The Schroeder-Bernstein theorem demonstrates the existence of a total order on these numbers, which are indexed by , an extension of a theorem which only defines a strict order; that theorem is due to Cantor.
A hypothesis about these numbers posits that the exponential of a singular strong limit one equals its successor; that hypothesis is a consequence of a certain generalized hypothesis about these numbers which with ZF implies the axiom of choice. In ZFC, if one sequence of these is termwise less than another sequence of these, then the sum of the first sequence is strictly less than the product of the second sequence, according to Konig's theorem. Multiplying one of these numbers by itself yields the (*) same number. The Schroeder-Bernstein theorem demonstrates the existence of a total order on these numbers, which are indexed by , an extension of a theorem which only defines a strict order; that theorem is due to Cantor. For 10 points, name these numbers that are cardinals, used to represent the size of infinite sets.
A hypothesis about these numbers posits that the exponential of a singular strong limit one equals its successor; that hypothesis is a consequence of a certain generalized hypothesis about these numbers which with ZF implies the axiom of choice.
aleph numbers [accept cardinal numbers until it is read]