What does a set contain?

Let us take the set of colours again...

{red, orange, yellow, green, blue, indigo, violet}

This clearly does not contain actual colours. "green" is not actually green.

It contains instead symbols that refer to colours.

If we have the set A = {1,2,3} and B = {4,5,6} we can easily see the process of setting symbols the mean things. A can be used instead of {1,2,3}.

1,2,3,4,5,6 are symbols that refer to numbers, they are not numbers themselves. In roman numerals we would write A = {I, II, III} and B = {IV, V, VI} to mean exactly the same.

So we make a distinction between the symbols that we see and the things they refer to.

So what of the rest of the symbols that we use to construct a set: { } , = what do they represent?

The basic idea in set theory is "containing" and we express containing in symbols by understanding that the brackets enclose the objects to be contained { } and each object is separated by a comma , and the whole set can be called something by using the = sign.

Obviously { } does not by itself "contain" anything without us knowing what the symbols mean, or how they are used. It would appear impossible then to construct the axiom within symbols and the question is raised how do we come by the concept of "containing"? This is running very close to what Wittgenstein was looking at and which he knowingly said he would never complete saying in his lifetime.

Returning to sets. On paper they are sequences of references, or symbols with specific uses. Thus { } we understand in the context to evoke the notion of containing, and the names that follow separated by spaces and commas we understand to evoke the things they are naming. The whole symbol system isolated and by itself is meaningless, but connected with the greater world of maths takes on the usual meanings and use. There is no containing at all within the symbols, indeed no meaning.

So let us turn to the reality of symbols and see if we can create a kind of existential self reference.

{x: x is a symbol for a counting number y & 0 < y < 4}

certainly gives us the set {1,2,3} but the context is different. This is a set of symbols, while A above is a set of numbers.

Furthermore there is a trickly problem... the entire expression is a sequence of symbols {, }, 1, 2, 3 so { and } are being used to articulate the notion of "containment" while 1, 2, 3 are not articulating anything ... they are just themselves... now sets contain references while this set is containing "real things". is that possible?

consider the following (none of this is new from recent posts I'm just seeking a clear expression of it)

The set of all symbols used in the expression for A

{A, =, {, ,, }, 1, 2, 3}

Clearly the symbols have become inadequate because in one place they are performing their function to construct the set, and in other places they are being operated upon.

Now this requirement of a system to be both subject and object is the question... does this have to be so? and why?