What exactly the word 'objects' does mean in the definition of a set, is it an indefinable concept? then how can we say it is a 'well defined object' ? Can I say 'objects' are 'being that exists' in the sence of existence? or Any difficulties to say it can be either a physical entity or an abstract concept?
Also, I have seen that objects are members of set and set is a collection of objects, Is it a circular reasoning?
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What exactly the word 'objects' does mean in the definition of a set
An Object of a set, is any entity (tangible or intangible, real or imaginary, concrete or abstract, any!) that "belongs to" the set and/or contributes to the definition of the set. Whereas, the set (or any set) is a concept of "collection of objects".
is it an indefinable concept? then how can we say it is a 'well defined object' ? Can I say 'objects' are 'being that exists' in the sence of existence?
In a very general sense, objects are undefined (as long as they are not). An object "can be" an indefinable concept, if one wants so or it can be as definable as a 'grape' or a 'wolf'. When one has taken an effort to define the object, it may be called "well defined".
Also, I have seen that objects are members of set and set is a collection of objects, Is it a circular reasoning?
There is a clear distinction between what an object "can be", and what a set "is". A set is a concept of collection of objects. Whereas, an object could be anything. (It could also be a set!)
A set, as a concept, remains undefined. We are just told, how it "relates" to objects i.e. set is a collection of objects.
A set is defined by the objects it holds. Even though the objects it contains is undefined. The set would become "defined" the instant a relation is drawn.
Whereas, an object is as defined as one can define it. What is love? What is regret? What is infinity?
So, is it a circular reasoning? You are making it seems so. Stop seeing objects in context of set and it would cease to be so.
We can do math without worrying about what the elements of a set "really are" in a metaphysical sense. What matters is that sets obey certain logical rules (i.e. the axioms of set theory, which are formalized as sentences of first order logic with the symbol $\in$). There's no circularity in stating those axioms and studying their consequences.