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Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Theorem in theoretical computer science
In lambda calculus, the Church–Rosser theorem states that, when applying reduction rules to terms, the ordering in which the reductions are chosen does
Church–Rosser_theorem
The nth prime number exceeds n log(n).
In number theory, Rosser's theorem states that the n {\displaystyle n} th prime number is greater than n log n {\displaystyle n\log n} , where log {\displaystyle
Rosser's_theorem
Method in mathematical logic
In mathematical logic, Rosser's trick is a method for proving a variant of Gödel's incompleteness theorems not relying on the assumption that the theory
Rosser's_trick
American logician (1907–1989)
known for his part in the Church–Rosser theorem in lambda calculus. He also developed what is now called the "Rosser sieve" in number theory. He was part
J._Barkley_Rosser
American mathematician and computer scientist (1903–1995)
Entscheidungsproblem ("decision problem"), the Frege–Church ontology, and the Church–Rosser theorem. Alongside his doctoral student Alan Turing, Church is considered one
Alonzo_Church
Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory) Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical
List_of_theorems
Property of rewriting systems in mathematics
basis. Matsumoto's theorem follows from confluence of the braid relations. β-reduction of λ-terms is confluent by the Church–Rosser theorem. Convergence (logic)
Confluence (abstract rewriting)
Confluence_(abstract_rewriting)
Characterization of how many integers are prime
( x ) {\displaystyle \log _{e}(x)} . In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of prime numbers among the
Prime_number_theorem
form of a term, if one exists, is unique (as a corollary of the Church–Rosser theorem). However, a term may have more than one head normal form. In the lambda
Beta_normal_form
Yes-or-no question that cannot ever be solved by a computer
Reference. Retrieved 2022-06-12. Aaronson, Scott (21 July 2011). "Rosser's Theorem via Turing machines". Shtetl-Optimized. Retrieved 2 November 2022.
Undecidable_problem
Problem in computer science
115 Lucas 2021. Kleene 1952, p. 382. Rosser, "Informal Exposition of Proofs of Gödel's Theorem and Church's Theorem", reprinted in Davis 1965, p. 223 letter
Halting_problem
Conjecture on zeros of the zeta function
hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!! Care should
Riemann_hypothesis
Automated theorem proving ACL2 theorem prover E equational theorem prover Gandalf theorem prover HOL theorem prover Isabelle theorem prover LCF theorem prover
List of mathematical logic topics
List_of_mathematical_logic_topics
Bound on the gaps between prime numbers
Nicholson's, C < D is Rosser's theorem, and A < D is Firoozbakht's conjecture. All have been verified to 264. Prime number theorem Andrica's conjecture
Firoozbakht's_conjecture
Dutch mathematician (1918–2012)
for automatic formula manipulation, with application to the Church-Rosser theorem." Indagationes Mathematicae (Proceedings). Vol. 75. No. 5. North-Holland
Nicolaas_Govert_de_Bruijn
Expression that cannot be rewritten further
University Press. ISBN 9780521779203. Ohlebusch, Enno (1998). "Church-Rosser theorems for abstract reduction modulo an equivalence relation". Rewriting Techniques
Normal form (abstract rewriting)
Normal_form_(abstract_rewriting)
function Referential transparency Currying Lambda abstraction Church–Rosser theorem Extensionality Church numeral Fixed point combinator SKI combinator
List of functional programming topics
List_of_functional_programming_topics
Impossible task in computing
impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of First-order logic, a statement is universally valid if and only if it
Entscheidungsproblem
Mathematical-logic system
equal if it is possible to α-convert one into the other). By the Church–Rosser theorem, any particular sequence of reduction steps starting from a given lambda
Lambda_calculus
Mathematical notation in lambda calculus
for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem" (PDF). Indagationes Mathematicae. 34: 381–392. ISSN 0019-3577. Archived
De_Bruijn_index
Indian computer scientist
Boyer–Moore theorem prover to prove metatheorems such as the tautology theorem, Godel's incompleteness theorem and the Church-Rosser theorem. He has contributed
Natarajan_Shankar
US domestic terror attack
full-time. Rosser was well known for his research in pure mathematics, logic (Rosser's trick, the Kleene–Rosser paradox, and the Church–Rosser theorem) and
Sterling_Hall_bombing
Mathematical formalism
into each other by α-conversion are defined to be equal. By the Church–Rosser theorem, the normal form is unique if it exists, regardless of the order in
Lambda_calculus_definition
definition of Rosser's provability predicate) T ⊩ P r o v R ( # ( ¬ ρ ) ) {\displaystyle T\Vdash Prov^{R}(\#(\neg \rho ))} (by condition no. 1 and theorem 1) Thus
Hilbert–Bernays–Löb provability conditions
Hilbert–Bernays–Löb_provability_conditions
American mathematician (1909–1994)
the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to describe
