The axiom of constructibility: guide for mathematician by K. J. Devlin

By K. J. Devlin

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8 (1962) ~16-~19. [15] .... , A remark on a paper b 2 N. G. de B r u i j n and P. Erdos, Nederl. Akad. We tensch. Proc. Set. A ~5 (i962) 343-345. 28 W. Luxemburg, A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras, Fund. Math. 55 (1964) 239-247. [~7] J. Mycielski, Some compactifications of general algebras, Coll. Math. 13 (1964) I-9. [~8] D. Scott, Prime Ideal Theorems for rinzs, lattices~ and boolean (abstract), Bull. Amer. Math. See. 60 (195~) ~8-~. Matematiczne 35 (1963J.

Of ultrapowers. of proving algebra Graph Colourin S Theorem. 3 we have Now an algebra B has an ultrafilter and 2 is compact. 4. Now every is a small proper algebra inclusion we subalgebra of B: into two atoms of B and it is then clear h o w on A to one on B. Hence any compact algebra, and 2, is injective. Let ~ be the variety it is easy to check of distributive that 2 is injective lattices. Following in D iff Stone's theorem 24 (see [22]) compact simple on the separation and Stone's theorem combinatorial (3) Let ~ be the variety compact divisible is injective group, is true.

2] R. D. Kopperman, On the axiomatizabilit~ of uniform spaces , J. Symb. Logic 32 (1967), 289-294. [3] W. Sierpi~ski, Sur un probl~me concernant les sous-ensembles croissant du continu, Fund. Math. 3 (1922), 109-112. [4] W. Sierpi~ski, Sur une propri@t@ des ensembles ordonn~s, Fund. Math. 36 (1949), 56£67. ' H~ MODELS AND H~-CATEGORICITY ~ Nlgel Cutland Hull, England Introduction 'Hyperarithmetlc model theory' was first investigated by Cleave ~2J; in thls paper we develop a theory of hyperarlthmetic and H~1 models of first order theories, obtaining analogues of results of classical model theory wlth the analogy: hyperarithmetlc H~ \ *--* countable Z~ <--~ of cardinallty ~ .

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