There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean http://www.formalontology.it/suszkor.htm), even nonequivalent logicians;-)
Are there nonequivalent randomnesses?
The main two theories we know dealing with randomness and probability is Kolmogorov Probability Theory ( by means of measure theory and Borel $\sigma$-algebras), and Bayesian a priori approach. Are they equivalent enough to say that they are the same in some deeper meaning?
@ Johannes Hahn - no, I am not asking about non isomorphic probability spaces as it would be trivial. rather I as about possible probability theories as different as different are geometries euclidean and noneuclidean. The obvious generalization is to change linearity in the second axiom of probability ($P(A U B) = P(A) + P(B)$ when A,B are independent.
In fact I mention about it after reading probability overview by Terence Tao http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/ He wrotes:
probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space.
Which in my opinion is something very deep ( but I am only a hobbyist;-). So probably if You have initial probability space, and You need to extent it to describe some additional phenomena, You have have done some kind of morphisms between structures of first space and another wider one. Is there a unique, canonical or any defined way of doing this? May we perform this kind of extension always in the same way or there are different ways of doing that? Gives it any predictable and interesting structure?
@sheldon-cooper Bayesian approach to probability is sometimes seen as alternative ( not very well defined) to Kolmogorov axiomatic probability system, because it do not require given a priori probability space. For example in this approach we may say that probability that tomorrow will be could day ( say temp<-10) is defined, whilst in Kolmogorov approach You probably cannot define proper space ( because You cannot have equivalent population of Wednesdays which are tomorrow days, with different temperatures). Agree - when You have possibility to use properly defined probability space this two approach coincides. From Wikipedia: http://en.wikipedia.org/wiki/Bayesian_probability
Bayesian probability interprets the concept of probability as "a measure of a state of knowledge", in contrast to interpreting it as a frequency or a physical property of a system.
@Qiaochu Yuan - of course randomness here is colloquialism. Yes You have right: maybe I just should ask about different probability theories, but note that non-euclidean geometries in analogy are just geometries but in different spaces, with some special properties. So in fact they share the same meaning of geometric set, figure, space, even so complicated objects as coordinate systems, and angle. But they have different relations between them. So I ask about something similar: different kinds of randomness which are in scope of probability theory but describes different relations between for example different classes of ways of extensions of probability spaces. If the last procedure changes anything in resultants;-) Agree - maybe this is not very interesting question. Maybe it would be more interesting in scope of algorithmic information theory and its randomness concept?