Suppose the beth function is defined as follows:
beth[a]=|V[a]|, for all ordinals a. Here V[a] is the ath level of the cumulative hierarchy, and || is the cardinality function defined as in for example the Levy Basic Set Theory book.
The above definition of beth is equivalent to the usual one in the presence of the axiom of choice, and of course that beth function is known to be normal and hence continuous.
My question is whether in the absence of the axiom of choice the above beth function is continuous. Continuity here does not require the cardinalities to be ordinals but means |V[a]| is the lub of the set of |V[b]| such that b<a, for each limit ordinal a, and according to Levy’s definition of lub of cardinalities.
The non existence of Lubs for some sets of cardinals makes me doubtful but I’m too naive to know for sure.
Best Answer
By the answer of Farmer S to my latest question, the answer to this question is no. Lub property on limit ordinal stages of cumulative hierarchy implies the axiom of choice. Thanks Farmer!