The answer is simple that "a set is a collection of well defined distinct elements ". In my opinion every definition of "something" is used to differentiate what can fall or be called as "something "and what can't . taking this analogy to "set" i will provide you my arguments and where i am getting confused.
As said the elements of set are well defined – which in my opinion means that elements of set have a definition (or in simple words "property") attached to them that makes them to belong in the set . for example {0,2,4,6…} – a set of even number , here each element is well defined because the property of being even is associated with it and thus why it belongs to set. now the second point is the elements are distinct – means they do not affect the existence of each other in simple words apart from belonging to set they do not know each other.
Now my confusion is that are there any elements exists that when combined can not form a set. well the answer is straight forward according to definition that if we can not assign the elements a well definition and they are not distinct . Here In my opinion this definition or why we call a set "a set" turns out to be quite blurry.Reason being take a set with definition associated with it as "All the things that exists in universe" turns out this set contain all things that could be listed out . And since a subset of a set is also a set , my question comes why we defined set in such a way if every collection of elements can share some property with it which in worst case could be traced back to be part of universe. What if set was just collection of objects .You may say that this definition helps us to differentiate between different set by associating properties to it . just for a second assume we will follow this thing then what this is the thing which we can call "No set"- by reversing a definition we can say that anything " that is not collection of well defined distinct object" . Now the question arises is the thing "NO set" exists and if not what what was the point of defining the set in this way if there is no existence of counter part to whom it should be separated, the analogy is same as between say living things and non-living things . now suppose , if there was no non-living things then it would be absurd to say we are living things because at that time we do not know what does makes a non-living .
The question is quite long because i tried to answer to my question but in the end i was not able to reach a conclusion so any help will be greatly appreciated.
Best Answer
In this phrasing, there appear to be two things that bother you: the well-defined and the distinct. Let us treat these in order.
Well defined
The phrase well defined appears to avoid sets like "the set of all foods that are tasty". With this 'definition' it's not actually obvious what is in this set. First off, it's not even clear what a food is: is water a food? I suppose not -- but then again I suppose soup is a food. What about a 50-50 mixture of water and soup? And so on. Even if we know which things exactly are food, it's still not clear what foods are "tasty": tasty to whom? How tasty do they have to be?
A more mathematical example might be "the set of all whole numbers that are very high". Is 10 very high? Is 1000000? Is $10^{10^{10^{10}}}$? Because the elements are not well defined, these do not form sets.
Distinct
A set traditionally can contain each element only once. So, for example, the set $\{1, 1, 1, 2, 2, 3, 3, 5, 6, 7\}$ is just the same thing as $\{1, 2, 3, 5, 6, 7\}$. An element can only be or not be in a set, it cannot be in there multiple times.
You could decide that this is not what you want: if you want an element to be allowed into a set multiple times, you get what is called a multiset.