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In Terry Tao's Analysis I, the following is stated:

Definition 3.1.1 (Informal) We define a set $A$ to be any unordered collection of objects, e.g., $\{3, 8, 5, 2\}$ is a set.

I'm wondering if the term "unordered" could be omitted from this informal definition. And the reasoning is that later, by Definition 3.1.4 given below,

Definition 3.1.4 (Equality of sets). Two sets $A$ and $B$ are equal, $A = B$, iff every element of $A$ is an element of $B$ and vice versa.

it follows that, say, $\{3, 8, 5, 2\}$ is the same as $\{3, 8, 5, 2, 5\}$ are equal. Am I right?

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    $\begingroup$ Why do you want an "informal definition" to be the minimal sequence of words that's logically consistent with the other definitions? Its purpose is to give you an intuition. $\endgroup$ Aug 31, 2022 at 22:51
  • $\begingroup$ @MishaLavrov Just to assure myself, I guess. $\endgroup$ Aug 31, 2022 at 22:56
  • $\begingroup$ As per comment above, the "definition" above is informal because it is not a "real" definition: it is introduced only for didactic purposes, and your question shows that it is not very useful. Simple question: what is a collection? If we do not know, the above definition is meaningless; if we know what it means, we already have the necessary "intuition". $\endgroup$ Sep 1, 2022 at 6:53

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No, we need the word unordered, otherwise $\{1,2\}$ and $\{2,1\}$ would be considered different sets.

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  • $\begingroup$ But after applying Definition 3.1.4, we have that $\{1,2\} = \{2,1\}$, isn't it? $\endgroup$ Aug 31, 2022 at 23:00
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    $\begingroup$ @QuantumHumanLearner Yes I think you're right. It's probably a question of whether you'd like to consider $\{1,2\}$ and $\{2,1\}$ as two different sets that are equal, or simply as the same set. $\endgroup$
    – Yanko
    Aug 31, 2022 at 23:04
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"Unordered" really means that the order of the set elements is not inherently important to the identity of the set. It's only because of this that definition 3.1.4 is even valid - if the order of the set is relevant, then we would need a concept of set equality that took the order into account.

It's possible to impose order on a set, but that's a separate property that has its own characteristics.

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  • $\begingroup$ I gave a thought to your second line but I'm not sure I follow. As I mentioned in another comment, if we simply said that a set is a collection of objects, and then someone asks whether $\{1,2\}$ equals $\{2,1\}$, then I would look at the definition of equality of sets, by which, they indeed are equal. Similarly, $\{1,1,2\} = \{1,2\}$. $\endgroup$ Sep 1, 2022 at 0:01
  • $\begingroup$ If the two sets are equal, then why are they different? $\endgroup$
    – ConMan
    Sep 1, 2022 at 0:02
  • $\begingroup$ Or to put it another way, what's the value in having an equality relation that doesn't distinguish between different sets, or conversely what's the value in being able to recognise sets as being different if they're considered equal to each other? $\endgroup$
    – ConMan
    Sep 1, 2022 at 0:05
  • $\begingroup$ I apologize if I don't follow. My whole point is that nothing is lost if I drop "unordered" from the informal definition given for a set. The equality definition does distinguish $\{1\}$ from $\{1,2\}$. $\endgroup$ Sep 1, 2022 at 0:14
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    $\begingroup$ Well, then you've answered your own question - the order of the elements of the set isn't an important property of the set, so its identity as a set is based on its unordered contents. The thing is, a priori, there's no reason why that has to be true, and so there's nothing stopping us from defining a set as an ordered collection of elements and building everything from there instead. $\endgroup$
    – ConMan
    Sep 1, 2022 at 0:31
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The point of “unordered” is to distinguish sets from “ordered” (or “indexed”) collections like vectors or sequences, for which we can write $S_i$ to refer to a specific item in the sequence. Indexing a set is meaningless.

It's also important to note that membership in a set is a Boolean operation: Either $x \in S$ or $x \notin S$; there's no concept of a value being multiply-included or partially-included in a set. (Those are called multisets and fuzzy sets, which have their uses, but are not “sets” in the conventional sense.)

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    $\begingroup$ Thanks Dan. Your first paragraph makes sense. Just so I understand it, we may omit the term "unordered" but we keep it distinct from an ordered set. I don't understand the significance of your second paragraph in this context (may be I'm missing something). $\endgroup$ Aug 31, 2022 at 23:21
  • $\begingroup$ @QuantumHumanLearner: I mean that it's not correct to say that $\{3, 8, 5, 2\}$ contains one 5 and $\{3, 8, 5, 2, 5\}$ contains two 5's. They both contain 5. The number of times you write it between the curly braces is irrelevant. $\endgroup$
    – Dan
    Aug 31, 2022 at 23:27
  • $\begingroup$ Of course, you'd probably never write a set literal with the same value twice like that, but it might come up with variables, like referring to a set $\{ x, y, z \}$ before determining that $y = z$. $\endgroup$
    – Dan
    Aug 31, 2022 at 23:30
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    $\begingroup$ Thanks for the clarification. My whole thought was that if we simply said that a set is a collection of objects, and then someone asks whether $\{1,2\}$ equals $\{2,1\}$, then I would look at the definition of equality of sets, by which, they indeed are equal. $\endgroup$ Aug 31, 2022 at 23:34

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