Quantities and real numbers

Here will be presented a draft of mathematical formalization (as an axiomatic theory) of the concepts of quantity (as used throughout physics) and real numbers and justification of their main properties.

I consider this as a quite needed work, usually neglected, with the habit of mathematicians to ignore the concept of quantity and focus instead on formalizing the system of real numbers, in a rather complicated way; and the absence of interest or inspiration by physics teachers to bring any mathematical rigor in their courses.
Still some claims here will be admitted without proof, as the full proofs would be a bit too difficult for the main purpose of this text which is to introduce the mathematical language for physics.

Ordered abelian groups

(abelian = commutative)

We start with the concept of ordered abelian group, that is a set G with :
Structures:

• a constant (0)
• a binary operation (+)
• a negation function (-)
• a subset (or unary relation of positivity) G₊⊂G called the positive cone of G (the set of positive elements of G).
  

Axioms:

• (G, 0,+,-) is a commutative group:
• ∀x∈G, x+0=x
• ∀x∈G, x+(-x)=0
• ∀x,y∈G, x+y=y+x
  
• ∀x,y,z∈G, (x+y)+z=x+(y+z)
• G₊ is a sub-monoid of G:
• 0∈G₊
• ∀x,y∈G₊, x+y∈G₊
• ∀x∈G₊, (-x)∈G₊ ⇒ x=0

We define the subtraction x−y = x+(-y) so that z = x−y ⇔ x = y+z.

We also define the product nx for x∈G and n∈ℕ, as equal to (x+...+x) (n times), also equal to the image of n by the unique morphism of monoid from ℕ to G that maps 1 to x.
An element x∈G is called negative if its opposite is positive: x∈G_- ⇔ -x∈G₊. Thus the last condition says that an element cannot be both positive and negative, unless it is 0.
The preorder relation defined by an action of such a G on a set E (by ∀x,y∈E, x≤y ⇔∃g ∈G₊ G, g+x=y) is then a order if every element of E belongs to the trajectory of a free element.
Thus it also defines an order relation on G, as G acts on itself by + and fits this condition.
It can be expressed as
∀x,y∈E, x≤y ⇔ y−x ∈G₊.

Now we introduce the more specific concept of totally ordered abelian group, by strengthening the last condition: every element is either positive or negative, while only 0 is both:

• ∀x∈G, (x∈G₊⇔ x∈G_-) ⇒ x=0.

Then, the order defined by an action of such a G is a total order.
Denoting the set of strictly positive elements as G₊₊ = {x∈G₊|x≠0}=G\G_-, the strict order is
x<y ⇔ x≤y∧x≠y ⇔  x−y ∉G₊⇔ y−x ∈G₊₊.

(Note: it would also be possible to formalize totally ordered abelian groups by means of the monoid G₊ alone, as any other element is negative so that it can represented by its opposite in G₊. To ensure the same properties, such a formalization would involve the operation of subtraction, with some axioms. )

Further possible properties of this system can be considered:

A totally ordered abelian group is

• Archimedean if ∀x,y∈G₊₊, ∃n∈ℕ, y≤ nx.
• Complete if every nonempty subset of G₊ has an infimum (see definition of infimum in the text on Galois connections)
• Discrete if it has a smallest strictly positive element (a unit 1∈G₊₊ such that ∀x∈G₊₊, 1≤ x) and is thus isomorphic with the set of integers; otherwise it is called dense.
  

Being complete implies being Archimedean, but the converse is not true (for example the set of rational numbers is Archimedean but not complete).

Definition. A type of quantities is a totally ordered abelian group that is dense and complete.
Definition. The set of real numbers is a type of quantities denoted ℝ, with another constant symbol 1∈ℝ₊₊ called the unit (thus, not the smallest element of ℝ₊₊).

This "definition" does not exactly specify the system ℝ, but it essentially does so in the sense that for any 2 systems satisfying this definition there exists a unique isomorphism between them. This will come from the following theorem, whose proof (not written were) actually involves a specifically constructed ℝ, but finally works for any type of quantity with a choice of unit:

Theorem. For any totally ordered abelian group G and any ∀x∈G₊₊ such that (∀y∈G₊, ∃n∈ℕ, y≤nx), there exists a unique morphism ("ratio by x") r_x ∈Mor(G,ℝ) such that r_x(x)=1. This morphism is an embedding if G is Archimedean, and it is an isomorphism if G is a type of quantities.

 (to be continued)

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