A new definition of subset

 1. I don't think it is necessary to have two words to indicate parts of a set, namely, subset and proper subset.

2. Let's drop "proper subset" and define "subset" to be the current definition of "proper subset". This is what mathematicians do in practice anyway. They use "subset" when they mean "proper subset". 

The above screenshot is a Mathematica function that lists all "subsets" of a set. But except the last one (equality) they are all "proper subsets". This means that mathematicians do not respect their own definitions of "subset" and "proper subset".

A new definition of subset   

1. Given two sets A and B, we say that B is a subset of A if every element of B is an element of A but A has at least one more element.

2. In other words, given that $\mathbf{A} \neq \mathbf{B}$, $\mathbf{B}$ is a subset of $\mathbf{A}$ if all the elements of $\mathbf{B}$ are elements of $\mathbf{A}$ and $\mathbf{A}$ has at least one more element.

3. This definition covers all possible situations a set can be partitioned except when $\mathbf{A}=\mathbf{B}$.

4. If $\mathbf{A}=\mathbf{B}$ it is silly to say that $\mathbf{A}$ includes $\mathbf{B}$ or that $\mathbf{B}$ includes $\mathbf{A}$. Since they are equal neither is a part of the other. No Cantorian rhetorical sophistry can change this fact. It's funny that mathematicians accept easily and never question rhetorical sophistry if it comes from a dead mathematician with fame and authority.

5. There are various ways to draw equal sets as subsets with Venn diagrams.

6. If $\mathbf{A}$ and $\mathbf{B}$ are equal we call them equal. There is no reason to define the word "subset" as a synonym to the word "equal". There is no subset relation between sets that are equal.

7. The "subset" and "proper subset" jargon is invented in order to claim that equal sets are subsets of each other, that is, one is a part of the other. But if $\mathbf{A}=\mathbf{B}$ neither is a part of the other.

8. Mathematicians corrupted the words "whole" and "part" by denying Euclid's Common Notion 5: The whole is greater than the part.

9. In this case we have two wholes: $\mathbf{A}=\{1,2,3\}$ and $\mathbf{B}=\{1,2,3\}$

10. $\mathbf{A}=\mathbf{A}$ must be different than $\mathbf{A}=\mathbf{B}$.

11. To say $\mathbf{A}=\mathbf{A}$ you need to clone $\mathbf{A}$ and create a new set. We must name this new set with a name other than $\mathbf{A}$, e.g., $\mathbf{A'}$ and then we say $\mathbf{A}=\mathbf{A'}$

12. $\mathbf{A}=\mathbf{A}$ doesn't make sense because a whole is not a part of itself.

13. No one is saying that equality implies whole/part relationship but when mathematicians say that every set is a subset of itself they are implying that the whole is part of itself.

14. To say that "a set is a subset of itself" is rhetorical sophistry and play on words "whole" and "part".

15. I respect Euclid's Common Notion 5 and I don't see any reason to deny that the whole is greater than the part. This means simply that the big is greater than the small. Only paradox loving set theoristas deny this Common Notion to save their beloved set theory.

16. If we accept and respect Euclid's Common Notion 5 that the whole is greater than the part, a set cannot be a subset of itself because the whole (the set) cannot be a part (subset) of itself. Equals will not contain one another except in the absurd world of the set theory.

17. The part cannot be equal to the whole. To say that the whole = the part is sophistry because the definitions of "whole" and "part" are being redefined on the fly and on the sly.

18. What justifications do mathematicians have to deny Euclid's Common Notion 5?

Does infinity belong to mathematics

 Bertrand Russell in the Preface to his Introduction to Mathematical Philosophy writes:

Much of what is set forth in the following chapters is not properly to be called "philosophy," though matters concerned were included in philosophy so long as no satisfactory science of them existed. The nature of infinity and continuity, for example, belonged in former days to philosophy, but belongs now to mathematics.

Russell must be the right person to make such a statement, he was both a philosopher and mathematician.

But how does he justify the claim that philosophical infinity is now mathematical infinity. 

I don't think these two can be reconciled. Philosophical infinity is something else and mathematical infinity is something else.

I'm still looking to find a satisfactory definition of the word infinite in mathematics.

