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