# Confusing Axiomatic Set Theory Proof

• darkchild
But these are not separate cases as you're asking the question. It's one case, and the proof just says that if x = y then x = u and y = v. That's all. So why is the proof bothering with u and v? Well, the thing is that the proof isn't told what x and y are. It's just told that they are elements of the set {x,y} and...that's it. It doesn't know anything else about x and y, so it has to work with u and v, which it does know something about.
darkchild
This proof makes no sense to me.

The theorem to be proved is

Theorem 44. {x,y} = {u,v} → (x = u & y = v) V (x = v & y = u)

where {x,y} and {u,v} are sets with exactly two members, which can be either sets or individuals. The proof relies on:

Theorem 43. z $\in$ {x,y} z = x V z = y.

The given proof is:

"By virtue of Theorem 43
u $\in$ {u,v},​
and thus by the hypothesis of the theorem
u $\in$ {x,y}.​
Hence, by virtue of Th. 43 again
(1)
u = x V u = y.​
By exactly similar arguments
(2)
v = x V v = y,​
(3)
x = u V x = v,​
(4)
y = u V y = v.​
We may now consider two cases.
Case 1: x = y. Then by virtue of (1), x = u, and by virtue of (2), y = v."

This is where I got lost. Couldn't I just as easily argue that x = v by virtue of (2) and y = u by virtue of (1)? Or by virtue of (3) and (4)? What's the rationale behind the assumed values of x and y, and couldn't any of the four propositions support it? And if x = y, how could I justify the argument that x and y were equal to two ostensibly different variables without showing that u = v? On one, hand I can sort of see that the assumption x = y and the conditions (1) - (4) would necessarily make it true that u = v...but, given that then either x or y could be said to be equal to either v or u, would there be any need for this part of the proof at all?

In the interest of completeness, the rest of the proof is:

Case 2: x ≠ y. In view of (1), either x = u or y = u. Suppose x ≠ u. Then y = u and by (3) x = v. On the other hand, suppose y ≠ u. Then x = u and by (4), y = v.

Yes, if x = y then basically the four propositions read (replace x and y by (x = y) everywhere):
(1)' u = x = y V u = x = y
(2)' v = x = y V v = x = y
(3)' x = y = u V x = y = v
(4)' x = y = u V x = y = v

Which reduces to
(1)'' u = x = y
(2)'' v = x = y
(3)'' x = y = u V x = y = v

Which reduces to
(1,2) u = x = y = v
(3)'' x = y = u V x = y = v

Which reduces to
(1-3) x = y = u = v

So indeed, you can use any of them to draw that conclusion.

Actually, x = y is a pretty boring case, but you have to handle it :-)

## 1. What is axiomatic set theory?

Axiomatic set theory is a branch of mathematics that deals with the study of sets and their properties based on a set of axioms or basic assumptions. These axioms serve as the foundation for all other mathematical theories and provide a rigorous and logical framework for understanding sets.

## 2. What makes a proof in axiomatic set theory confusing?

A proof in axiomatic set theory can be confusing due to its abstract nature and the use of complex mathematical symbols and notation. It often involves logical reasoning and arguments that may not be immediately intuitive for those unfamiliar with the subject.

## 3. How can one better understand a confusing proof in axiomatic set theory?

To better understand a confusing proof in axiomatic set theory, it is helpful to have a strong foundation in mathematical logic and set theory. It is also important to carefully follow each step of the proof and see how it relates to the axioms and definitions used. Seeking guidance from a professor or mentor can also be beneficial.

## 4. Are there any common mistakes made in proofs of axiomatic set theory?

Yes, there are some common mistakes that are made in proofs of axiomatic set theory. These include circular reasoning, using incorrect axioms or definitions, and making false assumptions. It is important to carefully check each step of a proof to avoid these types of errors.

## 5. Can axiomatic set theory be applied to real-world problems?

Yes, axiomatic set theory has many applications in various fields such as computer science, physics, and economics. It provides a powerful tool for understanding and analyzing complex systems and can be used to solve real-world problems. However, it is important to note that axiomatic set theory is a highly abstract and theoretical subject, and its applications may not always be immediately obvious.

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