Suppose that Freiling's axiom fails. Then fix some formula_26 to verify this fact. Define an order relation on formula_27 by formula_28 iff formula_29. This relation is total and every point has formula_15 many predecessors. Define now a strictly increasing chain formula_31 as follows: at each stage choose formula_32. This process can be carried out since for every ordinal formula_33, formula_34 is a union of formula_35 many sets of size formula_35; thus is of size formula_37 and so is a strict subset of formula_27. We also have that this sequence is "cofinal" in the order defined, "i.e." every member of formula_27 is formula_40 some formula_41. (For otherwise if formula_42 is not formula_40 some formula_44, then since the order is total formula_45; implying formula_46 has formula_47 many predecessors; a contradiction.) Thus we may well-define a map formula_48 by formula_49.