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Adding infinity to a set does not change it from numerable to non-numerable. There are plenty of ways to add infinity to a set, for example, you can take the natural numbers N and just add infinity: 0, 1, 2, 3, ..., ∞ It's also possible to add another set of natural numbers after infinity, but in this case infinity is usually call ω: https://en.wikipedia.org/wiki/Ordinal_number 0, 1, 2, 3, ..., ω, ω+1, ω+2, ω+3, ... Your example with an array is just: 0, 1, 2, 3, ..., ∞, ..., -3, -2, -1 It's possible to define a "<" relationship there and a topology and most of the other usual stuff, but it's a numerable set. There are similar constructions for non-numerable sets, like the real numbers. You can add one infinite on both sides and you get something that is topologically equivalent to the border of a circle, or add two infinites (one on each side) and you get something that is topologically equivalent to a closed interval, or add even more infinites https://en.wikipedia.org/wiki/Compactification_(mathematics) |
Indeed as you've illustrated the union of countable sets is countable, but unions aren't appropriate when order matters. The use of an array instead of a set data structure highlights this difference. The negative temperatures in the post begin after +∞. Because ordinals are an extension of enumerability we cannot simply drop ∞ into an array location and still call it enumerable. Speaking from turing recognizability / recursively enumerable languages there is no way for a machine to accept negative integers after all positive integers have been input.