[{"data":1,"prerenderedAt":1195},["ShallowReactive",2],{"subject:logic":3,"course-wordcounts":75,"nav:logic":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F07.logic\u002Findex.md","Logic","The mathematics of proof itself — propositional and first-order logic,\nwhat can be expressed, what can be deduced, and the completeness and\nincompleteness theorems that mark the boundary.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Logic turns the act of proving into something you can prove things about. The\nsequence starts with propositional logic — connectives, truth tables, and the\ndistinction between a formula's syntax and its meaning — then builds a proof\nsystem whose derivations are trees of inference rules. First-order logic adds\nquantifiers and the structures that interpret them, so a formula can finally\nassert genuine mathematics and a model can make it true or false. The soundness\nand completeness theorems bind provability to truth, resolution shows how to\nmechanise the search for a proof, and the study of decidability marks which\nquestions a machine can settle. It ends where Gödel did: with the incompleteness\ntheorems, which show that no consistent, sufficiently strong theory can prove\nevery truth it can express. These notes follow Enderton.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,30,32,34,38,40,42,46,48,50,54],{"p":20},"Logic makes \u003Cstrong>proof itself\u003C\u002Fstrong> the object of study. It fixes a\nformal language, a precise notion of truth, and rules of inference — then\nasks what those rules can reach and where they must stop.\n",{"fig":22,"n":23,"caption":24,"large":25},"log-derivation","001","A natural-deduction proof tree: premises at the leaves, each inference bar\ndischarging them toward the conclusion.\n",true,{"fig":27,"n":28,"caption":29},"log-truthtable","002","Propositional semantics: a truth table computing a connective, filled\ncolumn by column.\n",{"p":31},"Propositional logic comes first. A handful of connectives — \u003Cem>and\u003C\u002Fem>,\n\u003Cem>or\u003C\u002Fem>, \u003Cem>not\u003C\u002Fem>, \u003Cem>implies\u003C\u002Fem> — combine atoms into formulas,\nand a truth table settles every question about them by brute enumeration.\n",{"p":33},"Two views run in parallel. \u003Cstrong>Syntax\u003C\u002Fstrong> is the game of symbols\nand inference rules; \u003Cstrong>semantics\u003C\u002Fstrong> is what those symbols mean\nonce you interpret them. Keeping the two apart is the whole discipline.\n",{"fig":35,"n":36,"caption":37},"log-model","003","A structure interprets a language: a domain and its relations, made to\nsatisfy a first-order sentence.\n",{"p":39},"First-order logic adds \u003Cstrong>quantifiers\u003C\u002Fstrong>. With \u003Cem>for all\u003C\u002Fem>\nand \u003Cem>there exists\u003C\u002Fem> ranging over a domain, the language can finally\nstate real mathematics — orderings, arithmetic, the theory of a structure.\n",{"p":41},"The \u003Cstrong>soundness\u003C\u002Fstrong> and \u003Cstrong>completeness\u003C\u002Fstrong> theorems\nthen tie the two views together: a sentence is provable exactly when it is\ntrue in every model. Deduction and truth turn out to be the same reach.\n",{"fig":43,"n":44,"caption":45},"log-resolution","004","Resolution: clauses resolved pairwise on complementary literals, cascading\nto the empty clause ⊥.\n",{"p":47},"That equivalence makes proof \u003Cem>mechanical\u003C\u002Fem>. Resolution refutes an\nunsatisfiable set by resolving clauses until it derives the empty clause —\na contradiction — which is the engine underneath automated theorem provers.\n",{"p":49},"But mechanising proof runs into a wall. Validity in first-order logic is\nundecidable, and once a theory can talk about its own arithmetic, it can\ntalk about its own provability.\n",{"fig":51,"n":52,"caption":53},"log-incompleteness","005","The Gödel sentence: a formula that, by self-reference, asserts its own\nunprovability.\n",{"p":55},"Gödel's \u003Cstrong>incompleteness\u003C\u002Fstrong> theorems close the subject: any\nconsistent theory strong enough for arithmetic has true sentences it cannot\nprove, and it can never prove its own consistency — the boundary logic set\nfor itself.\n","math","Mathematical logic turns proof into an object of study. Propositional\nlogic fixes the connectives; first-order logic adds quantifiers and can\nstate real mathematics; the soundness and completeness theorems tie\ndeduction to truth; and Gödel's incompleteness theorems show what no\nconsistent theory can do. These notes follow Enderton.\n",false,"md",{},"\u002Flogic",[],"---\ntitle: Logic\nstatus: available\ncategory: math\nblurb: |\n  The mathematics of proof itself — propositional and first-order logic,\n  what can be expressed, what can be deduced, and the completeness and\n  incompleteness theorems that mark the boundary.