There are a couple of subtle pieces lurking just under the surface of this post, but diving into them requires a lot of time and words, so I don't mind them not

There are a couple of subtle pieces lurking just under the surface of this post, but diving into them requires a lot of time and words, so I don't mind them not being included. For example, proofs by contradiction require the Law of the Excluded Middle[1], which some reject. You could go into this for pages and pages, and it's very interesting, but it's also a complete tangent. However, there's one important part that I think deserves some more depth. One of the main points driving the logic on display here is that the mathematical concept of "implies" is different than the normal one, because in logic "A -> B" does not suggest any sort of relationship between A and B. The sentence "My name is Fred, and that implies water is wet" is a true statement in logic, but most non-logicians would consider the statement false because there's no causative relationship. A logic definition of implication is more like "there is no possible universe in which A is true and B is false," which can be satisfied by A being false (or B being true) in all universes, regardless of what B (or A) happens to be and what A and B have to do with each other. It's possible to see this as the crux of most arithmetic paradoxes. If you can slip in a false hypotheses, you can imply any conclusion, no matter how false it may be. All that's required is some sleight-of-hand so that you don't notice when the false axiom is being invoked. A common trick is dividing by an expression that evaluates to zero, without evaluating the expression. For related shenanigans, check out the Principle of Explosion: a single logical inconsistency suffices to prove all statements, including their negations. On an unrelated note, talking about the logical implication operator is difficult because you have to be careful not to use the word "implies." Good fun! [1] "A or ~A", or equivalently, "~~A -> A" for all logical propositions A.