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.
|
(Claim) |
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.
Read more about this topic: Freiling's Axiom Of Symmetry
Famous quotes containing the words relation, continuum and/or hypothesis:
“There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
—Blaise Pascal (1623–1662)
“The further jazz moves away from the stark blue continuum and the collective realities of Afro-American and American life, the more it moves into academic concert-hall lifelessness, which can be replicated by any middle class showing off its music lessons.”
—Imamu Amiri Baraka (b. 1934)
“The wheels and springs of man are all set to the hypothesis of the permanence of nature. We are not built like a ship to be tossed, but like a house to stand.”
—Ralph Waldo Emerson (1803–1882)