One Plus One Equals Two (2006)

The main point of the parent article is not about 1+1=2, but about the importance of the concept of ordered pair in mathematics and about how the introduction and use of this concept has simplified the demonstrations that were much too complicated before this.

While the article is nice, I believe that the tradition entrenched in mathematics of taking sets as a primitive concept and then defining ordered pairs using sets is wrong. In my opinion, the right presentation of mathematics must start with ordered pairs as the primitive concept and then derive sequences, sets and multisets from ordered pairs.

The reason why I believe this is that there are many equivalent ways of organizing mathematics, which differ in which concepts are taken as primitive and in which propositions are taken as axioms, while the other concepts are defined based on the primitives and other propositions are demonstrated as theorems, but most of these possible organizations cannot correspond to an implementation in a physical device, like a computer.

The reason is that among the various concepts that can be chosen as primitive in a mathematical theory, some are in fact more simple and some are more complex and in a physical realization the simple have a direct hardware correspondent and the complex can be easily built from the simple, while the complex cannot be implemented directly but only as structures built from simpler components. So in the hardware of a physical device there are much more severe constraints for choosing the primitive things than in a mathematical theory that only describes the abstract properties of operations like set union, without worrying how such an operation can actually be executed in real life.

The ordered pair has a direct hardware implementation and it corresponds with the CONS cell of LISP. In a mathematical theory where the ordered pair is taken as primitive and sets are among the things defined using ordered pairs, many demonstrations correspond to how various LISP functions would be implemented. Unlike ordered pairs, sets do not have any direct hardware implementation. In any physical device, including in the human mind, sets are implemented as equivalence classes of sequences, while sequences are implemented based on ordered pairs.

The non-enumerable sets are not defined as equivalence classes of sequences and they cannot be implemented as such in a physical device but at most as something of the kind "I recognize it when I see it", e.g. by a membership predicate.

However infinite sets need extra axioms in any kind of theory and a theory of finite sets defined constructively from ordered pairs can be extended to infinite sets with appropriate additional axioms.

What definition takes up fewer components in a digital circuit is a terrible reason. The whole point of math is we can reason about the most conceptually simple idea, rather than with engineering constraints. Sets existed before circuits! And before digital the only “hardware representation” was an analog voltage, which cannot easily represent a pair.

Also it’s not even true. There is no hardware representation for the ordered pair containing the earth and the moon. You now need a bit encoding of the information.

The distinctions of infinite constructions you mention are already well understood. See “recursively enumerable set”.

Ordered pairs are trivially definable in terms of sets. It’s a distinction which does not change any of the foundational proofs and gives you no new insight. This is like arguing that bounded vs counted ranges are foundationally important. We can show they are equivalent in one paragraph and move on.

An actually new ideas will give new results.