\ndescription: |\n  Mathematical logic turns proof into an object of study. Propositional\n  logic fixes the connectives; first-order logic adds quantifiers and can\n  state real mathematics; the soundness and completeness theorems tie\n  deduction to truth; and Gödel's incompleteness theorems show what no\n  consistent theory can do. These notes follow Enderton.\nbrief:\n  - p: |\n      Logic makes \u003Cstrong>proof itself\u003C\u002Fstrong> the object of study. It fixes a\n      formal language, a precise notion of truth, and rules of inference — then\n      asks what those rules can reach and where they must stop.\n  - fig: log-derivation\n    n: \"001\"\n    caption: |\n      A natural-deduction proof tree: premises at the leaves, each inference bar\n      discharging them toward the conclusion.\n    large: true\n  - fig: log-truthtable\n    n: \"002\"\n    caption: |\n      Propositional semantics: a truth table computing a connective, filled\n      column by column.\n  - p: |\n      Propositional logic comes first. A handful of connectives — \u003Cem>and\u003C\u002Fem>,\n      \u003Cem>or\u003C\u002Fem>, \u003Cem>not\u003C\u002Fem>, \u003Cem>implies\u003C\u002Fem> — combine atoms into formulas,\n      and a truth table settles every question about them by brute enumeration.\n  - p: |\n      Two views run in parallel. \u003Cstrong>Syntax\u003C\u002Fstrong> is the game of symbols\n      and inference rules; \u003Cstrong>semantics\u003C\u002Fstrong> is what those symbols mean\n      once you interpret them. Keeping the two apart is the whole discipline.\n  - fig: log-model\n    n: \"003\"\n    caption: |\n      A structure interprets a language: a domain and its relations, made to\n      satisfy a first-order sentence.\n  - p: |\n      First-order logic adds \u003Cstrong>quantifiers\u003C\u002Fstrong>. With \u003Cem>for all\u003C\u002Fem>\n      and \u003Cem>there exists\u003C\u002Fem> ranging over a domain, the language can finally\n      state real mathematics — orderings, arithmetic, the theory of a structure.\n  - p: |\n      The \u003Cstrong>soundness\u003C\u002Fstrong> and \u003Cstrong>completeness\u003C\u002Fstrong> theorems\n      then tie the two views together: a sentence is provable exactly when it is\n      true in every model. Deduction and truth turn out to be the same reach.\n  - fig: log-resolution\n    n: \"004\"\n    caption: |\n      Resolution: clauses resolved pairwise on complementary literals, cascading\n      to the empty clause ⊥.\n  - p: |\n      That equivalence makes proof \u003Cem>mechanical\u003C\u002Fem>. Resolution refutes an\n      unsatisfiable set by resolving clauses until it derives the empty clause —\n      a contradiction — which is the engine underneath automated theorem provers.\n  - p: |\n      But mechanising proof runs into a wall. Validity in first-order logic is\n      undecidable, and once a theory can talk about its own arithmetic, it can\n      talk about its own provability.\n  - fig: log-incompleteness\n    n: \"005\"\n    caption: |\n      The Gödel sentence: a formula that, by self-reference, asserts its own\n      unprovability.\n  - p: |\n      Gödel's \u003Cstrong>incompleteness\u003C\u002Fstrong> theorems close the subject: any\n      consistent theory strong enough for arithmetic has true sentences it cannot\n      prove, and it can never prove its own consistency — the boundary logic set\n      for itself.\n---\n\nLogic turns the act of proving into something you can prove things about. The\nsequence starts with propositional logic — connectives, truth tables, and the\ndistinction between a formula's syntax and its meaning — then builds a proof\nsystem whose derivations are trees of inference rules. First-order logic adds\nquantifiers and the structures that interpret them, so a formula can finally\nassert genuine mathematics and a model can make it true or false. The soundness\nand completeness theorems bind provability to truth, resolution shows how to\nmechanise the search for a proof, and the study of decidability marks which\nquestions a machine can settle. It ends where Gödel did: with the incompleteness\ntheorems, which show that no consistent, sufficiently strong theory can prove\nevery truth it can express. These notes follow Enderton.\n\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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as a Mathematical Model of Deduction","\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model",[989],"Symbolic logic models deductive reasoning the way probability theory models chance: it keeps the form of a correct deduction and discards its content. A deduction is valid when its conclusion follows from the form of the premises alone, independent of what the non-logical words mean. Two models carry the subject — coarse sentential logic and fine first-order logic — and four questions organize it: logical consequence, methods of proof, the gap between provable and true, and the link between logic and computability. Tuples, relations, functions, equivalence classes, and cardinality supply the set-theoretic vocabulary every later chapter uses.