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 develop
Development 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
Full text