Set Description and their Context (Context of a Set)

Question

When I was first introduced to sets the common "description" in place of a formal definition used to be something like a set is a collection of objects that do not have to be related in anyway prior to the formation of the set though the often are in mathematics. And a rule of sets was that an element of a set can be represented once and only once e.g. the following is not a set due to the repeat representation of $2$ $$\{ 1, 2, 3, (4-2) \} $$. Suppose I wanted to create a set which contained the representations (denotation) of the number 2: $$\{2, (\frac{2}{1}), (\frac{4}{2}), 5-3, 2k -2(k-1),...\}$$

In the sense(i.e. not necessarily intended as the philosophical Fregean term sense) of the first example the collection above is not a set but in the sense that it is a collection of representations of $2$ it is.

So it would seem that whether something is a set depends on another property, in addition to being a collection and having each element represented once : that is, context.

What would be a better description(not necessarily a definition) of a set if we want to describe its context? We could use the idea of a universe but that itself is conceived of as a set so the universe cannot provide the context exactly can it? For example, I can see how the universe being $\mathbb N$ or $\mathbb R$ can be of help but the universe of $\mathbb N \cup \ $ (Set of Representations of $\mathbb N$) does not seem like it provides much clearity.

Answer 1

If you want to distinguish between a set of numbers and a set of representations of numbers, the most obvious thing would be to describe the set as a subset of some well-understood (or at least better-understood) set. For example,

The set $\{1,1,2, \frac{2}{1},3\}\subset\mathbb N$ is a set of natural numbers with three distinct elements. Let $A$ be the set of strings of ASCII characters. The set $\{1,\text{I},\text{one},\text{uno}\}\subset A$ is a set of representations of the number $1$, with four distinct elements.

This ties closely to the use-mention distinction, between using the number $1$ and mentioning the number $1$. We often use quotation marks in English to mention words, and this could also be applied to distinguish a number from its representation providing we were clear about it:

The set $\{1, \text{`$1$'}\}$ contains the number $1\in\mathbb N$ and the representation $\text{`$1$'}\in A$.

Of course, if you want to define the set $\mathbb N$ from scratch this may not be very useful. But if all you are concerned about is specifying how the listed elements of a set should be regarded, this ought to be sufficient.

Answer 2

In the usual approach to numbers from (an empty) scratch, the von Neumann rank of a natural number $n$ is itself, that of a rational number is finite, while each real number defined as a Dedekind cut $(Q,Q')$ have von Neumann rank $\omega+1$. This means however that an identity like $\sqrt{2} \times \sqrt{2}=2$ has to be properly interpreted if we wish to view $2$ as a rational number. Thus context needs to be supplied if one is to interpret that identity correctly.

On the other hand, it seems curious that one can't make a distinction between $\mathbb R$ as a set of real numbers and $\mathbb R$ as an element of $\mathcal{P}(\mathbb R)$ as they are exactly identical in the von Neumann hierarchy. Here also context needs to be supplied to explain what you are doing.

All this leads me to a tentative conclusion that the concept of equality is in fact a vague one in modern mathematics. Sometimes this is handled by making a distinction between equality and equivalence, and further between natural and "un"natural isomorphisms, but these seem somewhat ad hoc.