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A set is defined as a collection of distinct objects.

Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? What is the benefit of defining a set, especially the way it is?

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We define a collection of potentially identical objects as a multiset –  Shaktal Apr 6 at 13:31
    
@MauroALLEGRANZA Let a set $A$ represents the number of soldiers in 3 different platoons. Suppose $A=\{1,2,1\}$, that is the number of soldiers in the first and the third platoon is same. Now can you say $A=\{1,2\}=\{1,2,1\}$. Infact the symbols $1$ ,$2$ etc are not "objects" these are the "occurrences" of the number of things in a finite group. –  user31782 Apr 6 at 17:07
    
@MauroALLEGRANZA I am very confused. We are discussing the definition of a set itself. We define the numbers like 1,2,3 as a set. So it does not make sense to talk about a set of numbers like A={1,3}. I have to go now, I will ask tommorrow. –  user31782 Apr 6 at 17:35

3 Answers 3

up vote 15 down vote accepted

In formal set theory, the closest thing to a definition of a "set" we get is "something which every objects either belongs to or doesn't belong to" -- in other words, if you have any object, you can ask the set whether the thing you have in your hand is one of its member or not and it will answer either yes or no. And it will give you the same answer each time you ask it about the same object.

Not that this description does not openly speak about "distinct" objects -- there just isn't any way for a set to claim to contain some object "more than once" or "only once". All we can do with it is ask whether something is in it or not, and get a yes/no answer.

If you find yourself in a situation where you need to reason about collections that may contain the same object more than once -- and sometimes we do find ourselves in such a situation -- you're free to do that. Such collections are usually known as multisets, and they need a somewhat different formalization than sets, but there's nothing bad about them.

It all depends on what you need, and the words just serve to communicate which of the concepts you're working with at the time. If you're talking about things that give a yes/no answer to "do you contain this?", you say "set". If you're talking about things that give a numeric answer to "how many of this do you contain?", you say "multiset".

The naming reflects that in practice sets turn out to be what you need rather more often than multisets. But don't let that stop you from using multisets when they are what you need.

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In mathematics definitions (and axioms) are the attempts to formalize some informal notions.

Sets come to formalize the notion of a "collection", so we can talk mathematically about collections of objects. The collection makes a distinction between two things which are not equal, but that's it. So if I open my wallet, and look at my coins, while I might have two coins of the same value, they are not the same coin.

Why this notion and not the notion of a multiset, where we also care about repetition? Because we want something bare, with as least structure as possible. You can always add structure to things which don't have any, but you can't remove structure from your atomic notion. (For example, a field is a ring, is an abelian group, is a group, is a set. But if the most basic objects in your world is a field, you can't strip it from structure anymore.)

In modern terms, sets are objects of a universe of set theory. It may sound circular, but only at the level of natural language where I used the term "set theory" to define "set". Where set theory is an informal, but rather well-understood term for theories whose concern is formalizing the notion of set into a mathematical object.

And why do we want them to be with the least structure possible? Because using the axioms of set theory we can prove that we can endow them with pretty much any structure we want (well, up to a certain limitation, but certainly we can endow them with the structure of a multiset).

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I suppose it is also due to historical reasons and the great effort to put mathematics on firm ground and overcome the foundational crisis of mathematics due to the paradoxon from Bertrand Russel and other paradoxa. Zermelo Fraenkel has placed the axiom of extensionality (German: Axiom der Bestimmtheit) stating two sets are equal if and only if they contain the same elements at the beginning of his axioms in 1908 (typically numbered as Zf1). This axiom implies, that $\{x, x\}=\{x\}$ and is a statement of the very standard connection between set equality and set membership.

Since the goal was to find axioms where consistency could be ascertained without any doubt the sets as the foundational objects had to be defined as simple as possible.

On the other hand multisets, intuitively sets with repeated elements were introduced later as natural mathematical objects and are heavily used e.g. in combinatorics. But they are not used to build the foundation of mathematics, at least not in standard set theory.

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Sir I haven't read Bertrand Russel, Zermelo Fraenkel etc. Could you explain how the consistency is ascertained by defining sets as the foundational objects. –  user31782 Apr 6 at 15:11
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@Anupam: Stop reading papers from 100+ years ago. It doesn't do you any good. Mathematics and presentation of topics like set theory has been developed and improved considerably over the course of those years. The only three reasons to read those papers are mathematical history; reading back references to semi-open or open problems; or bragging rights that you read "the masters". The first two you should do after having sufficient grasp of the modern approach of the topic, the last one shouldn't be done at all. –  Asaf Karagila Apr 6 at 15:36
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@Anupam: For historical perspective it's fine. But first you should develop a solid modern perspective on these topics. –  Asaf Karagila Apr 6 at 17:20
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@MauroALLEGRANZA: Do you perhaps mean Devlin's The Joy of Sets: Fundamentals of Contemporary Set Theory? –  J W Apr 6 at 19:08
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@Anupam - Reflecting your serious-minded comments above, here are some more recommendations: An accessible book from P.R. Halmos, Naive Set Theory with a great presentation of basics in small chapters could be valuable for you. I'm deeply impressed from Andre Weils classic Number Theory - An approach through history and from BL van der Waerdens Science Awakening covering ancient Math. Another little treasure is DE Knuths Surreal Numbers - Btw: Reading Stillwell and Devlin is always informative and a great pleasure! Have fun! :-) –  Markus Scheuer Apr 6 at 20:02

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