund. Math.
53 (1963/1964), 205–224. MR 0161787
(28 #4991)
 [MyS]
Jan
Mycielski and S.
Świerczkowski, On the Lebesgue measurability and the axiom
of determinateness, Fund. Math. 54 (1964),
67–71. MR
0161788 (28 #4992)
 [O]
John
C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in
Mathematics, vol. 2, SpringerVerlag, New York, 1980. A survey of the
analogies between topological and measure spaces. MR 584443
(81j:28003)
 [R]
Hartley
Rogers Jr., Theory of recursive functions and effective
computability, McGrawHill Book Co., New York, 1967. MR 0224462
(37 #61)
 [Sh]
Joseph
R. Shoenfield, Mathematical logic, AddisonWesley Publishing
Co., Reading, Mass.LondonDon Mills, Ont., 1967. MR 0225631
(37 #1224)
 [St]
Jacques
Stern, Some measure theoretic results in effective descriptive set
theory, Israel J. Math. 20 (1975), no. 2,
97–110. MR
0387057 (52 #7904)
 [Bu]
 D. R. Busch, Some problems connected with the axiom of determinacy, Ph. D. Thesis, Rockefeller Univ., 1972.
 [D]
 M. Davis, Infinite games of perfect information, Advances in Game Theory, Ann. of Math. Studies, no. 52, Princton Univ. Press, Princeton, N.J., 1964, pp. 85101. MR 30 #965. MR 0170727 (30:965)
 [Fe]
 J. E. Fenstad, The axiom of determinateness, Proc. Second Scandinavian Logic Sympos., NorthHolland, Amsterdam, pp. 4161. MR 0332479 (48:10806)
 [Fr]
 H. Friedman, Borel sets and hyperdegrees, J. Symbolic Logic 38 (1973), 405409. MR 49 #30. MR 0335248 (49:30)
 [Fr]
 , A basis theorem for L (circulated notes).
 [Gu]
 D. Guaspari, Thin and wellordered analytical sets, Ph.D. Thesis, Univ. of Cambridge, 1972.
 [H]
 S. Hechler, On the existence of certain cofinal subsets of , Proc. Sympos. Pure Math., vol. 13, part 2, Amer. Math. Soc., Providence, R.I., 1974, pp. 155173. MR 50 #12716. MR 0360266 (50:12716)
 [Ke]
 A. S. Kechris, Lecture notes on descriptive set theory, M.I.T., Cambridge, Mass., 1973.
 [Ke]
 , The theory of countable analytical sets, Trans. Amer. Math. Soc. 202 (1975), 259298. MR 0419235 (54:7259)
 [Ke]
 , Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1972/73), 337384. MR 51 #5308. MR 0369072 (51:5308)
 [Ku]
 K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966. MR 36 #840. MR 0217751 (36:840)
 [Ma]
 D. A. Martin, determinacy implies determinacy, 1973 (circulated notes).
 [Ma]
 , Countable sets, 1972 (circulated notes).
 [Mo]
 Y. N. Moschovakis, Descriptive set theory, NorthHolland (to appear). MR 561709 (82e:03002)
 [Mo]
 , Determinacy and prewellorderings of the continuum, Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), NorthHolland, Amsterdam, 1970, pp. 2462. MR 43 #6082. MR 0280362 (43:6082)
 [Mo]
 , Analytical definability in a playful universe, Proc. Fourth Internat. Congr. Logic, Methodology and Philosophy of Science, (Bucharest, 1971), NorthHolland, Amsterdam, 1973. MR 0540769 (58:27486)
 [Mo]
 , Uniformization in a playful universe, Bull. Amer. Math. Soc. 77 (1971), 731736. MR 44 #2609. MR 0285390 (44:2609)
 [My]
 J. Mycielski, On the axiom of determinateness, Fund. Math. 53 (1964), 205224. MR 28 #4991. MR 0161787 (28:4991)
 [MyS]
 J. Mycielski and S. Swierczkowski, On the Lebesgue measurability and the axiom of determinateness, Fund. Math. 54 (1964), 6771. MR 28 #4992. MR 0161788 (28:4992)
 [O]
 J. C. Oxtoby, Measure and category, Springer, New York, 1971. MR 584443 (81j:28003)
 [R]
 H. Rogers, Jr., Theory of recursive functions and effective computability, McGrawHill, New York, 1967. MR 37 #61. MR 0224462 (37:61)
 [Sh]
 J. R. Shoenfield, Mathematical logic, AddisonWesley, Reading, Mass., 1967. MR 37 #1224. MR 0225631 (37:1224)
 [St]
 J. Stern, Some measuretheoretic results in effective descriptive set theory, Israel J. Math. 20 (1975), 97110. MR 0387057 (52:7904)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
04A15,
54H05
Retrieve articles in all journals
with MSC:
04A15,
54H05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704500701
PII:
S 00029947(1977)04500701
Article copyright:
© Copyright 1977 American Mathematical Society
