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about (purpose and author) - foundations of physics - other topics and links other languages : fr − ru − tr − es 1. first foundations of mathematics (details) - all in 1 file (35 paper pages) - obsolete pdf in 13 + 7 pages . 1.1. introduction to the foundations of mathematics 1.2. variables, sets, functions and operations 1.3. form of theories: notions, objects, meta-objects 1.4. structures of mathematical systems 1.5. expressions and definable structures 1.6. logical connectives 1.7. classes in set theory 1.8. bound variables in set theory 1.9. quantifiers 1.10. formalization of set theory 1.11. set generation principle philosophical aspects time in model theory truth undefinability time in set theory interpretation of classes concepts of truth in mathematics 2. set theory (continued) - all in one file (17 paper pages; obsolete pdf in 11 pages ) 2.1. tuples, families 2.2. boolean operators on families of sets 2.3. products, graphs and composition 2.4. uniqueness quantifiers, functional graphs 2.5. the powerset axiom 2.6. injectivity and inversion 2.7. properties of binary relations ; ordered sets 2.8. canonical bijections 2.9. equivalence relations and partitions 2.10. axiom of choice 2.11. galois connection 3. algebra 1 ( all in one file )(updated, march 2018) 3.1. relational systems and concrete categories 3.2. algebras 3.3. special morphisms 3.4. monoids 3.5. actions of monoids 3.6. invertibility and groups 3.7. categories 3.8. initial and final objects 3.9. algebraic terms 3.10. term algebras (still incomplete) 3.11. integers and recursion 3.12. presburger arithmetic 4. model theory 4.1. finiteness and countability (draft) 4.2. the completeness theorem 4.3. non-standard models of arithmetic 4.4. development of theories : definitions 4.5. constructions 4.6. second-order logic 4.7. well-foundedness 4.8. ordinals and cardinals (draft) 4.9. undecidability of the axiom of choice 4.10. second-order arithmetic 4.11. the incompleteness theorem (draft) more philosophical notes (uses part 1 with philosophical aspects + recursion) : gödelian arguments against mechanism : what was wrong and how to do instead philosophical proof of consistency of the zermelo-fraenkel axiomatic system 5. geometry (draft) 5.1. introduction to the foundations of geometry 5.2. first-order invariants in concrete categories 5.3. second-order invariants 5.4. affine spaces 5.5. duality 5.6. vector spaces and barycenters beyond affine geometry euclidean geometry 6. algebra 2 (draft) products of systems varieties polymorphisms and invariants relational clones abstract clones rings (to be continued - see below drafts) 7. galois connections (11 pdf pages). rigorously it only uses parts 1 (without complements) and 2. its position has been moved from 3 for pedagogical reasons (higher difficulty level while the later texts are more directly interesting). the beginning was moved to 2.11. monotone galois connections (adjunctions) upper and lower bounds, infimum and supremum complete lattices fixed point theorem transport of closure preorder generated by a relation finite sets generated equivalence relations, and more well-founded relations drafts of more texts, to be reworked later dimensional analysis : quantities and real numbers - incomplete draft text of a video lecture i wish to make on 1-dimensional geometry introduction to inversive geometry affine geometry introduction to topology axiomatic expressions of euclidean and non-euclidean geometries cardinals well-orderings and ordinals (with an alternative to zorn's lemma). diverse texts ready but not classified pythagorean triples (triples of integers (a,b,c) forming the sides of a right triangle, such as (3,4,5)) resolution of cubic equations outer automorphisms of s 6 contributions to wikipedia i wrote large parts of the wikipedia article on foundations of mathematics (sep. 2012 - before that, other authors focused on the more professional and technical article mathematical logic instead; the foundations of mathematics article is more introductory, historical and philosophical) and improved the one on the completeness theorem .

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