\n",{"module":999,"moduleNumber":16,"slug":1000,"lessons":1001},"Sentential Logic","sentential-logic",[1002,1007,1012,1018,1024,1030,1036],{"title":1003,"path":1004,"lessonNumber":990,"topics":1005,"summary":1006},"Formal Languages and Well-Formed Formulas","\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas",[999],"The language of sentential logic has an alphabet of sentence symbols, five connectives, and two parentheses, with formation rules that pick out the well-formed formulas. The wffs are the least set of expressions closed under the five formula-building operations, and every such generated set carries an induction principle.\n",{"title":1008,"path":1009,"lessonNumber":16,"topics":1010,"summary":1011},"Truth Assignments, Tautologies, and Consequence","\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies",[999],"A truth assignment fixes the sentence symbols true or false, and a recursion extends it uniquely to every formula. Satisfaction, tautologies, and tautological implication — one formula following semantically from others — rest on that extension, and the truth-table procedure decides implication for finite premise sets.\n",{"title":1013,"path":1014,"lessonNumber":1015,"topics":1016,"summary":1017},"Unique Readability and a Parsing Algorithm","\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing",3,[999],"Parentheses keep a formula from being read two ways. The parenthesis lemmas and a top-down parsing algorithm recover a formula's structure and yield unique readability: every wff has exactly one formation tree, which is what makes the truth recursion well defined.\n",{"title":1019,"path":1020,"lessonNumber":1021,"topics":1022,"summary":1023},"Induction and Recursion on Formulas","\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion",4,[999],"Two principles govern any set generated from initial elements by operations: prove a property of all its members by covering the initial elements and the closure steps, and define a function on it by recursion on structure. The recursion theorem needs the set to be freely generated, and unique readability supplies that condition for the well-formed formulas.\n",{"title":1025,"path":1026,"lessonNumber":1027,"topics":1028,"summary":1029},"Sentential Connectives and Normal Forms","\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms",5,[999],"Every formula computes a Boolean function of its atoms, and Post's theorem gives the converse: every Boolean function is realized by a wff in disjunctive normal form, so the five connectives are more than enough. Minimal complete sets follow, down to the single connectives NAND and NOR, together with a method for proving a set of connectives incomplete.\n",{"title":1031,"path":1032,"lessonNumber":1033,"topics":1034,"summary":1035},"Switching Circuits","\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits",6,[999],"A memoryless two-valued circuit computes a Boolean function, so every formula names a gate network and every network a formula. Cost and delay are read off the formula by recursion, and tautological equivalence and normal forms design and simplify circuits realizing a given specification.\n",{"title":1037,"path":1038,"lessonNumber":1039,"topics":1040,"summary":1041},"Compactness and Effectiveness","\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness",7,[999],"The compactness theorem reduces satisfiability of an infinite set of formulas to its finite subsets, proved by extension to a maximal finitely satisfiable set and applied to color infinite graphs. Effectiveness fixes what \"decidable\" and \"effectively enumerable\" mean and settles the decidability of tautologyhood.\n",{"module":1043,"moduleNumber":1015,"slug":1044,"lessons":1045},"First-Order Languages and Structures","first-order-languages",[1046,1051,1056,1061],{"title":1047,"path":1048,"lessonNumber":990,"topics":1049,"summary":1050},"First-Order Languages","\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages",[1043],"Sentential logic cannot see inside a simple statement, so it misses valid arguments that turn on quantifiers and predicates. A first-order language adds a quantifier, variables, and a chosen vocabulary of predicate, function, and constant symbols. Terms and well-formed formulas are built by recursion over this alphabet, and a variable occurs free or bound according to the quantifiers that reach it.