combinatorial

combinatorial - 1 COMBINATORIAL SET THEORETIC PRINCIPLES OF...

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1 COMBINATORIAL SET THEORETIC PRINCIPLES OF GREAT LOGICAL STRENGTH PRELIMINARY REPORT by Harvey M. Friedman Department of Mathematics Ohio State University (for mathematicians with some interest in set theory) (Results obtainedin 1995). We present a surprisingly simple statement of combinatorial set theory of tremendous logical strength. We follow the set theorist's convention that an ordinal α is identified with its set of predecessors. Let j: β β , where β is an ordinal. Let R α x α , where β α . We define j[R] = {(j(c),j(d)): R(c,d)}. We say that j is a nonidentity function if and only if j is not the identity function on β . Consider the following property of an ordinal α : P 1 ( α ). For all R α x α there exists a nonidentity j: β β , β < α , such that j[R] R. Observe that this properties are preserved upwards; i.e., if α < β then P 1 ( α ) implies P 1 ( β ). We introduce the combinatorial statement A
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combinatorial - 1 COMBINATORIAL SET THEORETIC PRINCIPLES OF...

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