I’ve been thinking about zero, none, nothing and the empty set time to time… And the infinite.
Can the opposite of zero be infinite? No. Why? Zero is a number, infinite is categorically different concept than a number. In the set N (all whole numbers) is infinite amount of numbers, but none of those is infinite.
Therefore, the opposite of zero is not infinite. And as far as I can see at the moment, semantically zero is none, not nothing.
Does zero really in terms of mathematics have an opposite? Is it neutral in a way, that it doesn’t have an opposite?
As to the empty set, it is an empty collection, one could say ”collection of nothing”. Poetically one could ask: Does the empty set exist? The empty set is ”collection of nothing”. If there is a collection of nothing, a collection that consists of nothing, the collection seems non-existent.
Can non-existent exist? Though, to ask, that does some kind of mathematical concept (the empty set) exist, is quite meaningless…
So, perhaps one can say, that the empty set is some kind of nothing… What’s the opposite to nothing, to something completely non-existent? Everything? Everything of what? Everything of everything that exists.
Can one say that there is an opposite to the empty set? If it would be the set of all sets, there is a problem: Also the empty set would be included in the set of all sets — if the empty set exists in same sense than non-empty sets. If the empty set exists, the set of all sets couldn’t be an opposite to the empty set.
The infinite is difficult concept. I’ve read, that Gauss himself objected at first to bring the actual concept of infinite to the mathematics. He would at first wanted to keep it only in philosophy and religion.
As to the infinite, perhaps, to be precise, one really can’t find an opposite to the infinite, not in terms of mathematics nor by the terms of semantics.