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In my notes the axiom of extensionality is that 'two equal sets contain the same elements', this language suggests to me that there are two different sets that happen to be equal, which suggests there is an infinite number of sets with the same elements. I see this definition given many places and it causes me some confusion.

This seems to defeat the point of equality, which usually defines the 'same value', the above seems incorrect then, is it just a choice of language and clearly there exists only one set with a certain group of elements, or can we have two seperate sets with the same elements?

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    $\begingroup$ Yes............ $\endgroup$ May 9, 2023 at 7:33
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    $\begingroup$ Just as if $A,B$ are numbers then $A=B$ means they are the same number. $\endgroup$
    – dxiv
    May 9, 2023 at 7:34

2 Answers 2

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Generally in math, you can always consider two things $A,B$ without it being implicit that $A\neq B.$ Another way to phrase the definition is: if $A$ has the same elements as $B,$ then $A=B.$ This avoids the potentially misleading “two sets” terminology.

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These are Different View-Points / Interpretations of the Same thing , at the foundational/fundamental level , rarely affecting the higher level.


( View-Point 1 ) Exactly 1 Set with Same Elements :

Imagine having a BlackBox with all the Sets you are currently considering.
You put your hand into the BlackBox , to take out a Set $A$ , then note the elements on Paper , then put it back.
Then you again put your hand in , to take out some Set $B$ , then note the elements on Paper.
If the elements are Exactly Same , then you can claim that you picked the Exact Same Set on both attempts , that is , $A=B$.
That is what the Set Equality Definition is stating.

This Interpretation claims that there can be Exactly 1 Set with Same Elements , no mater what the Set Definition was.


( View-Point 2 ) Multiple Sets with Same Elements :

In the BlackBox , you do not have the Sets with elements listed. You have the Sets with the Definition ( Set rule ) listed.
Then you put your hand in the BlackBox to pick a Set $A$ , then use the attached Definition ( Set rule ) to list the elements of Set $A$ & DO NOT PUT the set back.
Then you again put your hand in the BlackBox to pick a Set $B$ , then use the attached Definition ( Set rule ) to list the elements of $B$.

If Set $A$ had the Definition $A=\{\text {Set of Integers between -1 & +1}\}$ & Set $B$ had the Definition $B=\{\text{Set of Integer Solutions to Equation X^3-X=0}\}$ , then your notes will be like this :
$A=\{-1,0,+1\}$
$B=\{-1,0,+1\}$
Again that Set Equality Definitions states that $A=B$ , even though these two had Different Definitions ( Set rules )

This Interpretation claims that there can be Multiple Sets with Same Elements , which Differ by the Set Definition , with no Change in Elements.


View-Point 2 is more general & more useful than View-Point 1.

We can check the analogy with Integers , rational numbers , functions , fractions , Etc :

(Eg 1) Consider Distinct fractions $1/2,2/4,3/6,4/8$ , which are all Equal to rational number 0.5 , which itself is Equal to no other rational number.
Our Definition of fraction Equality is : $(x_1/y_1=x_2/y_2) \iff (\exists r \in Q : x_1/y_1=r \land x_2/y_2=r)$
Our Definition of fraction Sameness is : $(x_1/y_1 \equiv x_2/y_2) \iff ( x_1 = x_2 \land y_1 = y_2 )$

Like-wise , we can have Sameness of Sets & Equality of Sets :
Two Sets $A$ & $B$ are Same $\iff$ they have the Same Defining Predicate & Same Universal Set.
Two Sets $A$ & $B$ are Equal $\iff$ they have the Same Elements.
With this , it is easy to see that the claim "2 Sets are Equal ...." makes sense : We can take 2 sets which are not "Same" , check the elements to conclude whether they are Equal or not.

(Eg 2) In trigonometry , we have $X = \sin^2 x = 1 - \cos^2 x = Y$ : Why do we have Identity / Equation when talking about Same Number ?
Well , the numbers are Equal , though Definitions are not Same !
We are claiming that the Definitions give Same & Equal Numbers !

Like-wise , we have $X= \sin {2 \theta} = 2 \sin{\theta} \cos {\theta} = Y$ , where Equality always holds even though we have Different Expressions on the 2 Sides.
Putting in Sets , we can claim that the Set of Points given by $X$ is Equal to the Set of Points given by $Y$.

(Eg 3) We have $f(x)=2x+10$ & $g(x)=3x-4$ , then we may want to know when $f(x)=g(x)$ : we will get $x=14$ , where $f(14)=38$ & $g(14)=38$.
We arrive at $38$ via Different Calculations.
We can then say that though $f(x)$ & $g(x)$ are not Same , they have Equal value at $x=14$.

This type of thinking is at the foundational/fundamental level , thus we can not Casually Detect much effect at the higher level.

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  • $\begingroup$ No, this is all just incorrect. Equal sets with different defining predicates are equal as sets. The predicates aren’t equal, and that’s fine. You say this correctly in your own first example: numbers can be equal even though the definitions are the same. It’s no different for sets. To answer your questions in the following three examples: yes of course, numbers that are equal are the same; no of course, different people aren’t equal and nobody would ever suppose they are just because their ages are equal; similarly, of course functions can have some equal values without being equal. $\endgroup$ May 9, 2023 at 15:32
  • $\begingroup$ (1) I agree with "Equal sets with different defining predicates are equal as sets." , whereas It is a matter of Interpretation / Philosophy whether they are the Same Set. (2) We can have a Set theory where we allow many Different Predicates to give Equal but not Same Set , though we must Define what we mean by Sameness. (3) Putting in analogy : there are many Distinct fractions $1/2,2/4,3/6,4/8$ , which are all Equivalent & Equal to rational $1/2$. Among rational numbers , $1/2=1/2$ , nothing else is Equal to $1/2$. @KevinArlin $\endgroup$
    – Prem
    May 9, 2023 at 15:57

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