Talk:Freiling's axiom of symmetry
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The page says:
- Given a function f in A, and some arbitrary real numbers x and y, it is generally held that x is in f(y) with probability 0, i.e. x is not in f(y) with probability 1.
How is this probability counted? My intuity would be that x is in f(y) with chance 0.5, since my method of getting a random subset of real numbers would be by giving each real number chance 0.5 to be in the set. Apparently there is another way of defining 'random' set of real numbers being used here, but which is it? Andre Engels 14:11, 27 Jan 2004 (UTC)
- f(y) is not a "random" set of real numbers at all (the article says nothing about f being random). However, it is countable; this means that f(y) covers a very small fraction of all possible real numbers and that a "random" real number will be in f(y) (or any other countable set) with probability 0. Cwitty 22:15, 27 Jan 2004 (UTC)
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- Oops... I missed that 'countable' in the definition. Thanks. Andre Engels 00:50, 29 Jan 2004 (UTC)
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[edit] Reference?
I seem to recall that this appeared in the Journal of Symbolic Logic in about 1985. I'll add the reference if I find it, unless someone beats me to it. Michael Hardy 23:34, 23 Jan 2005 (UTC)
[edit] Stewart Davidson
The article claims the argument is based on "Stewart Davidson"'s intuition. Who is he? --Aleph4 18:04, 28 May 2006 (UTC)
- Never mind. Stuart Davidson is mentioned in the abstract of Freiling's article. --Aleph4 19:13, 28 May 2006 (UTC)
[edit] smallest non-zero set
Define κ as the largest cardinal such that
- For all sets C of cardinality less than κ it is virtually certain that a random x is not in C. Equivalently, κ is the smallest cardinal such that there is a set D for which the statement
-
- a randomly selected x will be outside D
- will not be almost surely true.
(In traditional mathematical language this is read as "κ is the smallest size of a set which is not of measure zero". This cardinal is usually called the "uniformity of the Lebesgue null ideal", unif(null) or non(null)).
Let B be the set of all functions mapping numbers in the unit interval [0,1] to subsets of the same interval of cardinality smaller than κ. Let A'X be the axiom stating:
- For every f in B, there exist x and y such that x is not in f(y) and y is not in f(x).
Replacing "countable" in Freiling's argument by "of cardinality less than κ" now justifies the axiom A'X. From axiom A'X one can derive (using the function that assigns to each element of D the set of its predecessors, and to all other reals the empty set) that after throwing two arrows at the unit interval, it is virtually certain that not both arrows are in the set D. But as the two arrows are independent, we must be certain that both arrows land outside D. This contradicts the definition of D. -- June 10, 2006. Aleph4
- It is well known that such κ = continuum. You seem to think that a proper subset of a set of given cardinality must have smaller cardinality - it is not so for infinite sets, so your argument fails. Leocat 17:44, 22 October 2006 (UTC)
[edit] Where is the contradiction with the axiom of choice?
If we replace "countable" by any other statement which implies "of Lebesgue measure zero" we will still get probability = 1 that x is not in f(y) and that y is not in f(x). I do not see any contradiction with the axiom of choice. Leocat 21:26, 21 October 2006 (UTC)
