>>16700423>0! = 1 is just a mathematical convention, not metaphysics. It’s defined this way.No, its not just some arbitrary definition that we could simply change willy nilly over night and still have a functional system, it is logically defined because it is the only thing that makes sense in the greater context of the number system and trying to change the definition would break the rest of the factorial function and falsify the whole of combinatorics.
>It’s a formal trick and doesn't imply anything about "nothingness". No, it is a necessary definition that establishes and confirms that nothing is a unit of 0, it is a physically measurable and countable amount.
>"nothing" is a label for absence.No, it is a description of emptiness (hence {0}= empty set) negativity is the label for an absence.
>There is no logical proof that "nothing" is foundational.The empty set is the foundational set. 0 is the origin number. So, logically nothing is the foundational quantity.
>But a thing can't both exist and not exist at the same time in the same way.It can if it is nothing which is why 0 and -0 are the same thing at the same time in the same way despite being opposite numbers by definition.
>It is like saying zero is a numberIt is.
>numbers are thingsThey are.
>zero is somethingYes, it is a number, the origin number, the smallest possible absolute value in existence.
>absence is presenceTotal emptiness is the presence of total absence which is why {0} = |0|.