What do we mean by the word "distinct" in the first line of the first article Number in this book?
This is the line: "We say of certain distinct things that they form a group when we
make them collectively a single object of our attention."
Example: Suppose a collection of three coins(may be of same or different value) in my pocket. I call the collection of these three coins A. Let's name the three coins C_1,C_2 and C_3. So A can be written as A={C_1,C_2,C_3}. Let C_1 and C_3 be of same value(say 1 rupee each). Are C_1 and C_2 still distinct from each other?
Does the word "distinct" have some other meaning in Fine's book as compared to the general meaning of the English word "distinct"?
Note: this question is also being asked at Math.SE
https://math.stackexchange.com/questions/732370/what-do-we-mean-by-the-term-number-of-things/743660#743660
Best Answer
In set theory, the only concern for "distinctness" is whether or not items have different names. Different names = Distinct.
Items can share a property (such as male/female or the value of a coin) and still be distinct items. Consider the following:
Set1 = {John, Joe, Mary}
We know John and Joe are "male" and "Mary" is female. Do we have three distinct "things"? Yes.
On the other hand, consider 3 people: Joe McDonald, Joe Harris, Jane Fonda. What is the set of all first names? It's {Joe, Jane}. There are only two distinct first names.
You have three different coins. They are distinct because each is a different coin and you gave each one a different name.