Stephen_Cole_Kleene
Certain kind of modal formula
modal formula with remarkable properties. The Sahlqvist correspondence theorem states that every Sahlqvist formula is canonical, and corresponds to a
Sahlqvist_formula
Formal system for transcribing expressions into equivalent terms
sometimes called weak confluence. Theorem. For an ARS the following three conditions are equivalent: (i) it has the Church–Rosser property, (ii) it is confluent
Abstract_rewriting_system
Thesis on the nature of computability
Physics. Springer Verlag. Rosser, J. B. (1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". The Journal of Symbolic Logic
Church–Turing_thesis
Number of integers coprime to and less than n
Chebyshev's theorem (Hardy & Wright 1979, thm.7) and Mertens' third theorem is all that is needed. Hardy & Wright 1979, thm. 436 Theorem 15 of Rosser, J. Barkley;
Euler's_totient_function
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
music critic, complications from surgery. J. Barkley Rosser, 81, American logician (Church–Rosser theorem), aneurysm. Jessie Mae Brown Beavers, 66, American
Deaths_in_September_1989
five color theorem. The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976. Schröder–Bernstein theorem. In 1896 Schröder
List_of_incomplete_proofs
1931 paper by Kurt Gödel
to prove the incompleteness theorems. The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact
On Formally Undecidable Propositions of Principia Mathematica and Related Systems
On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems
Paradox in set theory
named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Georg
Burali-Forti_paradox
American logician (1933–2019)
forcing, a forcing notion based on perfect sets and the Sacks Density Theorem, which asserts that the partial order of the recursively enumerable Turing
Gerald_Sacks
Relationship between programs and proofs
state or prove that a morphism is normalizing, establish a Church-Rosser type theorem, or speak of a "strongly normalizing" cartesian closed category.
Curry–Howard_correspondence
Branch of mathematical logic
proof-theoretic semantics, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications
Proof_theory
German mathematician (1909–1945)
calculus, building off of previous work of Paul Hertz. His cut-elimination theorem is the cornerstone of proof-theoretic semantics, and some philosophical
Gerhard_Gentzen
3 (Sep. 1936), pp. 103–105. Rosser. J. B., 1939, An informal exposition of proofs of Gödel's Theorem and Church's Theorem, The Journal of Symbolic Logic
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Product of the first "n" prime numbers
Griffiths (2015) proved that it is irrational. Euclid's proof of his theorem on the infinitude of primes can be paraphrased by saying that, for any
Primorial
Mathematical analysis
all the frameworks share a broad common core of results that are also theorems of classical analysis. Constructive frameworks for its formulation are
Constructive_analysis
Concept in model theory
substructures of a larger one. This property plays a crucial role in Fraïssé's theorem, which characterises classes of finite structures that arise as ages of
Amalgamation_property
This approach has been used successfully for (interactive) automated theorem proving. The first logical framework was Automath; however, the name of
Logical_framework
Mathematical function
theorem relates the two quotients ψ ( x ) x {\displaystyle {\frac {\psi (x)}{x}}} and ϑ ( x ) x {\displaystyle {\frac {\vartheta (x)}{x}}} . Theorem:
Chebyshev_function
from the axioms of ZFC. In 1931, Kurt Gödel proved his incompleteness theorems, establishing that many mathematical theories, including ZFC, cannot prove
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Pair of mathematical objects
b} = {c, d}, and so: {b} = {a, b} \ {a} = {c, d} \ {c} = {d}, so b = d. Rosser (1953) employed a definition of the ordered pair due to Quine which requires
Ordered_pair
Book by Wilhelm Ackermann
that logic was complete (i.e., whether all semantic truths of FOL were theorems derivable from the FOL axioms and rules). The former problem was answered
Principles of Mathematical Logic
Principles_of_Mathematical_Logic
Proof by Alan Turing
to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture
Turing's_proof
Fundamental physical law of electromagnetism
Where the last equality follows by the mean value theorem for integrals. Using the squeeze theorem and the continuity of ρ {\displaystyle \rho } , one
Coulomb's_law
Function representing the number of primes less than or equal to a given number
\infty }{\frac {\pi (x)}{x/\log x}}=1.} This statement is the prime number theorem. An equivalent statement is lim x → ∞ π ( x ) li ( x ) = 1 {\displaystyle
Prime-counting_function
Paradox in set theory
already realized that his theory would lead to a contradiction (to Cantor's theorem), as he told Hilbert and Richard Dedekind by letter. Hilbert also formulated
Russell's_paradox
Inequality relating the primorial to square of the next prime number
2 . {\displaystyle p_{n}\#=\prod _{i=1}^{n}p_{i}>p_{n+1}^{2}.} Barkley Rosser showed an upper bound where n # ≤ 2.83 n {\displaystyle n\#\leq 2.83^{n}}
