Abstract

In set theory without the axiom of choice (AC), we study the relative strength of the principle “No decreasing sequence of cardinals,” that is, “There is no function f on ω such that |f(n+1)|<|f(n)| for all n∈ω” (NDS) with regard to its position in the hierarchy of weak choice principles. We establish the following results: (1) The Boolean prime ideal theorem plus countable choice does not imply NDS in ZF; (2) “Every non-well-orderable set has a well-orderable partition into denumerable sets” (Dℵ0) plus the axiom of choice for well-ordered families of nonempty finite sets does not imply NDS ∨ “The axiom of choice for countable families of nonempty countable sets” in ZFA; and (3) The axiom of choice for well-ordered families of nonempty sets, each of cardinality not greater than or equal to 2ℵ0, does not imply NDS ∨ “countable union theorem” in ZFA. The above results answer corresponding questions left open in works by Howard and Rubin (1998) and Howard and Tachtsis (2016). We also show the following: (4) “Every non-well-orderable set has a well-orderable partition into infinite well-orderable sets” (Dw) is strictly weaker than D ℵ0 in ZFA. This resolves an open problem from Keremedis and Tachtsis (2003).