In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic...

ongoing rehabilitation of Frege's logicism. Boolos, George, 1998. Logic, Logic, and Logic. MIT Press. — 12 papers on Frege's theorem and the logicist approach...

centuries saw the development of modern logic and formalized mathematics. Frege's Begriffsschrift (1879) introduced both a complete propositional calculus...

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories...

motivate Frege's later works in logicism. The book was also seminal in the philosophy of language. Michael Dummett traces the linguistic turn to Frege's Grundlagen...

known as Frege's theorem, which is the foundation for a philosophy of mathematics known as neo-logicism. Hume's Principle appears in Frege's Foundations...

In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf...

ISBN 978-0-387-08417-6. Frege, Gottlob (1893). Grundgesetze der arithmetik. Jena, H. Pohle. p. 69. Zalta, Edward N. (2024), "Frege's Theorem and Foundations for...

{\displaystyle P\to \bot } , the principle is as a special case of Frege's theorem, already in minimal logic. Another derivation makes use of A → ¬ B...

philosopher Gottlob Frege. Boolos proved a conjecture due to Crispin Wright (and also proved, independently, by others), that the system of Frege's Grundgesetze...

Dedekind–Peano axioms. Both results were proven informally by Gottlob Frege (Frege's Theorem), and would later be more rigorously proven by George Boolos and...

Foundations of mathematics Frege's theorem Goodstein's theorem Neo-logicism Non-standard model of arithmetic Paris–Harrington theorem Presburger arithmetic...

a specific value assigned to it within the context of the formula. Frege's theorem A result in logic and mathematics demonstrating that arithmetic can...

question marks, boxes, or other symbols. In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle...

which lacks both excluded middle and the principle of explosion. Frege's theorem states ( B → ( C → D ) ) → ( ( B → C ) → ( B → D ) ) {\displaystyle...

However, Frege's work was short-lived, as it was found by Bertrand Russell that his axioms lead to a contradiction. Specifically, Frege's Basic Law V...

Commons has media related to Begriffsschrift. Zalta, Edward N. "Frege's Logic, Theorem, and Foundations for Arithmetic". In Zalta, Edward N. (ed.). Stanford...

B} . A second equivalent to ¬ B {\displaystyle \neg B} follows from Frege's theorem, ( B → ¬ B ) ↔ ¬ B {\displaystyle (B\to \neg B)\leftrightarrow \neg...

by Frege, Richard Dedekind, and Georg Cantor. Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929...

strongly recommended to examine Frege's argument on the point" (Russell 1903:522); The abbreviation Gg. stands for Frege's Grundgezetze der Arithmetik. Begriffsschriftlich...

Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century. The two-dimensional notation Frege developed...

thereby qualifies as a Hilbert system dates back to Gottlob Frege's 1879 Begriffsschrift. Frege's system used only implication and negation as connectives...

incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies...

foundational systems, including ZFC and category theory, and from the system of Frege's Grundgesetze der Arithmetik using modern notation and natural deduction...

"a formula language, modeled on that of arithmetic, of pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's...

article gives a sketch of a proof of Gödel's first incompleteness theorem. This theorem applies to any formal theory that satisfies certain technical hypotheses...

to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization...

formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the...

paradox in 1901 by Bertrand Russell. This proved Frege's naive set theory led to a contradiction. Frege's theory contained the axiom that for any formal...

X. Frege's propositional calculus is not a Frege system, since it used axioms instead of axiom schemes, although it can be modified to be a Frege system...