Bonse's_inequality
Sequence of operations for a task
Press. ISBN 978-0-262-68052-3. Rosser, J.B. (1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". Journal of Symbolic Logic
Algorithm
Apparent contradiction in metamathematics
Curry's paradox List of self–referential paradoxes Kleene–Rosser paradox List of paradoxes Löb's theorem Ordinal definable set, a set-theoretic concept of definability
Richard's_paradox
American mathematician (1900-1982)
1933, he learned of the Kleene–Rosser paradox from correspondence with John Rosser. The paradox, developed by Rosser and Stephen Kleene, had proved the
Haskell_Curry
Use of functions that call themselves
where proving the base case(s) and the inductive step ensures that a given theorem holds for all valid inputs. The base case specifies input values for which
Recursion_(computer_science)
List of statements that appear to contradict themselves
excusable, it is not negligence. Gödel's incompleteness theorems – and Tarski's undefinability theorem Ignore all rules – To obey this rule, it is necessary
List_of_paradoxes
Idea that the universe is fundamentally computational or informational
within the class of local hidden-variable theories constrained by Bell's theorem and the experiments that test it. Two claims are often run together but
Digital_physics
Systematic endeavour to gain knowledge
formal systems. A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. It includes mathematics, systems
Science
introduced without the name in the proof of the first of Gödel's incompleteness theorems (Gödel 1931). The β function lemma given below is an essential step of
Gödel's_β_function
American economist (born 1942)
1969-70 2010 "What's wrong with the fundamental existence and welfare theorems?", Journal of Economic Behavior and Organization 75 2009, “The economic
Duncan_K._Foley
Law of classical electromagnetism
Electrodynamics (3rd ed.). Prentice Hall. pp. 222–224, 435–440. ISBN 0-13-805326-X. Rosser, W. G. V. (1968). Classical Electromagnetism via Relativity. pp. 29–42.
Biot–Savart_law
Property of space that quantifies the magnetic influence at a given location
10.15 & 10.16. Griffiths 1999, p. 422. Griffiths 1999, §. 12.3.2. Rosser, W. G. V. (1968). Classical Electromagnetism via Relativity. Boston, MA:
Magnetic_field
Sum of an (infinite) geometric progression
convergence of 1. This could be seen as a consequence of the Cauchy–Hadamard theorem and the fact that lim n → ∞ a n = 1 {\displaystyle \lim _{n\rightarrow
Geometric_series
Model of concurrent computation
prove a generalization of the Church-Turing-Rosser-Kleene thesis [Kleene 1943]: A consequence of the above theorem is that a finite actor can nondeterministically
Actor_model
Mathematical theory of data types
driven by proof checkers, interactive proof assistants, and automated theorem provers. Most of these systems use a type theory as the mathematical foundation
Type_theory
Relationship in which one statement follows from another
originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935). Syntactic consequence does not depend on any interpretation
Logical_consequence
Axiomatic set theory devised by W.V.O. Quine
comprehension as a theorem. The precise set of axioms can vary, but includes most of the following, with the others provable as theorems: Extensionality:
New_Foundations
Paradox involving a game with repeated coin flipping
Nonlinear Science, 2016; 26 (2): 023103 DOI: 10.1063/1.4940236 2. J. Barkley Rosser Jr. 'Reconsidering ergodicity and fundamental uncertainty'. In: Journal
St._Petersburg_paradox
Difference between logarithm and harmonic series
mathématiques pures et appliquées. 63: 187–213. Weisstein, Eric W. "Mertens Theorem". mathworld.wolfram.com. Retrieved 2024-10-08. Weisstein, Eric W. "Mertens
Euler's_constant
Mathematical paradox
Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem. Claims of the form "if A, then B" are called conditional claims. Curry's
Curry's_paradox
(metalogic) A ⊢ B {\displaystyle A\vdash B} says " B {\displaystyle B} is a theorem of A {\displaystyle A} ". In other words, A {\displaystyle A} proves B
List_of_logic_symbols
American philosopher (1927–1999)
"Linear reasoning. A new form of the Herbrand-Gentzen theorem", "Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory" by William
Burton_Dreben
Large number used in number theory
Roughly speaking, Littlewood's proof consists of Dirichlet's approximation theorem to show that sometimes many terms have about the same argument. In the
Skewes's_number
Informal set theories
they do exclude some paradoxes, like Russell's paradox. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at
Naive_set_theory
American philosopher and logician (1908–2000)
cryptic. The last chapter, on Gödel's incompleteness theorem and Tarski's indefinability theorem, along with the article Quine (1946), became a launching
Willard_Van_Orman_Quine
Physical field surrounding an electric charge
ISBN 978-1-139-01297-3. Retrieved 2022-07-04. {{cite book}}: |website= ignored (help) Rosser, W. G. V. (1968). Classical Electromagnetism via Relativity. pp. 29–42.
