hardmathhardmath 37.5k2020 gold badges7979 silver badges147147 bronze badges $endgroup$ nine 2 $begingroup$ No need to use that there are infinitely numerous primes: the subgroups $nmathbb Z$ for every $n in mathbb N$ are all unique as they have distinct indices. $endgroup$
In the situation of a set of genuine numbers with all of its limit details (a shut set), Cantor confirmed that the rest established can be a set of limit details of exactly the same dimension because the set of serious figures (identified as a "ideal" established). The technique is usually generalised to sets the place branches transfinite sequences and (dropping the usage of trees) to metric spaces and specific topological spaces. For more reading through on Cantor's mathematics I'd advise the traditional books by J. Dauben and M. Hallett, and for a readable tackle what would now be referred to as descriptive set theory, File. Hausdorff's Established Principle (through the 1930s).
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Why "An infinite cyclic group is isomorphic for the set of integers" and never "there is bijection $varphi: mathbb Z to langle g rangle$" 0
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one $begingroup$ The result is sort of counter-intuitive. How can summing up merchandise of finite numbers (the values on the random variable) with finite numbers (the chance in the random variable taking over that worth) be infinite? $endgroup$
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Extra right when you used Conway's surreal figures. In the surreals, It might be organic to associate $one+one+ldots$ with $omega$, While there remains an ambiguity as pointed out by Karolis. $endgroup$
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