The title is pretty self-explanatory. Published December 2021.
Here is one of the issues discussed:
You may have heard before that the empty set is a subset of every set. I disagree with this claim. It leads to a self-contradiction.
If the empty set is a subset of every set then it must be a subset of itself, since it is a set. But if the empty set has a member or subset (itself) then it is not really empty. This is verified by the fact that in math {} is treated differently than {{}}; one is compared to an empty container, the other is like an empty container inside of an empty container. If these are different then the container that is inside of the other one must be ‘something’ not ‘nothing’, and thus the outer container is not completely empty. So is the empty set empty or not? Perhaps we could call this The Empty Set Paradox.
Now suppose that we have the set of non-empty sets; if the empty set is a subset of that one then the categorical statement ‘An empty set is a non-empty set’ or ‘All things identical to the empty set are non-empty sets’ would have to be true because the empty set would be a species or type of non-empty set. But how could any set be a species of its opposite? An empty set, or the empty set, is not a type of non-empty set (its class complement) in the way that equilateral triangles are a specific type (or subset) of triangle, or black bears are a subset of bears. That is also self-contradictory. Perhaps we could call this The Non-Empty Set Paradox.
So how might one resolve these paradoxes? My proposed solution is stated in the text.