\n",{"title":1052,"path":1053,"lessonNumber":16,"topics":1054,"summary":1055},"Structures, Truth, and Satisfaction","\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction",[1043],"A structure interprets a language: a nonempty universe plus a meaning for every predicate, function, and constant symbol. Tarski's recursion defines when a structure satisfies a formula under a variable assignment, and hence when a sentence is true. From satisfaction we recover logical implication, validity, and logical equivalence for first-order logic.\n",{"title":1057,"path":1058,"lessonNumber":1015,"topics":1059,"summary":1060},"Definability and Elementary Equivalence","\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence",[1043],"Fix a structure and ask which relations a formula can pick out: the definable ones. A set of sentences picks out a class of structures, the elementary classes. Homomorphisms and isomorphisms compare structures, and the homomorphism theorem shows isomorphic structures satisfy the same sentences. Automorphisms bound what first-order logic can distinguish, giving a tool for proving relations undefinable.\n",{"title":1062,"path":1063,"lessonNumber":1021,"topics":1064,"summary":1065},"Parsing, Substitution, and Substitutability","\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing",[1043],"Every recursion on first-order syntax rests on unique readability. A parenthesis-counting function proves that terms and formulas decompose in exactly one way, and a parsing algorithm recovers the decomposition. Substituting a term for a free variable can capture it under a quantifier; the substitutability condition rules that out, and the substitution lemma trades syntactic substitution for a change of assignment.\n",{"module":1067,"moduleNumber":1021,"slug":1068,"lessons":1069},"The Deductive Calculus and Its Metatheorems","deductive-calculus",[1070,1075,1080,1085],{"title":1071,"path":1072,"lessonNumber":990,"topics":1073,"summary":1074},"A Deductive Calculus for First-Order Logic","\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus",[1067],"A proof must be finite and mechanically checkable. A Hilbert-style calculus meets both demands: six schemas of logical axioms, a single rule of inference (modus ponens), and the syntactic consequence relation they generate. Substitution and substitutability are defined by recursion, and the bridge theorem reduces deducibility to tautological implication from the axioms.\n",{"title":1076,"path":1077,"lessonNumber":16,"topics":1078,"summary":1079},"The Deduction Theorem and Derived Rules","\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules",[1067],"Raw deductions from axioms are unusable by hand. The generalization theorem, the deduction theorem, contraposition, reductio ad absurdum, and rule T reduce the calculus to the moves of ordinary mathematics, each proved once to license a block of axiom-level steps. Generalization on constants and alphabetic variants handle the quantifier and substitution bookkeeping.\n",{"title":1081,"path":1082,"lessonNumber":1015,"topics":1083,"summary":1084},"The Soundness Theorem","\u002Flogic\u002Fdeductive-calculus\u002Fsoundness",[1067],"Soundness is the easy half of the match between proof and truth. Whatever the calculus deduces is logically implied, by an induction on deduction length that rests on one lemma: every logical axiom is valid. The only hard case, quantifier instantiation, needs the substitution lemma. The contrapositive corollary states that every satisfiable set is consistent.\n",{"title":1086,"path":1087,"lessonNumber":1021,"topics":1088,"summary":1089},"The Completeness Theorem","\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency",[1067],"Gödel's completeness theorem is the deep converse of soundness: whatever is logically implied can be deduced. Equivalently, every consistent set has a model. The Henkin proof manufactures that model out of syntax alone: add witnessing constants, extend to a maximal consistent set, and read a term model off the formulas it contains. Compactness and the enumerability theorem drop out.\n",{"module":1091,"moduleNumber":1027,"slug":1092,"lessons":1093},"Models, Compactness, and Theories","models-and-theories",[1094,1102,1107,1112],{"title":1095,"path":1096,"lessonNumber":990,"topics":1097,"summary":1101},"Compactness and the Löwenheim–Skolem Theorems","\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem",[1098,1099,1100],"Models","Compactness","and Theories","A set of first-order sentences has a model whenever each of its finite subsets does. This compactness theorem follows from completeness and yields the finiteness limitation, the downward and upward Löwenheim–Skolem theorems, models of every infinite cardinality, and nonstandard models of arithmetic.