Set theory and Foundations of Mathematics1. First foundations of mathematicsUpdates first apply to the below separate pages, then to the one page of all, then the PDF version (24 pages).Some pages focus on model theory, others focus on set theory, others are philosophical |
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| 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 About of ZF set theory Formalizing types and structures as objects of one-model theory |
| 1.5. Expressions and definable structures |
Terms and formulas The diverse kinds of symbols Root and sub-expressions 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 Russell's Paradox The function definer Relations |
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1.9. Axioms and proofs | Statements Realistic vs. axiomatic theories in mathematics and other sciences Provability Logical validity Refutability and consistency |
| 1.10. Quantifiers | Bounded vs open quantifiers Both main 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 order between interpretations
of expressions
The infinite time between models The metaphor of the usual time The strength order of theories Strengthening axioms of set theory The main foundational theories |
| 1.B. Truth undefinability | Standard objects and quotes Truth undefinability theorems The hierarchy of formulas Refined versions Truth predicates Properties of models |
| 1.C. Introduction to incompleteness | Existential classes Provability predicates The diversity of non-standard models First incompleteness theorem Second incompleteness theorem Proving times |
| 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 Zeno's Paradox |
| 2. Set theory | |
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3. Algebra 4. Arithmetic 5. Second-order foundations |
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Other languages :
FR : Théorie des ensembles et fondement des mathématiques : Premiers fondements