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A further weird problem with this, even given the one-dimensional universe with only six locations, is to form the power set of points in the universe. Now it's
by schoen 9y ago
A further weird problem with this, even given the one-dimensional universe with only six locations, is to form the power set of points in the universe. Now it's true that there's nowhere in the universe to write down the elements of the power set because no representation of them fits anywhere in the universe at a single time. But one might still feel that this power set is well defined. (You could try to describe it as "How many groups of places in the universe are there?", although indeed no useful representation of that question can fit into the universe either.)
It contains 64 elements. ... oops!
- opportune 9y agoIn a sense it's not a weird problem so much as the entire concept of forbidding a 6 is inconsistent with the axioms of mathematics. Reasoning about a universe without 6 will inherently include contradictions because you can always redefine something equivalent to 6 given our axioms.
- schoen 9y agoI think that's a very reasonable way to put it. For example, https://en.wikipedia.org/wiki/Peano_axioms#Set-theoretic_models https://en.wikipedia.org/wiki/Peano_axioms#Set-theoretic_mod... does not depend on having any pre-existing set of 6 objects that you can point out by ostension. Instead, you can construct one, given the concepts of ∅, ∪, and {}!