\n",{"title":1103,"path":1104,"lessonNumber":16,"topics":1105,"summary":1106},"Theories, Elementary Classes, and Categoricity","\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity",[1098,1099,1100],"A theory is a set of sentences closed under logical consequence. Theories correspond to classes of models; a theory may be complete, axiomatizable, or finitely axiomatizable, and completeness together with axiomatizability yields decidability. The Łoś–Vaught test derives completeness from categoricity in a cardinal, applied to dense linear orders and to algebraically closed fields.\n",{"title":1108,"path":1109,"lessonNumber":1015,"topics":1110,"summary":1111},"Interpretations Between Theories","\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories",[1098,1099,1100],"An interpretation translates the vocabulary of one theory into formulas of another, relativizing quantifiers to a definable domain and mapping symbols to defining formulas. Defined function symbols meet a noncreativity criterion; the syntactic translation of formulas carries theoremhood forward, and a faithful interpretation transfers decidability and undecidability between theories.\n",{"title":1113,"path":1114,"lessonNumber":1021,"topics":1115,"summary":1116},"Nonstandard Analysis","\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis",[1098,1099,1100],"Compactness builds a model of the real ordered field containing infinite elements and nonzero infinitesimals. The transfer principle carries every first-order truth from the reals to this extension, the standard-part map collapses finite hyperreals back onto the reals, and continuity and the derivative are rederived by working with infinitely small quantities directly.\n",{"module":1118,"moduleNumber":1033,"slug":1119,"lessons":1120},"Number Theory and Definability","arithmetic-and-definability",[1121,1126,1131,1136],{"title":1122,"path":1123,"lessonNumber":990,"topics":1124,"summary":1125},"The Structure of Arithmetic and Definability","\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic",[1118],"Number theory is the theory of one fixed structure, the natural numbers under successor, order, addition, multiplication, and exponentiation. Every number is named by a numeral, and a relation is definable when a single formula picks out exactly its tuples. The central gap separates the sentences true in that structure from those any reasonable set of axioms can prove.\n",{"title":1127,"path":1128,"lessonNumber":16,"topics":1129,"summary":1130},"Natural Numbers with Successor","\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor",[1118],"The weakest reduct keeps only zero and successor. Its models are a standard chain together with disjoint copies of the integers, which makes the theory categorical in every uncountable power, hence complete and decidable. A quantifier-elimination procedure gives a practical decision method and shows a subset is definable if and only if it is finite or cofinite.\n",{"title":1132,"path":1133,"lessonNumber":1015,"topics":1134,"summary":1135},"Reducts: Order, Addition, and Multiplication","\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts",[1118],"Adding order to the successor reduct keeps decidability and makes the theory finitely axiomatizable; adding addition gives Presburger arithmetic, still decidable by quantifier elimination once congruence predicates are included, with definable sets exactly the eventually periodic ones. Multiplication is the break point: neither addition nor order can define it, and once it joins addition the theory stops being decidable.\n",{"title":1137,"path":1138,"lessonNumber":1021,"topics":1139,"summary":1140},"A Subtheory of Number Theory and Representability","\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability",[1118],"A finite set of eleven axioms, the recursion equations for successor, order, addition, multiplication, and exponentiation, already proves every true quantifier-free and existential sentence. Representability asks a theory to prove the right instances of a formula rather than merely make them true, and a relation is defined to be recursive exactly when some consistent finite theory represents it. Church's thesis identifies that with decidability, and closure under composition, minimization, and primitive recursion builds the catalog the incompleteness proofs need.\n",{"module":1142,"moduleNumber":1039,"slug":1143,"lessons":1144},"Arithmetization and the Incompleteness Theorems","incompleteness",[1145,1150,1155],{"title":1146,"path":1147,"lessonNumber":990,"topics":1148,"summary":1149},"Arithmetization of Syntax","\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax",[1142],"Gödel numbering assigns a natural number to every symbol, expression, formula, and deduction, turning statements about syntax into statements about numbers. The syntactic operations — substitution, \"is a wff\", \"is an axiom\", \"d codes a deduction of a\" — come out primitive recursive and hence representable in the subtheory, which lets a formula of arithmetic talk about formulas, including itself.