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Why the set $\{1\}$ is equal to the set $\{1,1,1\}$? A box with 3 equal elements is NOT the same as a box with only one of those elements.

This just doesn't seems right, i can't explain it further than the title.

I do know why there isn't multiplicity in sets due to the axiom of extensionality by the way, but that's not the point!

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    $\begingroup$ Sets are defined to ignore multiplicity. The object you want is called a multiset. Sets focus on the yes/no question of inclusion, not how many. $\endgroup$ – Michael Burr Aug 22 '18 at 4:12
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    $\begingroup$ In mathematics, there aren't three different 1s, there is only one 1. I think of $\{1,1,1\}$ as saying, "1 is in the set, 1 is in the set, 1 is in the set, nothing else is in the set". $\endgroup$ – Rahul Aug 22 '18 at 4:28
  • $\begingroup$ Rahul, that's actually helpful, thanks. $\endgroup$ – Karine Silva Aug 22 '18 at 4:35
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    $\begingroup$ Just because you say someone's name three times doesn't mean you are talking about three people. $\endgroup$ – Lee Mosher Aug 22 '18 at 20:17
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    $\begingroup$ @Lee: In some cases mentioning someone's name three times can summon them in demon form. $\endgroup$ – Asaf Karagila Aug 23 '18 at 3:09
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You ignore equality, in its strongest sense.

If you have three apples, you don't have three of the same apple. You have three apples. Even if you cloned the apple perfectly three times, you still don't have the same apple, you have three copies of the same apple.

This is where the analogy of "sets are like boxes" fails our intuition. Because in real life we often replace "equality" by "sufficiently the same, even if not really the same".

One could also try and make the argument that if I put "you" and "yourself" inside a box, it won't have two copies of you, just the one. But this analogy is weird and awkward, because it seems unnatural to put someone in a box and then put them into the box again.

 

In mathematics, two objects are equal means that they are just the one object. So $\{1\}$ and $\{1,1,1\}$ are the same set. To see why, note that every element of $\{1\}$, namely $1$, is an element of $\{1,1,1\}$. But on the other hand, every element of $\{1,1,1\}$ is either $1$ or $1$ or $1$, and $1\in\{1\}$. So the two sets have the same elements, and are therefore equal, which means that they are the same.

On the other hand, there is a concept of a multiset where repetition matters. You might want to read up on that.

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Sets define a concept that corresponds to the concept of a classification of all things as either in the set or not in the set. If I have e.g. "the set of positive integers less than 10", it doesn't make sense to say how many times "3" is in that set. It's either in the set or it isn't. This concept is useful for many things.

If you want to create a system of your own which differentiates between {1} and {1,1,1} then you can do that. Many, many different ways in fact. Some of those will also correspond to useful things, but each of them is a distinct concept from that of a set. Is your problem just the use of a word that means something different to you?

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Sets are not boxes. Boxes are an analogy that shows some of the features of sets -- but the analogy does not define how sets behave. Or in other words, the purpose of set theory is not to be a mathematical theory of how boxes behave. When you have a situation where sets behave differently than boxes, that just means you've reached the point where the box analogy is not helpful anymore. That is not set theory's problem.

$\{1\}$ is a useful shorthand for the set $$ \{ x \mid x = 1 \}. $$ $\{1,1,1\}$ is a (useful?) shorthand for the set $$ \{ x \mid x = 1 \lor x = 1 \lor x = 1 \} $$ Since $x=1\lor x=1\lor x=1$ is true for exactly the same $x$s as $x=1$ is (namely, $1$ and nothing else), the two sets have the same elements, and therefore they are the same set.

Do not be confused by the fact that $\{1,1,1\}$ appears multiple times in the notation. That just means that we're writing down the condition for being in the set in an unnecessarily redundant way. But the condition still means the same, and therefore the set is the same.

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