I’m curious how such a debate even started and what the purpose of even using an empty-set even is.
I'll tell you what happened. It's not too complicated.
@an anarchist started a thread, asking a question, "Is religion dying?" I was one of the first responses. I answered, "Don't worry, religious Jews are having lots of babies." Another person, the one who is arguing in favor of true-lies, said "All the Jews I know are Atheists." I asked, "How many Jews do you know?" They said "it's true even if I know only 1." I objected, "You said Jews plural." They responded "It's true even if I don't know any Jews."
I think the person arguing for true-lies thought I was making an empty assertion, and I didn't know what I was talking about. And that triggered them to make their own empty assertion. which they thought was funny. It was like a little inside joke with themself. But, I wasn't making an empty assertion. It's actually true, and people who actually, umm, know stuff about religious Jews knows it's true. Another poster, an atheist confirmed it, and spoke about it ( religious Jews are having lots of babies ) as if it's common knowledge. And, it is.
I've mentioned this several times, and asked the person arguing for true-lies to simply admit it that this is what happened. But they have neither confirmed nor denied it. Which is kind of a good thing. It is a sign of a person who has
some integrity. Maybe they don't want to say something that is blatantly untrue.
So that's the story of the debate and how it got started.
Regarding the empty-set and its purpose: Way-way-way back mathematicians were philosophers. And this makes sense because math is just another way of describing reality using symbols. In the 1600s, one of these math/philosophers named Leibnitz wanted to develop a math system that was all inclusive and could be used to prove all math theorums. So, he began with what was known to exist, things he could point to with his finger and moved out from there eventually reaching infinity. Or more accurately, his best effort to derive infinity from things that exist. Nothing is empty in this system yet. It's just growing and growing and growing.
After that, he started taking these things that exist and comparing them. Are they the same? Are they different? How similar are they? How different are they? When he compared two things which have nothing in common, the result was peculiar. It was unlike anything that existed in the system that was being developed. It wasn't zero, it wasn't infinity, it wasn't anything. It contradicted everything.
This idea of contradiction, though, can be useful. The example I'm aware of is subtraction and addition. 1 + 1 = 2. The first 1 is positive, the second 1 is positive, the plus sign is positive ( which is collecting the quanitity, it is inclusive ). Everything is positive, there is no contradiction, it is evaluated in an intuitive manner. 1+1 = 2. But! What about -1 + 1 = 0. That's a little wierd. The negative 1 is contradicting the positive sign. Because of this, the positive sign cannot be evaluated intuitively, it must be evaluated counter-intuitively. This equation is not collecting the quanitity, it is not inclusive. It is instead removing the quantity, it is exclusive, it is excluding 1 from the quantity.
And that is how subtraction can be derived. Yes, it can be derived using only simple ideas where the first number is always bigger than the second number. And this forces the result to be a positive number. And positive numbers are easy to see and point to. One can imagine a primitve person with a basket moving the apples from one basket to another. Engaging in commerce, etc.. But how does a person derive a debt? And how is that understood? How can I prove that a debt, or a deficit exists? I can't. I can't derive it. Unless I have this peculiar thing, this thing which always contradicts. Now I can start to make negative numbers, and contradicting operators. "-" is the contradiction of "+".
This thing, the peculiar contradiction was given a name. It was called NULL. Math as we know it was born. And it was all based on starting from known existing things.
Fast forward to the 1900s. Some people thought they had discovered some flaws in the system of Math that had been developed. It, in theory, threatened the entire system. If there were right, nothing could be proven to exist. Nothing. The underlying dilemma, in theory, could be resolved by starting instead with defining "nothing" and forcing it to exist in this system. What we have today that is called "the empty-set" is a hybrid of the original NULL combined with some contradicting ideas in order to ward off any threats to the foundation of Math.
Taking "nothing" and forcing it to exist is itself a contradiction. But, it kind of works... until it doesn't. I'll even say that it mostly works. One of the nice things about this approach is, when evaluating different sets in set theory, the notation, the symbols used, and the resulting symbol for the result match with what one would expect in algebra. A person doesn't have to think too much to get the correct answer. And so, if a person is a math teacher for children, they will defend their axioms to the grave because they appreciate how simple it is.
The only real problem is when the symbolic notation is translated into real world phenomena. Or when real world phenomena is being translated into symolic notation. In other words... the problem is "word problems" or "story problems". Since "nothing" was forced to exist, the concept of "NULL" and "EMPTY" are conflated. But they are not the same. Denying NULL means a person is denying that anything can threaten the foundations of Math. It doesn't have to be that way. In the parellel thread I proposed a very simple way to resolve these issues. NULL remains null. Empty remains empty. All I ask is that the symbolic notation is consistent and that "word problems" are translated accurately.
The only casualty is that these fakey-fakey nonsense statements which claim things like "All I know" = "I know nothing" dissappear. And I cannot yet find a good reason for maintaining them.
