# zbMATH — the first resource for mathematics

Selctive ultrafilters and homogeneity. (English) Zbl 0649.03036
This paper consists, roughly speaking, of three parts: in the first part the author refines older results on the existence of homogeneous sets for partitions of $$[\omega]^{\omega}$$. An interesting byproduct of these investigations is the result that: if every ccc forcing adds either a Cohen or a random real, then there are no P-points. This sheds some light on the question of Laver and Prikry whether the if-clause is consistent with ZFC.
The second part investigates what happens after Lévy-collapsing a Mahlo cardinal to $$\omega_ 1$$. In that model non-isomorphic selective ultrafilters are radically different: they are mutually generic on $$[\omega]^{\omega}$$ over the class of sets hereditarily definable over the ground model and $${\mathbb{R}}.$$
In the third part, the author shows that under CH there is a group of autohomeomorphisms of $$\beta\omega\setminus \omega$$ that distinguishes non-isomorphic ultrafilters. It is a result of W. Rudin [Duke Math. J. 23, 409-419 (1956; Zbl 0073.396)] that under CH any two P-points can be mapped onto each other by an autohomeomorphism of $$\beta\omega\setminus \omega$$.
Reviewer: K.P.Hart

##### MSC:
 03E05 Other combinatorial set theory 03E55 Large cardinals 03E35 Consistency and independence results 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
Full Text:
##### References:
 [1] Aczel, P., Quantifiers, games, and inductive definitions, (), 1-14 [2] Baumgartner, J.; Taylor, A., Partition theorems and ultrafilters, Trans. amer. math. soc., 241, 283-309, (1978) · Zbl 0386.03024 [3] Blass, A., Orderings of ultrafilters, Thesis, (1970), Harvard University [4] Blass, A., The rudin-keisler ordering of P-points, Trans. amer. math. soc., 179, 145-166, (1973) · Zbl 0269.02025 [5] Blass, A., Amalgamation of non-standard models of arithmetic, J. symbolic logic, 42, 372-386, (1977) · Zbl 0381.03050 [6] Blass, A., Kleene degrees of ultrafilters, (), 29-48, Lecture Notes in Math. · Zbl 0573.03020 [7] A. Blass and S. Shelah, Ultrafilters with small generating sets, to appear. · Zbl 0681.03033 [8] Booth, D., Ultrafilters on a countable set, Ann. math. logic, 2, 1-24, (1970) · Zbl 0231.02067 [9] Daguenet, M., Propriété de Baire de βN muni d’une nouvelle topologie et application à la construction des ultrafiltres, Sém. Choquet. 14e année, (1974/5), Exp. 14 [10] Ellentuck, E., A new proof that analytic sets are Ramsey, J. symbolic logic, 39, 163-165, (1974) · Zbl 0292.02054 [11] Galvin, F.; Prikry, K., Borel sets and Ramsey’s theorem, J. symbolic logic, 38, 193-198, (1973) · Zbl 0276.04003 [12] Grigorieff, S., Combinatorics on ideals and forcing, Ann. math. logic, 3, 363-394, (1971) · Zbl 0328.02041 [13] Henle, J.; Mathias, A.R.D.; Woodin, W.H., A barren extension, (), 195-207, Lecture Notes in Math. · Zbl 0594.03030 [14] Kunen, K., Ultrafilters and independent sets, Trans. amer. math. soc., 172, 299-306, (1972) · Zbl 0263.02033 [15] Kunen, K., Some points in βN, Math. proc. Cambridge phil. soc., 80, 385-398, (1976) · Zbl 0345.02047 [16] Kuratowski, C., Topologie, Vol. I, (1933), Państwowe Wydawnictwo Naukowe Warsaw [17] Louveau, A., Une méthode topologique pour l’étude de la propriété de Ramsey, Israel J. math., 23, 97-116, (1976) · Zbl 0333.54022 [18] Martin, D.A., Borel determinacy, Ann. math., 102, 363-371, (1975) · Zbl 0336.02049 [19] Mathias, A.R.D., Happy families, Ann. math. logic, 12, 59-111, (1977) · Zbl 0369.02041 [20] Moschovakis, Y., Descriptive set theory, (1980), North-Holland Amsterdam · Zbl 0433.03025 [21] Puritz, C., Skies, constellations, and monads, (), 215-243 · Zbl 0257.02047 [22] P.Ramsey, F., On a problem of formal logic, Proc. London math. soc., 30, 2, 264-286, (1930) · JFM 55.0032.04 [23] E. Rudin, M., Types of ultrafilters, (), 147-151, Ann. Math. Studies · Zbl 0158.20104 [24] E. Rudin, M., Partial orders on the types in βN, Trans. amer. math. soc., 155, 353-362, (1971) · Zbl 0212.54901 [25] Rudin, W., Homogeneity problems in the theory of čech compactifications, Duke math. J., 23, 409-419, (1956) · Zbl 0073.39602 [26] Shelah, S., Proper forcing, Lecture notes in math., 940, (1982), Springer Berlin · Zbl 0495.03035 [27] Sikorski, R., Boolean algebras, Ergebnisse der Mathematik, 25, (1969), Springer Berlin · Zbl 0191.31505 [28] Silver, J., Every analytic set is Ramsey, J. symbolic logic, 35, 60-64, (1970) · Zbl 0216.01304 [29] M. Solovay, R., A model of set theory in which every set of reals is Lebesgue measurable, Ann. math., 92, 1-56, (1970) · Zbl 0207.00905 [30] Szpilrajn, E., Sur certains invariants de l’operation (A), Fund. math., 21, 229-235, (1933), (=Marczewski) · JFM 59.0093.01 [31] A. Taylor, P-points and Ramsey subsets of ω, mimeographed. [32] Wimmers, E., The Shelah P-point independence theorem, Israel J. math., 43, 28-48, (1982) · Zbl 0511.03022 [33] Wolfe, P., The strict determinateness of certain infinite games, Pacific J. math., 5, 841-847, (1955) · Zbl 0066.38003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.