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1001 mathematical worlds.

Can we exchange the order of integration of a bounded positive function of two variables 0<x,y<1 assuming just that both of the iterated integrals exist? Is there a space filling curve such that at each point one of its coordinates has a derivative?It is possible to prove that these problems cannot be solved.First we will explain what it means and that by the K. Godel completeness theorem this impossibility is equivalent to the fact that the answers to these questions depend on the mathematical world in which we consider them. The method of forcing invented by P. Cohen can produce an infinite number of such worlds (models) where different mathematics is true. The Godel incompleteness theorem implies that the situation is irreparable by a simple addition of a list of new axioms.Putting aside the philosophical aspects of this situation we will describe some of the mathematical advances in recent decades: 1) building constellations of models with prescribed configurations of true statements 2) Searching for models with few pathological objects and where structurally elegant mathematics is true (e.g., every function from the real line into itself is monotone on an uncountable set etc.,) 3) Providing mathematics for non-standard models in physics.