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Usually in other books, a family is defined as a set whose elements themselves are sets. But I don't think $a$ in a set $\{a, a, a\}$ with repetetion of $a$ and $2$ in $\{1, 2\}$ or $\{2, 4\}$ are sets.

Indexed Families of Sets
Recall that a set is a collection of elements that are all distinct.
Roughly speaking, a family is to be considered as a collection of non necessarily distinct objects called members. For example, $\{a, a, a\}$ is a family with three members $a, a,$ and $a.$ But the same family $\{a, a, a\}$ considered as a set is just the singleton set $\{a\}$ with only one element, $a.$ $$(. . .)$$
For example, the family of sets: $$\{1, 2\}, \{2, 4\}, \{3, 6\} \cdots, \{n, 2n \}, \cdots$$ may be considered as an indexed family of sets indexed by the set $N$ of natural numbers, where $A_n =\{n, 2n\}$ for each $n \in N$.
Source: Set Theory by You-Feng Lin, Shwu-Yeng T.Lin

[Added]

a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class rather than a set. ...The power set $P(S)$ is a family of sets over $S.$ The $k$-subsets $S^{(k)}$ of a set S with n elements form a family of sets. Let $S =\{a, b, c, 1, 2\}$, an example of a family of sets over $S$ (in the multiset sense) is given by $F =\{A_1, A_2, A_3, A_4\}$ where $A_1 = \{a, b, c\}$, $A_2 = \{a, 2\}$, $A_3 = \{1, 2\}$ and $A_4 = \{a, b, a\}$
Source: e-Study Guide for: Passage to Abstract Mathematics: Mathematics, Mathematics by Cram101 Textbook Reviews

"A set does not change if one or more of its elements are repeated." Source: Comprehensive Discrete Mathematics & Structures

[Found a similar explanation as the You-Feng Lin's book]

"A set is a collection of elements that are all distinct.
A family is a collection of members which are not necessarily distinct.
$$\text{For example} \{a, a\}$$
is a family with two members $a,$ and $a.$ On the other hand $\{a, a\}$ considered as a set is $\{a, a\} = \{a\}$, the singleton set. Let $K$ denote a set such that with each element $k \in K$ there is associated a set $A_k$.

Source: Set Theory Essentials by Emil G. Milewski, p.15

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  • $\begingroup$ This extract seems to be mainly about the distinction between sets which cannot have repeated members and families which can. $\endgroup$ – almagest May 27 '16 at 14:44
  • $\begingroup$ See Def and comment page 36: "since elements in a set are distinct, $\{ a, a, b \}$, for example, is not a proper notation of a set and should be replaced by $\{ a, b \}$." $\endgroup$ – Mauro ALLEGRANZA May 27 '16 at 20:21
  • $\begingroup$ The collection $\{ \{ n, 2n \} \mid n \in \mathbb N \}$ is called a family of sets bacause it is made of sets. In this case, the members of the family are themselves sets. $\endgroup$ – Mauro ALLEGRANZA May 27 '16 at 20:24
  • $\begingroup$ @MauroALLEGRANZA But a's in {a, a, a} are not sets, aren't they? $\endgroup$ – buzzee May 28 '16 at 4:49
  • $\begingroup$ See page 35: the book is not about axiomatic set theory and starts with an "intuitive" concept of set: "a set is any collection into a whole of definite, distinguishable objects, called elements". Thus, it is not stated that elements are sets, and the "specification" about "distinguishable objects" implies that in $\{ a, a, a \}$ there is only one element $a$, because we cannot distinguish the different "occurrences" of it into the list. $\endgroup$ – Mauro ALLEGRANZA May 28 '16 at 9:56
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The meaning of $S=\{a,b,c,....\} $ is that the members of the set $S$ are all those and only those things that are each equal to something inside the brackets. So $S=\{a,a,a\}$ means $\forall x\; (x\in S \iff (x=a\lor x=a\lor x=a)\;),$ which is equivalent to $\forall x\;(x\in S\iff x=a)$, which is equivalent to $S=\{a\}.$

Also, when $P(x)$ is some (any) assertion about $x$ then $S=\{x: P(x)\}$ and $S=\{x|P(x)\}$ both mean that the members of the set $S$ are all those and only those $x$ for which $P(x)$ is true.

With regards to whether $1,2,3,4$... are sets, once we have sufficient axioms to prove that there is a unique empty set $\phi$ and that $\{s\}$ and $s\cup \{s\}$ are sets whenever $s$ is a set, we may use the set-theorists' definitions : $0=\phi,$...$1=0 \cup\{0\}=\{0\}=\{\phi \},$...$2=1\cup \{1\}=\{0,1\},$...$3=2\cup \{2\}=\{0,1,2\},$... etc.

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