A set is an idealized, abstract concept - no notion of an ordering of elements \[\{1, 2\}\]

A product of sets is another set. \[\{1\} \times \{2\} \times \{3\} = \{\{1, 2, 3\}\}\]

As long as a particular ordering is present (in any representation) we suddenly have the notions of a symmetry and rotations.

The pure abstraction is traditionally called a tuple, while its representation is called a Pair (and it is a fundamental type).

The same information, different ordering of elements \[(1, 2) \cong (2, 1)\] This can be seen as just rotation of a structure (or of an observer!) around the vertical axis.

And this is where the notion of an isomorphic “objects” comes from.

Whether or not these are 2 “images” of the same “object” is another question, which, in principle, requires additional knowledge.

Ideally, it is just matter of where an observer is located.

Another kind of an isomorprhism is when we (external observers) ignore (or abstract out) actual structure and consider only “information”.

Yes, it is basically irrelevant how exactly the information in an RNA is represented (aside of being a string of a certain set of symbols (an alphabet) with explicit begin- and end-makers - which is a universal pattern.)

This is not a rotation, but a re-nesting \[((1, 2), 3) \cong (1, (2, 3))\]

Abstractly, different kinds of nesting are isomorphic to each other (same information, different representations).

Thus nested pairs are isomorphic to n-tuples \[((1, 2), 3) \cong (1, 2, 3)\]

In this view (which abstracts out any representations) \(()\) is a unit for a Product (which, structurally, is a particular arrangement) which is what an n-tuple is.

\[\{1\} \times \{2\} \times \{3\} = (1, 2, 3)\]

When we consider a Product of \(n\) sets as a set of all ordered n-tuples, we usually ignore (abstract out) any representation whatsoever.

Lists can be represented as deeply-nested pairs.

\[(1 . (2 . (3 . '()))) \cong '(1 2 3)\] or in another classical language \[(1 : (2 : (3 : []))) \cong '[1, 2, 3]\]

2 is not just a number, it is a minimal compound structure - one more than one.

It could be viewed as (or represent something) structurally different from “ones” which are parts of it.

For instance, a product of two coordinates is structurally different from coordinates themselves (and naturally requires a data-structure to represent it).

The same stuff is going on in underlying Reality, and this is where the patterns and observer’s generalizations came from.