# 4. Arithmetic and first-order foundations

4.1. Algebraic terms

Algebraic drafts

Sub-drafts and terms

Categories of drafts

Intepretations of drafts in algebras

Operations defined by terms

4.2. Term algebras
Condensed drafts

Term algebras

Role of term algebras as sets of all terms

Free monoids

4.3. Integers and recursion

The set ℕ

Recursively defined sequences

Addition

Multiplication

Inversed recursion and relative integers

4.4. Presburger Arithmetic

First-order theories of arithmetic

Presburger arithmetic

Parity

The order relation

Arithmetic with order

Trajectories of recursive sequences

4.5. Finiteness and countability

Axiom of infinity

Finite cardinalities
Countability of ℕ×ℕ

Countability of finite sequences of integers

Rebuilding recursion

4.6. The Completeness
Theorem

Existence of countable term algebras

Interpretation of first-order formulas

The Completeness Theorem

Skolem's Paradox

4.7. Non-standard
models of Arithmetic
Standard and non-standard numbers

Existence of non-standard models

Non-standard models of bare arithmetic

Non-standard models of Presburger Arithmetic

Non-standard models of full first-order arithmetic

4.8. How mathematical
theories developDevelopment levels : proofs, definitions, constructions

The Galois connection (Mod,Tru)

Schemes of definitions

Extending models by undefined structures

Definitions extend models

Definitions preserve sets of isomorphisms

4.9. Constructions

Construction schemes

A development scheme at each level looks like a component at the next level

How constructions preserve isomorphisms

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