Definition of a Set - equivalence endowed?

Question

I've read through a few posts on the definition of Sets and that ZFC is the most commonly used.

If it is fair to say that a set is a Collection of Distinct Object then what must be imposed for distinction to be enforced? It seems that the definition of a set in this sense has an equivalence binary operator to evaluate distinction, i.e. S = { some objects | no two objects in the set (object1 = object2) }

And then, in the evaluation of object1 = object2 do we need a Boolean Algebra first?

Sorry if this is all hand-wavy and wish washy with respect to Sets. I'm very much a novice trying to make sense of the basics (where I can).

If anyone knows of any good resources to study algebraic structures (in particular if they start primitive and build their way up).

Thanks in Advance

Answer 1

A set is something that has an answer, either "yes" or "no", whenever you ask "is such-and-such one of your elements?" This is the only thing it can do.

If you ask the set about the same thing more than once, you will get the same answer -- either yes, yes or no, no. If you ask it about things that are not the same, you can get different answers.

When semi-formal introductions to set theory say that a set is a collection of "distinct" elements, what they mean is neither more nor less than there is no way to ask the set "how many of this particular thing do you have?" -- because it either has that thing or does not have it, yes or no.

So, for example, the notations $\{2,3\}$ and $\{2,2,3\}$ both describe the same set, namely the one that answers yes when we ask it about the number $2$, and answers yes when we ask it about the number $3$, and answers no to all other questions. Once we've decided that our set will answer yes to $2$, making that same decision once again will not actally change the answers we get.

You may be thinking about a particular equivalence relation, but sets don't care about that. If you have two things that are not the same thing, yet your equivalence relation says they're related, there can (and will) still be sets that contain one but not the other.

Answer 2

There are very interesting questions along the lines of what you ask — e.g. see setoid.

However, the specific notion I think you are asking about is not that; the point of emphasizing "distinct objects" is to highlight that membership is a "yes/no" question: "Is this element a member of that set?", rather than a "how many" question: "How many copies of this element does that set have?"

The latter notion — where membership is a "how many" type question — is generally called a multiset, or a bag.

Answer 3

In the formal language of set theory ("L.o.s.t") with the axiom system ZFC there is no definition of "set" nor of "$\in$". There are only the binary-relation symbols $=$ and $\in$ and no function-symbols nor symbols for constants. (Symbols such as $\emptyset$ are, formally, abbreviations.)

If you take an arbitrary binary relation $R$ there is no reason to expect that $(\;\forall z\; (zRx\iff zRy)\;)$ implies $x=y.$ E.g. if $xRy\iff (x,y\in \Bbb N \land |x-y|/2\in \Bbb N)$ then $\forall z (1Rz\iff 3Rz),$ but $1\ne 3.$

The Axiom of Extensionality in ZFC declares that $x=y$ iff $x,y$ have the same members: $\forall x\;\forall y (\;( \forall z\;(z\in x\iff z\in y)\iff x=y\;).$

I recommend an introductory book on set theory, preferably a modern one. There are many. And for fun, Stories About Sets, by Vilenkin.