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
Interpretation of first-order formulas
Covering of equational systems by drafts
Equations
4.2. Quotient systemsQuotients
of relational systems
Quotients in concrete categories
Congruences
Quotients
of modules
Intersections of congruences
Generating congruences
Minimal congruences
Sums of partial algebras
Quotients of algebras by equational systems
Condensates of injective systems
4.3. Term algebras
Synonymous terms
Term algebras
Term
algebras in injective algebras
Particular languages
Long composition
Trajectories by transformation sets
Free modules
Free monoids
4.4. Integers and recursion
The set ℕ
Recursively defined sequences
Addition
Inversed recursion and integers
Multiplication
Commutativity properties
Counter-examples of multiplication structures
Formal description of free monoids
4.5. Presburger Arithmetic
First-order theories of arithmetic
Presburger arithmetic
Equivalence of axiom schemas
The order relation
Parity
Images of recursive sequences
4.6. Finiteness
Equinumerosity and cardinals
Finiteness
ℕ and finite cardinals
Cancellativity in finite monoids
Free commutative monoids
Sums in commutative monoids
Coproducts of commutative monoids
4.7. Countability and Completeness
Numbers of injections
Numbers of combinations
Countable sets
Sums of total orders
Lexicographic order
Countable term algebras
Proof of the Completeness Theorem
Skolem's Paradox
4.8. Limits and colimits
Axiom of dependent choice (DC)
Countable downward Löwenheim–Skolem theorem
Limits in categories
Projective limits
Projective limits of sets
Inductive limits
Birkhoff's variety theorem
4.9. More recursion tools
Rebuilding recursion
Morphisms as subalgebras
Another proof of
recursion
Injectivity lemma
A more general form of recursion
4.10. 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 : definitions and constructions
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
Construction schemes
A development scheme at each level looks like a component at the next level
How constructions preserve isomorphisms
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