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Perhaps this is related. I learned that in fact you do not need for "God made the integers". The definition of every object in mathematics can be bootstrapped f
by ummwhat 6y ago
Perhaps this is related. I learned that in fact you do not need for "God made the integers". The definition of every object in mathematics can be bootstrapped from the empty set. 0 is the set containing the empty set. 1 is the set containing 0 and the empty set. Etc. Perhaps the two symbols you are thinking of are equivalent to "{" and "}".
- JadeNB 6y ago> 0 is the set containing the empty set. 1 is the set containing 0 and the empty set. Etc. Perhaps the two symbols you are thinking of are equivalent to "{" and "}". This is not the usual von Neumann encoding (https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_def...), so I think you may have started one level too encoded. The usual encoding puts 0 equal to the empty set, and n + 1 equal to the union of n (a set with n elements) and {n} (a singleton with 1 element). In other words, if I understand the meaning of 'Etc.', your encoding puts 0 = {Ø}, 1 = {0, Ø} = {{Ø}, Ø}, 2 = {1, Ø} = {{{Ø}, Ø}, Ø}, …, whereas the usual encoding puts 0 = Ø, 1 = 0 ∪ {0} = {Ø}, 2 = 1 ∪ {1} = {Ø, {Ø}}, …. One advantage of this latter encoding is that the set n has n elements, and that the set n is n-times nested, in the sense that longest chain of elements x ∈ y ∈ z ∈ … ∈ n has length n. It also generalises nicely to ordinals (a transfinite generalization of the natural numbers), as explained in the above link (https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_def...).