Le 29-avr.-05, à 02:32, Stathis Papaioannou a écrit :

Norman Samish writes:

Jonathan,

If it is true that “In infinite time and infinite space, whatever can

happen, must happen, not only once but an infinite number of times,” then

what does probability mean? In your example below, there must be an

infinity of worlds where Colin Powell is president and an infinity of worlds

where your 6-year old niece is president. Are you saying that the Colin

Powell infinity is bigger than the 6-year old niece infinity?

Norman

Yes, there are different sizes of infinity. For example, there are "more" integers than there are even numbers, even though there are an infinite number of both, and there is a 50% chance that a random integer is even. Cantor and all that.

I guess you were in some hurry ... I agree that the measure of "Colin Powell" is bigger than the "niece". But it is a question of measure (cf Jesse post) not of cardinality. Also the cardinality of the set of integers, even integers, prime numbers, square, rationnals .... are all the same (omega), but they are all less than the cardinality of the reals, which is itself less than the cardinality of the set of function from real to <any set with at least 2 elements>. This is proved by the diagonalisation technic (like the incompleteness result) as I explain in the list here : http://www.escribe.com/science/theory/m3079.html and here http://www.escribe.com/science/theory/m3344.html

I guess you were in some hurry ... I agree that the measure of "Colin Powell" is bigger than the "niece". But it is a question of measure (cf Jesse post) not of cardinality. Also the cardinality of the set of integers, even integers, prime numbers, square, rationnals .... are all the same (omega), but they are all less than the cardinality of the reals, which is itself less than the cardinality of the set of function from real to <any set with at least 2 elements>. This is proved by the diagonalisation technic (like the incompleteness result) as I explain in the list here : http://www.escribe.com/science/theory/m3079.html and here http://www.escribe.com/science/theory/m3344.html

Bruno

http://iridia.ulb.ac.be/~marchal/