Reals and Positive Partition Relations

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

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On Finite Polarized Partition Relations

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-040-1 ◽

1969 ◽

Vol 12 (3) ◽

pp. 321-326 ◽

Cited By ~ 17

Author(s):

V. Chvátal

Keyword(s):

Cardinal Numbers ◽

Partition Relations ◽

Image Position

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write(1)if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is writtenand means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

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FORCING AND THE HALPERN–LÄUCHLI THEOREM

Journal of Symbolic Logic ◽

10.1017/jsl.2019.59 ◽

2019 ◽

Vol 85 (1) ◽

pp. 87-102

Author(s):

NATASHA DOBRINEN ◽

DANIEL HATHAWAY

Keyword(s):

Partition Relations ◽

Finite Products

AbstractWe investigate the effects of various forcings on several forms of the Halpern– Läuchli theorem. For inaccessible κ, we show they are preserved by forcings of size less than κ. Combining this with work of Zhang in [17] yields that the polarized partition relations associated with finite products of the κ-rationals are preserved by all forcings of size less than κ over models satisfying the Halpern– Läuchli theorem at κ. We also show that the Halpern–Läuchli theorem is preserved by <κ-closed forcings assuming κ is measurable, following some observed reflection properties.

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