Аннотация:
The set of fusible numbers is the least set of rationals such that 0 is fusible and for fusible x,y if |x-y|<1, then the number (x+y+1)/2 is fusible. There is a less formal description of this set as the set of all intervals of time measurable by certain procedure using fuses. Recently Erickson, Nivasch, and Xu have proved that the true statement "for any rational q there is the nearest to the right fusible number" is independent from first-order Peano arithmetic. For any monotone function f(x_1,..,x_n) we consider the set G_f that is the least set such that 0∈G_f and for any x_1,...,x_n∈G_f if f(x_1,...,x_n)>max(x_1,...,x_n), then f(x_1,...,x_n)∈G_f. The sets G_f could be regarded as generalizations of the set of fusible numbers that itself corresponds to the case of f(x,y)=(x+y+1)/2. Using Kruskal's theorem we show that all sets G_f are well-ordered. Furthermore we obtain some results about the order types of the sets G_f. The talk is based on the joint research with A. Bufetov.
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