4. Arithmetic and first-order foundations
4.1. Algebraic terms
Introduction
Algebraic drafts
Sub-drafts and terms
Categories of drafts
Images of drafts
Intepretations of drafts in algebras
4.2. Quotient systemsQuotients
of relational systems
Quotients in concrete categories
Congruences
Quotients
of modules
Intersections of congruences
Minimal congruences
Condensates of injective systems
4.3. Term algebras
Synonymous terms
Term algebras
Term
algebras in injective algebras
Particular languages
Trajectories by transformation sets
Free monoids
4.4. Integers and recursion
The set ℕ
Recursively defined sequences
Addition
Inversed recursion and integers
Multiplication
4.5. Presburger Arithmetic
First-order theories of arithmetic
Presburger arithmetic
Parity
The order relation
Arithmetic with order
Trajectories of recursive sequences
4.6. Finiteness
Equinumerosity and cardinals
Some binary predicates on the class of sets
Structures relating cardinals
Finiteness
Some properties of cardinal
ordering
Equivalent expressions of the axiom of infinity
More results
4.7. Countability and Completeness
Countability of ℕ×ℕ
Countability of finite sequences of integers
Existence of countable term algebras
Interpretation of first-order formulas
The Completeness Theorem
Skolem's Paradox
4.8. More recursion tools
Rebuilding recursion
Morphisms as subalgebras
Another proof of
recursion
Injectivity lemma
A more general form of recursion
4.9. 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
4.11. Developing theories : definitionsDevelopment 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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