## 1. First foundations of mathematics

The last updated version is as the below list of separate html pages and also all in 1 html page.
Previous versions in PDF (13 pages for main text + 7 pages for philosophical aspects).
Some pages focus on model theory, others focus on set theory, others are philosophical
1.1. Introduction to the foundations of mathematics What is mathematics
Theories
Foundations and developments
Platonism vs Formalism
The cycle of foundations
1.2. Variables, sets, functions and operations Constants
Free and bound variables
Ranges and sets
Functions
Operations
1.3. Form of theories The variability of the model
Notions and objects
One-model theory
The diversity of logical frameworks
Examples of notions from various theories
Meta-objects
Components of theories
Set-theoretical interpretation
1.4. Structures of mathematical systems First-order structures
Structures of set theory

Formalizing types and structures as objects of one-model theory
1.5. Expressions and definable structures Terms and formulas
Display conventions
Variable structures
Structures defined by expressions
Invariant structures
1.6. Logical connectives Tautologies
Negation
Conjunctions, disjunctions
Implication
Chains of implications and equivalences
1.7. Classes in set theory Classes, sets and meta-sets
Definiteness classes
Extended definiteness
1.8. Binders in set theory The syntax of binders
set-builder
The function definer
Relations
1.9. Axioms and proofs
Statements
Realistic vs. axiomatic theories in mathematics and other sciences
Provability
Logical validity
Refutability and consistency
1.10. Quantifiers The quantifiers ∃,∀
Inclusion between classes
Rules of proofs for quantifiers
Completeness of first-order logic
1.11. Second-order universal quantifiers Second-order quantification
Second-order Universal Introduction
Second-order Universal Elimination
Incompleteness of second-order logic
Axioms of equality
Defining new binders
1.A. Time in model theory The time of interpretation
The metaphor of the usual time
The infinite time between models
The main foundational theories
1.B. Truth undefinability Truth undefinability
Non-standard models of foundational theories
The hierarchy of formulas
1.C. Introduction to incompleteness Existential classes and provability
First incompleteness theorem
Second incompleteness theorem
The time of proving
1.D. Set theory as a unified framework Structure definers in diverse theories
The unified framework of theories
Set theory as its own unified framework