Freiling's Axiom of Symmetry - Relation To The (Generalised) Continuum Hypothesis

Relation To The (Generalised) Continuum Hypothesis

Fix an infinite cardinal (e.g. ). Let be the statement: there is no map from sets to sets of size for which either or .

Claim: .

Proof: Part I :

Suppose . Then letting a bijection, we have clearly demonstrates the failure of Freiling's axiom.

Part II :

Suppose that Freiling's axiom fails. Then fix some to verify this fact. Define an order relation on by iff . This relation is total and every point has many predecessors. Define now a strictly increasing chain as follows: at each stage choose . This process can be carried out since for every ordinal, is a union of many sets of size ; thus is of size and so is a strict subset of . We also have that this sequence is cofinal in the order defined, i.e. every member of is some . (For otherwise if is not some, then since the order is total ; implying has many predecessors; a contradiction.) Thus we may well-define a map by . So which is union of many sets each of size . Hence and we are done.

Note that so we can easily rearrange things to obtain that the above mentioned form of Freiling's axiom.

The above can be made more precise: . This shows (together the fact that the continuum hypothesis is independent of choice) a precise way in which the (generalised) continuum hypothesis is an extension of the axiom of choice.