1. This is set notation. It doesn't matter what kind of bracket we use. Curly brackets are fine too.
2. This notation shows a set with three elements in it.
3. If we describe a set this way we would be assuming that the set and its elements are two different objects and that elements are in a container we call a "set". To me this is not the case. Elements are the set.
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4. We have two possibilities:
1. Brackets are part of the set
1. When we remove all the elements we are left only with brackets. We call these brackets "an empty set".
2. In this case when we remove all the elements only the brackets remain with no elements in them.
3. In order to be able to say that such a thing as «empty set» exists we must assume that brackets are part of the set and when we remove the elements the brackets remain with nothing in them.
2. Brackets are not part of the set
1. In this case we assume that the elements are the set.
2. When we remove the elements nothing remains.
3. Brackets are not part of the set, they are only a visual help for humans.
4. When the elements a b c are removed from the set [ a b c ] nothing is left: [Here we are supposed to put a symbol of "nothing" but we don't have such a symbol because nothing is nothing. "Zero" is not a symbol for nothing.]
5. When the elements are removed nothing remains and there is no empty set: [ ]. These brackets "[ ]" do not represent "nothing."
6. Therefore if we accept that brackets are not part of the set there cannot be something called "empty set". There may be a metaphysical and mystical object called "nothing set" that is only visible to set theory fanatics.
7. We have a similar case in geometry. In geometric figures dotted lines are not part of the figure. These dotted lines used to be called "occult" lines.
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| Figure with occult lines from Newton's Principiu |
8. I.B. Cohen explains the occult lines like this in his translation of Newton's Principia:
In Newton's day, the adjective "occult" was used to denote "a line drawn in the construction of a figure, but not forming part of the finished figure and also to denote a dotted line.
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| Close-up of the occult line |
5. So mathematicians need to decide if the brackets delimiting the elements are occult or not. One or the other. To uphold both is casuistry.
6. Unfortunately for mathematics, mathematicians chose casuistry. They make the brackets occult or not occult case by case as they wish depending on the situation at hand. This love of casuistry turns their set theory into laughable mathematical junk.
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| Set notation with occult brackets |
7. So, for non-empty sets brackets are occult but for the empty set brackets are real and part of the set. This is called casuistry or rhetorical sophistry and it has no place in mathematics.
8. In short, mathematicians are reifying symbols, namely, the brackets when this suits their needs.
9. To me, elements are the set. When there are no elements there is no set.
10. Mathematicians talk about "elements in a set" as if elements were inside a container.
11. Dear mathematicians, you have to make up your mind and choose one of the options. If the set is the elements then when there are no elements there is no set, and consequently, there is no empty set. ("Empty set" is a set, right? If there is no set, there is no empty version of a set either.)
12. Of course, you may have a placeholder as in computer languages and keep the brackets as placeholders to indicate that you deleted a set but you intend to put it back. But a placeholder is not a set. A placeholder is not the «empty set». A placeholder cannot have properties of a set because it is not a set. You cannot say "a placeholder is a subset of every set". But this is exactly what you are saying when you say, "the empty set is a subset of every set."
13. A nonsense like «empty set» exists to make another nonsense work, namely «mathematical induction».
An example from sports: empty team
1. We are watching a game of football.
2. The players of team A break all the rules of the game and as the game progresses the referee ejects all of the 11 players.
3. Now no players from team A is in the pitch and the referee decides to stop the game.
4. He is right. A team is made of players, when there is no player there is no team.
5. But a set theorist objects. Even if there is no players from team A in the pitch the team as an "empty team" is still in the pitch and the game must go on.
6. So according to the rules of the set theory the team A is still in the pitch because removing the players does not remove the team. Team is still present as an empty team of 0 cardinality.
7. There is no doubt that an empty set is still a set even though it is empty so a team is still a team even though it has no players.
8. Sounds like absurd reasoning to me.
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