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Yes, the empty set is a subset of all sets. I think some wires got crossed somewhere because whether zero is a subset of some other set doesn’t fit into the que
by klodolph 2y ago
Yes, the empty set is a subset of all sets. I think some wires got crossed somewhere because whether zero is a subset of some other set doesn’t fit into the questions we are trying to answer.
You could pick a construction where the natural numbers are a subset of the integers. This is trivial, but this is a poor strategy overall, because you can always find a bigger set of numbers to work with. You can’t take the “biggest” set of numbers and then define all other sets of numbers as subsets of that. It would be kind of like trying to count down from infinity.
- Affric 2y agoI see where I have gone wrong. Element vs subset. Classic error when thinking fast. Thank you for the thought provoking comments.
- SAI_Peregrinus 2y agoThe Surreal Numbers manage to be a sort of "biggest" ordered field, in NBG, though they're a proper class not a set. All other ordered fields are subfields of the Surreals. Of course "ordered" is doing a lot, since it excludes the complex numbers, vectors, bivectors, etc. Whether elements of some object that isn't an ordered field should be considered "nubmers" is related but different question.