With the Axiom of Choice, for any infinite cardinal but, without, we cannot conclude any relationship between and for an arbitrary infinite cardinal. In this paper, we give some properties of in the absence of and compare them to those of for an infinite cardinal. Among our results, we show that “ for any infinite cardinal and any natural number n” is provable in although “ for any infinite cardinal ” is not.

We write for the cardinality of the set of finite sequences of a set which is of cardinality. With the Axiom of Choice (), for every infinite cardinal where is the cardinality of the permutations on a set which is of cardinality. In this paper, we show that “ for every cardinal ” is provable in and this is the best possible result in the absence of. Similar results are also obtained for : the cardinality of the set of finite (...) sequences without repetition of a set which is of cardinality. (shrink)

We study independent families of functions and permutations and associated cardinal characteristics and. In this paper, we show that these cardinals lie between the cardinals and and give some related consistency results.