Consequences of Arithmetic for Set Theory

  title={Consequences of Arithmetic for Set Theory},
  author={Lorenz Halbeisen and Saharon Shelah},
  journal={J. Symb. Log.},
In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider seq1 1 (A), the set of all sequences of A without repetition. We compare seq1 1 (A) , the cardinality of this set, to P(A) , the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is… CONTINUE READING


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