This question has its origin in this post. More specifically, in giving answers to the following questions,
- In what way does the collection of all sets consist of sets?
- What are collections of sets actually?
It was written here that,
The answers to 1. and 2. is that you have to set up some theory that allows you to talk about collections. The standard theory do this with is first-order logic, in which every formula with free variables is a description of a collection (aka a class), and these collections may also be described by auxillary functions and relations.
answer it has been commented that,
The point is that you can view first-order logic itself as a primitive "Theory of Collections", hence avoiding infinite regress. We are not describing or postulating how actual collections behave, but we are describing the language by which we express or refer to the collections. The potential for regress lies in disagreeing on what it takes to specify a formal language (since of course, we never actually have access to ALL formulas unless there were only finitely many), but certainly specifying the rules for producing well-formed statements and the rules for judging them justified is considered sufficient description: most humans seem to be expected to apply some sort of "pattern-matching" to the rules and to comprehend from them how to form well-formed statements (i.e. how to speak) and how to judge whether a statement is justified (what a proof is). So you don't to actually go ahead and develop a formal theory of languages before you can specify a formal language, but there is an important philosophical shift from mental to social activity in the level of justification required.
However when I asked my teacher the same question (the question linked above) he replied that,
We actually even don't need to worry about a Theory of Collection at all. When you learn a language, say English, do you ever worry whether the alphabets a,b,c,... belong to some collection or not? Surely not.
But I think that we do not worry about that mainly because we have an explicit list of what my alphabets are and also we have a precise criteria (so, we are inclined to believe) to determine whether a particular symbol is our alphabet or not.
My teacher also said that the correct way to view the situation is, "We have some symbols, whose meaning we don't know and we don't need to know. We also don't know and neither need to know where the symbols exist, whether in a collection or not." But when I asked him what he exactly meant when he said that we "have some symbols", he said that it is simply a variant of saying that the symbols exist.
If the symbols exist, then they exist where? (In case of alphabets, if I want to be painfully philosophical, I may be tempted to say that the alphabets exist in some "imaginative world" of my mind - which will in this case be "collection".)
Is it even meaningful to ask 'where the symbols exist'?
I think that it is necessary to have a Theory of Collection because the phrase "[w]e have some symbols" is very much vague. Consider the following question for example,
- Does the meaning (or sense) of "have" in "have some symbols" the same as that of "have" in "have a doll"? How can you say that?