\n",{"title":1151,"path":1152,"lessonNumber":16,"topics":1153,"summary":1154},"Incompleteness, Undecidability, and Church's Theorem","\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability",[1142],"The fixed-point lemma manufactures a sentence that talks about its own Gödel number. Pointed at truth it gives Tarski's theorem — arithmetic truth is not arithmetically definable; pointed at provability it gives Gödel's first incompleteness theorem and the undecidability of the theory of the natural numbers, and, applied to validity, Church's theorem that first-order logic is undecidable. The set of theorems of a recursive theory is only recursively enumerable — the gap between provable and true.\n",{"title":1156,"path":1157,"lessonNumber":1015,"topics":1158,"summary":1159},"The Second Incompleteness Theorem","\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem",[1142],"Consistency of a recursively axiomatized theory is itself an arithmetic sentence, built from a provability predicate. When the theory is strong enough to formalize its own reflection and modus ponens — the Hilbert–Bernays–Löb derivability conditions — it cannot prove that sentence unless it is inconsistent. Löb's theorem is the companion result, and set theory is the case that closes Hilbert's program.\n",{"module":1161,"moduleNumber":1162,"slug":1163,"lessons":1164},"Recursive Functions and Representability",8,"computability-and-representability",[1165,1170],{"title":1166,"path":1167,"lessonNumber":990,"topics":1168,"summary":1169},"Recursive Functions and Church's Thesis","\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions",[1161],"The recursive functions are the formal counterpart of the effectively computable ones: built from three initial functions by composition, primitive recursion, and minimization, and equivalently the functions representable in a finitely axiomatized arithmetic. Church's thesis identifies the class with effective calculability; Kleene's normal form theorem and the unsolvable halting problem place the recursive sets strictly inside the recursively enumerable ones.\n",{"title":1171,"path":1172,"lessonNumber":16,"topics":1173,"summary":1174},"Representing Exponentiation and the β-Function","\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation",[1161],"Coding finite sequences by prime-power exponents already assumes exponentiation, so representing exponentiation from addition and multiplication alone needs a different encoder. Gödel's β-function, built from a pairing function and the Chinese remainder theorem, reads back arbitrary finite sequences using only plus and times. This represents exponentiation in the addition-multiplication arithmetic and closes the last gap in the representability of every recursive syntactic operation.\n",{"module":1176,"moduleNumber":1177,"slug":1178,"lessons":1179},"Second-Order Logic and Beyond",9,"second-order-logic",[1180,1185,1190],{"title":1181,"path":1182,"lessonNumber":990,"topics":1183,"summary":1184},"Second-Order Languages","\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages",[1176],"Second-order logic quantifies over relations and functions, not just individuals. Second-order Peano arithmetic and the second-order theory of the reals become categorical, and finiteness is definable by a single sentence. Compactness, completeness, and the Löwenheim–Skolem theorems all fail for the standard semantics.\n",{"title":1186,"path":1187,"lessonNumber":16,"topics":1188,"summary":1189},"Skolem Functions and Many-Sorted Logic","\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic",[1176],"Skolem functions replace existential quantifiers with named witnesses, putting any first-order formula into a prenex form with all existentials — now over functions — pulled to the front. The Skolemized formula is equisatisfiable with the original, which reduces satisfiability to universal sentences and, through Herbrand expansions, to sentential logic. Many-sorted logic then adds several universes at once and reduces cleanly to ordinary one-sorted logic.\n",{"title":1191,"path":1192,"lessonNumber":1015,"topics":1193,"summary":1194},"General (Henkin) Structures","\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures",[1176],"General semantics reinterprets second-order logic by letting the predicate and function quantifiers range over a designated collection of relations and functions rather than all of them. Recast as many-sorted first-order logic with comprehension axioms, general second-order logic recovers a sound and complete calculus together with compactness and Löwenheim–Skolem, giving up the categoricity of the standard semantics. The ω-models of analysis show the trade.\n",1786059479672]