1. We denote a set like this: [ a b c ]. It doesn't matter what type of brackets we use. Curly brackets will do too. And if we define "space" as our separator we don't need to use commas either.
2. [ a b c ] shows a set with three elements in it.
3. By defining a set this way, we assume that the set and its elements are two different objects and that these elements are in a container that we call a "set". I don't agree. For me, elements are the set.
4. We have two options:
1. Brackets are part of the set
1. When we remove all the elements, we are left with brackets only. We call these brackets "empty set".
2. According to the assumption that "brackets are part of the set", when we remove all elements, we are left with brackets that do not contain any elements.
3. In order to say that there is such a thing as an "empty set", we need to assume that the brackets are part of the set.
2. Brackets are not part of the set
1. In this case we assume that the elements are the set: Set = Elements
2. When we remove the elements, there is nothing left.
3. Brackets are not part of the set, they are just a visual aid for people.
4. When the elements a b c are removed from the set [ a b c ], nothing remains: [We need to put the symbol for "nothingness" here, but we do not have such a symbol because nothingness is nothing. "Zero" is not a symbol of nothingness.]
5. When the elements are removed, nothing is left and there is no empty set. These brackets "[ ]" do not represent "nothingness". Mathematicians call two brackets an empty set!
6. Therefore, if we accept that brackets are not part of the set, there can be no such thing as an "empty set". An "empty set" can only be a metaphysical and mystical object, such as the "set of nothingness" that only set theory fanatics can see.
7. We have a similar situation in geometry. Dotted lines drawn as visual aids in geometric figures are not part of the figure. These dotted lines used to be called "occult" lines.
8. I.B. Cohen explains the occult lines in his translation of Newton's Principia:
In Newton's time, the adjective "occult" was used to denote "a line drawn in the construction of a figure but not forming part of the finished figure, as well as a dotted line."
9. Therefore, mathematicians need to decide whether the brackets delimiting the elements of a set are occult or not. One or the other. Upholding both violates the certainty principle* of mathematics.
10. But mathematicians chose to define brackets as occult or non-occult as they wish, case by case. This is called casuistry and turns the set theory into a garbage dump of nonsense.
11. So, for non-empty sets the brackets are assumed not to exist, but for empty sets the brackets are real and are assumed to be part of the set. This is called a rhetorical sophistry and has no place in mathematics.
12. In short, mathematicians reify the brackets when it suits their needs, and ignore them when it does not.
13. To me, elements are the set. When there are no elements, there is no set.
14. Mathematicians talk about "elements in a set" as if the elements were in a container.
15. Dear mathematicians, you must make up your mind and choose one of the options. If the set consists of elements only, then when there are no elements, there is no set, and therefore there is no empty set. ("An empty set" is a set, right? If there is no set, there is no empty version of the set.)
16. Computer languages also have the concept of sets. The empty set also exists as a placeholder. Likewise, in mathematics, you can use the empty set as a placeholder. You can replace this placeholder symbol later with an actual set if you wish. However, a placeholder is not a set. A placeholder is not an "empty set" either. A placeholder cannot have the properties of a set because it is not a set. For example, you can't say "a placeholder is a subset of every set". But when you say "the empty set is a subset of every set", you exactly do that.
17. A non-mathematical absurd and nonsensical concept such as the "empty set", is a "patch" created to save another nonsense called "mathematical induction".
* Certainty principle: One word, one definition. Each word is defined once.
Set representation without brackets
1. We can even go one step further and write sets without brackets: a b c
2. This way, when we remove the elements a b c, we can clearly see that no "empty set" remains.
An example from football: Empty team
1. Teams A and B are playing.
2. The players of team A constantly commit hard fouls and one by one receive red cards. And then we see that there are no players from team A left on the field.
3. Since no players from team A are left on the field the referee decides to stop the game.
4. He's right. A team consists of players, without players there is no team.
5. However, a set theorist watching the game objects to referee's decision to stop the game. Even though no players from team A are on the field, the team is still on the field as an "empty team" and the game must continue, says he, this genius set theorist.
6. So, according to the rules of set theory, team A is still in the field because removing players does not mean removing the team from the field. Even if there are no players, there is a team. The team still exists on the field as an empty team with zero cardinality.
7. There is no doubt that according to the set theory, an empty set is still a set even though it has no elements, similarly, a team is still a team even if it has no players.
8. This seems like absurd reasoning to me.
Note: Of course, according to the rules, if a team remains with 7 players, the game will stop anyway. In other words, it is not realistic for 11 players to receive red cards. But isn't it even more absurd to have a team with zero players on the field?
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