Unordered or ordered

1. I'm commenting from this article by James D. Fearon of Stanford University

2. I'm curious to know how we can define a set as an unordered collection then we can talk about "ordered sets." If a set is defined as unordered there cannot be ordered sets.

3. I know that, I'm sure that, there is an explanation, in fact I suspect that there exists a "Zermalo's axiom" that lets us write ordered sets even though we defined sets as unordered. Or we may say "ordered sets are not really ordered in the sense of the original definition so there is no contradiction."

4. So I'd like to know how mathematicians rationalize ordered sets even though an ordered collection in mathematics is called a sequence.

5. Let's look at an explicit example.

6. On page 1, Fearon writes:

The order of elements in the set does not matter.

7. So the set $\mathbf{A} = \{\; 0, 1, 2, 3\;\}$ and $\mathbf{A} = \{\; 1, 2, 0, 3\;\}$ are the same set.

8. A set is defined as an unordered collection of elements.

9. So far so good. A set is unordered. Order of elements does not matter. But now let's look at the next page where he explains infinite sets.

10. He first gives an example with the set construction notation:

$$\mathbf{S}= \{\; s : 0 < s < 1\; \}$$

11.  So, $\mathbf{S}$ is a set containing elements $s_n$ between $0$ and $1$.

12. Fearon explains:

In this method a set is defined by specifying the property or properties that characterizes or is true of all elements of the set.

13. This is standard mathematics too.

14. Then, he gives another example:

Occasionally, you will also see notation like the following for an infinite set:

$$\mathbf{T} = \{\; 0, 1, 2, 3\; \ldots \;\}$$

where it is understood that the ellipsis means that the set proceeds as indicated.

15. But $\mathbf{T}$ is an ordered collection, so it cannot be a set. To call $\mathbf{T}$ a set contradicts the standard definition of a set.

16. $\mathbf{T}$ is not a set because it is ordered. Ordered collections are called "sequences". So why are mathematicians calling a sequence a set?

17. If $\mathbf{T}$ were to be a set I could write it like

$$\mathbf{T} = \{\; 3, 0, 2, 1\; \ldots \;\}$$

18. But now ellipsis makes no sense. We cannot say "the ellipsis means that the set proceeds as indicated."

19. $\mathbf{T} = \{\; 3, 0, 2, 1\; \ldots \;\}$ has no order so we cannot know how to continue this set. Ellipsis implies and assumes order, when there is no order ellipsis makes no sense.

20. So, which one of Zermalo's famous axioms allows us to write ordered sets even though we defined a set to be an unordered collection?

21. I can almost hear the answer. "An ordered set is not really ordered, it is Zermalo-ordered, or Dedekind-ordered or Cantor-ordered which is according to $N$'th order logic is not really order at all. Where is this nasty odor of scholastic sophistry is coming from?

Beauty of the particular

 The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. 

Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance.

[...]

Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact.

From: Concrete Mathematics: A foundation for computer science, by Graham, Knuth and Patashnik

What's in an empty set?

 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? 

Mathematical casuistry: The nothingness of the empty set

 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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Instead of reading watch the video...

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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.

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.

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.

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.

Opting out of the set theory

1. It seems that several prominent mathematicians including Kroenecker objected to the set theory.
2. Mathematics, physics and philosophy are academic and scholastic fields practiced by « learned doctors ». These professional are in competition to own academic subjects.
3. The word « infinity » was owned by philosopher for millennia, but in the 19th century mathematicians « stole » infinity from philosopher and made infinity a mathematical object. Or at least they tried by using the set theory as false witness. To me they failed.
4. The culprit is Herr Cantor. He initiated the sick process of incorporating the metaphysical concept of infinity into the precise science of mathematics. Mathematicians could not quantify infinity instead they transformed mathematics into metaphysics.
5. Mathematics is about definitions.
6. Any symbol in mathematics can be defined arbitrarily by other symbols.
7. Mathematical definitions do not attribute « meaning » to the defined word. If they did, that would be physics not mathematics. In mathematics « a » can mean anything. It only has mathematical meaning as a quantity. In physics « a » can be « acceleration » or some other named quantity.
8. For instance, we have a definition of number by Euclid as multiplicity of units...