



 
Justin, The phrasing "infinity from real numbers" is potentially confusing. We are not (or should not be) talking about the "infinity" that we adjoin to the real numbers as an upper bound in calculus. In fact, we don't usually use this word to talk about numbers like Beth_{2}: we call them "transfinites". Before we go on, you need to know (perhaps you do already?) that there is a theorem ("Cantor's theorem") that says that every set, empty, finite or infinite, has strictly more subsets than it has elements, in the sense of cardinality. "Cardinality" is a concept that generalizes "number" or "size" of a set to include infinite cases; it is defined in terms of the existence of 11 pairings. The proof of the theorem shows that any supposed such paring must break down, by a sort of "Spanish Barber" paradox. Suppose we have such a pairing: consider the set of all numbers that are not in the set they are paired with. Is it paired with itself or isn't it? [Evil supercomputer freezes and smoke starts to come out...] So there cannot be such a pairing. Beth_{1} is equal to the cardinality of the real numbers, at least in standard axiomatizations. This is because Beth_{0} is the cardinality of the natural numbers, and each [nonlimit] transfinite in the Beth sequence is the number of subsets of the one below it. Real numbers correspond (more or less; there are some technicalities about place value and 0.111... = 1.000... but these can be worked around) to binary (or decimal, but binary is easier) expansions, and every subset of the natural numbers corresponds to an expansion in which exactly those digits equal 1. Beth_{2} is the number of subsets of Beth_{1}  by Cantor's theorem it must be bigger than Beth_{1}. Good Hunting!
 


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