Electric_field
Physical phenomenon in electromagnetic field theory
electronics engineers broke out in the 1960s after Richard Feynman's textbook. Rosser's book Classical Electromagnetism via Relativity was popular, as was Anthony
Relativistic_electromagnetism
Basic notion of sameness in mathematics
scientists like John Alan Robinson in their work on resolution and automated theorem proving. The substitution property can produce false statements when applied
Equality_(mathematics)
theories prove theorems (and nothing else). So saying that a theory allows the construction of a certain object means that it is a theorem of that theory
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Replacing subterm in a formula with another term
convergent or canonical. Important theorems for abstract rewriting systems are that an ARS is confluent iff it has the Church–Rosser property, Newman's lemma (a
Rewriting
Computation model defining an abstract machine
2000. Minsky 1967, p. 104. Hennie & Stearns 1966. Arora & Barak 2009, theorem 1.9. Grötschel, Lovász & Schrijver 1993, p. 32. Gandy 1995, p. 54. Gandy
Turing_machine
American mathematician
Boas with dissertation Uniqueness, Interpolation and Characterization Theorems for Functions of Exponential Type. For three years he was an assistant
Robert_Creighton_Buck
Type of economic system
ISBN 978-1579580919. Stiglitz criticizes the first and second welfare theorems for being based on the assumptions of complete markets (including a full
Market_economy
Atkinson–Stiglitz theorem Where the utility function is separable between labor and all commodities, no indirect taxes need be employed. Aumann's agreement theorem If
Glossary_of_economics
propositional logic can be defined using only conjunction and negation. Rosser J. Barkley created a system based on conjunction and negation { ∧ , ¬ }
List of axiomatic systems in logic
List_of_axiomatic_systems_in_logic
Game class in game theory
[Accessed 30 October 2020]. Vernengo, Matias; Caldentey, Esteban Perez; Rosser Jr, Barkley J, eds. (2020). U-M Weblogin. doi:10.1057/978-1-349-95121-5
Simultaneous_game
Framework in lambda calculus
B':s} in the Conversion rule is a convenience; one could prove a meta-theorem that Γ ⊢ A : B ∧ B = β B ′ ⇒ Γ ⊢ B ′ : s {\displaystyle \Gamma \vdash A:B\land
Lambda_cube
Graph of space and time in special relativity
JSTOR 20022840. Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived 2009-10-27 at the Wayback Machine
Spacetime_diagram
Contributions of women to the field of science
algebra, filled in gaps in relativity, and was responsible for a critical theorem about conserved quantities in physics. One notes that the Erlangen program
Women_in_science
School of thought in philosophy of mathematics
incompleteness theorems are 'proved with logic just like any other theorems'. However, that argument appears not to acknowledge the distinction between theorems of
Logicism
Concept in contract theory and economics
Modigliani–Miller theorem, which states that the valuation of a firm is unaffected by its financial structure. It challenges the theorem as one of the key
Information_asymmetry
Architectural element similar to the hollow upper half of a sphere; there are many types
three dimensions to a catenary curve for a two-dimensional arch. The safe theorem (in the formulation of Jacques Heyman [de]) states that when a thrust line
Dome
2023 edition of film festival
Film Festival]. Ten Asia (in Korean). Naver. Retrieved September 19, 2023. Rosser, Michael (September 5, 2023). "Busan film festival unveils 2023 line-up
28th Busan International Film Festival
28th_Busan_International_Film_Festival
Category of formal programming language semantics
that imperative extensions of this calculus satisfy these theorems. Consequences of these theorems are that the equational theory—the symmetric-transitive-reflexive
Operational_semantics
Ability of bacteria to move independently using metabolic energy
its trajectory and it's back where it started". In light of this scallop theorem, Purcell developed approaches concerning how artificial motion at the micro
Bacterial_motility
Generalization of a magic square
been constructed by J. R. Hendricks. Marian Trenkler proved the following theorem: A p-dimensional magic hypercube of order n exists if and only if p > 1
Magic_hypercube
Academic association dedicated to the use of mathematics in industry
Joseph P. LaSalle (1962–1963) Alston Householder (1963–1964) J. Barkley Rosser (1964–1966) Garrett Birkhoff (1966–1968) J. Wallace Givens (1968–1970) Burton
Society for Industrial and Applied Mathematics
Society_for_Industrial_and_Applied_Mathematics
Attempts to formalize the concept of algorithms
converse appears as his Theorem XXVIII. Together these form the proof of their equivalence, Kleene's Theorem XXX. With his Theorem XXX Kleene proves the
Algorithm_characterizations
travel, tourism, insurance
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