higher-order logic - University of Amsterdam
... new characterization results. For instance, Lindström himself proved that elementary logic is also the strongest logic with an effective finitary syntax to possess the Löwenheim-Skolem property and be complete. (The infinitary language L!1 ! has both, without collapsing into elementary logic, howe ...
... new characterization results. For instance, Lindström himself proved that elementary logic is also the strongest logic with an effective finitary syntax to possess the Löwenheim-Skolem property and be complete. (The infinitary language L!1 ! has both, without collapsing into elementary logic, howe ...
What is "formal logic"?
... mathematics (cf Halmos, 1970). Symbols in mathematics do not reduce to simple signs as the ones used for numbers, they include visual representation of functions, commutative diagrams (category theory), fractals, etc. Frege’s ideography (1879) is obviously symbolic in this sense. The wave of formali ...
... mathematics (cf Halmos, 1970). Symbols in mathematics do not reduce to simple signs as the ones used for numbers, they include visual representation of functions, commutative diagrams (category theory), fractals, etc. Frege’s ideography (1879) is obviously symbolic in this sense. The wave of formali ...
Lectures on Proof Theory - Create and Use Your home.uchicago
... arithmetical truths might be inconsistent—but that was simply an empty skepticism. On the other hand, as we now know, in view of Gödel’s incompleteness theorems, there is no relevant sense in which we can refute it. Surely, if we are to judge by how long it took for the various successive extension ...
... arithmetical truths might be inconsistent—but that was simply an empty skepticism. On the other hand, as we now know, in view of Gödel’s incompleteness theorems, there is no relevant sense in which we can refute it. Surely, if we are to judge by how long it took for the various successive extension ...
in every real in a class of reals is - Math Berkeley
... sorted logics (N ; M; : : :) in the usual sense of having two types of variables one ranging over the elements of N and the other over those of M in addition to the usual apparatus of function, relation and constant symbols of ordinary …rst order logic. While formally merely a version of …rst order ...
... sorted logics (N ; M; : : :) in the usual sense of having two types of variables one ranging over the elements of N and the other over those of M in addition to the usual apparatus of function, relation and constant symbols of ordinary …rst order logic. While formally merely a version of …rst order ...
Action Logic and Pure Induction
... and the relation R∗ as the ancestral or reflexive transitive closure of R. The implications have a quite natural meaning: for example if R and S are the relations loves and pays respectively then R→S is the relation which holds of (v, w) just when every u who loves v pays w. The sentence “king(loves ...
... and the relation R∗ as the ancestral or reflexive transitive closure of R. The implications have a quite natural meaning: for example if R and S are the relations loves and pays respectively then R→S is the relation which holds of (v, w) just when every u who loves v pays w. The sentence “king(loves ...
Model Theory of Second Order Logic
... be the set of A ⊆ N with A∆G finite. We show that M = (ω ∪ F G , ω, <, E), where nEx ⇐⇒ n ∈ X , is not characterizable by a second order theory. Let M0 = (ω ∪ F −G , ω, <, E). Since M M0 , it suffices to prove that M and M0 are second order equivalent. In fact more is true: If Φ(x) is any formula ...
... be the set of A ⊆ N with A∆G finite. We show that M = (ω ∪ F G , ω, <, E), where nEx ⇐⇒ n ∈ X , is not characterizable by a second order theory. Let M0 = (ω ∪ F −G , ω, <, E). Since M M0 , it suffices to prove that M and M0 are second order equivalent. In fact more is true: If Φ(x) is any formula ...
The Emergence of First
... To most mathematical logicians working in the 1980s, first-order logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on first-order logic, it was a radical and u ...
... To most mathematical logicians working in the 1980s, first-order logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on first-order logic, it was a radical and u ...
Frege, Boolos, and Logical Objects
... explicit assertion of the existence of numbers embodied by Numbers is a way of making clear the commitment implicit in the use of the definite article in ‘the number of F s’.5 In his papers of [1986] and [1993], Boolos returned to the idea of salvaging Frege’s work by using biconditionals which are w ...
... explicit assertion of the existence of numbers embodied by Numbers is a way of making clear the commitment implicit in the use of the definite article in ‘the number of F s’.5 In his papers of [1986] and [1993], Boolos returned to the idea of salvaging Frege’s work by using biconditionals which are w ...
Mathematical Logic. An Introduction
... intended to denote an n-ary function. A symbol set is sometimes called a type because it describes the type of structures which will later interpret the symbols. We shall denote variables by letters like x, y, z, , relation symbols by P , Q, R, , functions symbols by f , g , h, and constant sy ...
... intended to denote an n-ary function. A symbol set is sometimes called a type because it describes the type of structures which will later interpret the symbols. We shall denote variables by letters like x, y, z, , relation symbols by P , Q, R, , functions symbols by f , g , h, and constant sy ...
P,Q
... Clear communication of logical arguments in any area of study. Discovery and elucidation, through proofs, of interesting new mathematical theorems. Theorem-proving has applications in program verification, computer security, automated reasoning systems, etc. Proving a theorem allows us to re ...
... Clear communication of logical arguments in any area of study. Discovery and elucidation, through proofs, of interesting new mathematical theorems. Theorem-proving has applications in program verification, computer security, automated reasoning systems, etc. Proving a theorem allows us to re ...
Intuitionistic Type Theory - The collected works of Per Martin-Löf
... Mathematical logic and the relation between logic and mathematics have been interpreted in at least three different ways: (1) mathematical logic as symbolic logic, or logic using mathematical symbolism; (2) mathematical logic as foundations (or philosophy) of mathematics; (3) mathematical logic as l ...
... Mathematical logic and the relation between logic and mathematics have been interpreted in at least three different ways: (1) mathematical logic as symbolic logic, or logic using mathematical symbolism; (2) mathematical logic as foundations (or philosophy) of mathematics; (3) mathematical logic as l ...
Intuitionistic Type Theory
... Mathematical logic and the relation between logic and mathematics have been interpreted in at least three different ways: (1) mathematical logic as symbolic logic, or logic using mathematical symbolism; (2) mathematical logic as foundations (or philosophy) of mathematics; (3) mathematical logic as l ...
... Mathematical logic and the relation between logic and mathematics have been interpreted in at least three different ways: (1) mathematical logic as symbolic logic, or logic using mathematical symbolism; (2) mathematical logic as foundations (or philosophy) of mathematics; (3) mathematical logic as l ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.