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the connectivity theorems. Two basic sets of theorems exists, one for flux and another for concentrations. The concentration connectivity theorems are
Connectivity_theorems
Mathematical model of biochemical pathways
pathway. Two basic sets of theorems exists, one for flux and another for concentrations. The concentration connectivity theorems are divided again depending
Metabolic_control_analysis
Theorem in graph theory
Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and
Menger's_theorem
Basic concept of graph theory
most important facts about connectivity in graphs is Menger's theorem, which characterizes the connectivity and edge-connectivity of a graph in terms of the
Connectivity_(graph_theory)
Graph which remains connected when k or fewer nodes removed
connected whenever fewer than k vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected
Vertex_connectivity
Symbol connecting formulas in logic
for the English connectives. Some logical connectives possess properties that may be expressed in the theorems containing the connective. Some of those
Logical_connective
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
In mathematics, a statement that has been proven
called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the
Theorem
Common pattern in metabolism
sets of theorems exist, the summation and connectivity theorems. Branched pathways have an additional set of branch centric summation theorems. When combined
Branched_pathways
Graph which remains connected when fewer than k edges are removed
edges are removed. The edge-connectivity of a graph is the largest k for which the graph is k-edge-connected. Edge connectivity and the enumeration of k-edge-connected
Edge_connectivity
complement of st-connectivity, known as st-non-connectivity, is also in the class NL, since NL = coNL by the Immerman–Szelepcsényi theorem. In particular
St-connectivity
Results on triad homotopy groups
theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces. This connectivity result
Blakers–Massey_theorem
Equivalence between strongly orientable graphs and bridgeless graphs
is already included (with the assumption of 2-vertex-connectivity rather than 2-edge-connectivity) in the seminal earlier work of Hopcroft & Tarjan (1973)
Robbins'_theorem
Gives a homomorphism from homotopy groups to homology groups
homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. The Hurewicz theorems are a key link between
Hurewicz_theorem
Second-smallest eigenvalue of a graph Laplacian
(vertex) connectivity of a graph, algebraic connectivity ≤ connectivity {\displaystyle {\text{algebraic connectivity}}\leq {\text{connectivity}}} , unless
Algebraic_connectivity
Theorem that arithmetical truth cannot be defined in arithmetic
Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Subfield of automated reasoning and mathematical logic
automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a
Automated_theorem_proving
Closure of nondeterministic space under complementation
principle used to prove the theorem has become known as inductive counting. It has also been used to prove other theorems in computational complexity
Immerman–Szelepcsényi_theorem
Mathematical model for deduction or proof systems
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
Formal_system
Branch of geometry
perspective on non-integrability is through the Chow–Rashevskii connectivity theorem, which states that any two points in a contact manifold can be connected
Contact_geometry
On the connectivity of a group of diffeomorphisms of a manifold
In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold. Given
Pseudoisotopy_theorem
Fundamental theorem in mathematical logic
of these theorems can be proven in a completely effective manner, each one can be effectively obtained from the other. The compactness theorem says that
Gödel's_completeness_theorem
homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates
Homotopical_connectivity
Branch of mathematical logic
prove theorems of mathematics. The field was founded by Harvey Friedman. Its defining method can be described as "going backwards from the theorems to the
Proof_theory
Type of logical system
has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to
First-order_logic
Reasoning for mathematical statements
of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle
Mathematical_proof
Characteristic of some logical systems
"semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically
Completeness_(logic)
On Hamiltonian cycles in planar graphs
Grinberg used his theorem to find non-Hamiltonian cubic polyhedral graphs with high cyclic edge connectivity. The cyclic edge connectivity of a graph is the
Grinberg's_theorem
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
Mathematical theorem
{\displaystyle f'(a)>0} . It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for b ∈ C ∖ G {\displaystyle b\in \mathbb {C}
Riemann_mapping_theorem
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Branch of mathematical logic
are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast
Reverse_mathematics
Algebra concept
Therefore, homological connectivity is equivalent to the graph having a single connected component, which is equivalent to graph connectivity. It is similar to
Homological_connectivity
Problem in computer science
Minsky notes: ...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness [of] the state diagram
Halting_problem
Basic framework of mathematics
theorem that is proved from true premises by means of a sequence of syllogisms (inference rules), the premises being either already proved theorems or
Foundations_of_mathematics
Mathematical game
Meshulam's game is a game used to explain a theorem of Roy Meshulam related to the homological connectivity of the independence complex of a graph, which
Meshulam's_game
Measure of algorithmic complexity
impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program P computing a
Kolmogorov_complexity
Statement that is taken to be true
knowledge. They are accepted without demonstration. All other assertions (theorems, in the case of mathematics) must be proven with the aid of these basic
Axiom
Theorem in mathematical logic
Chapter XVIII: Compactness, Embeddings and Definability. 645--716, see Theorems 4.5.9, 4.6.12 and Proposition 4.6.9. For compact logics for an extended
Compactness_theorem
Axiom used in logic and philosophy
law allows one to enhance the technique of using the deduction theorem to prove theorems. Suppose one is given a set of premises Γ and one wants to deduce
Peirce's_law
On forbidden minors in planar graphs
topological minor. Wagner published both theorems in 1937, subsequent to the 1930 publication of Kuratowski's theorem, according to which a graph is planar
Wagner's_theorem
Theorem for proving more complex theorems
April 2023. Doron Zeilberger, Opinion 82: A Good Lemma is Worth a Thousand Theorems This article incorporates material from Lemma on PlanetMath, which is licensed
Lemma_(mathematics)
Branch of logic
the sense that all and only the classical propositional tautologies are theorems, may be derived using only disjunction and negation (as Russell, Whitehead
Propositional_logic
Concept in mathematical logic
In logic, a functionally complete set of logical connectives or Boolean operators is one that can be used to express all possible truth tables by combining
Functional_completeness
Concept in mathematics
supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative
Conservative_extension
In mathematical logic, a well-formed formula with no free variables
sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate
Sentence_(mathematical_logic)
Whether a decision problem has an effective method to derive the answer
the theorems of a system. A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the
Decidability_(logic)
Mathematical principle
{\displaystyle T_{1}} prove the same theorems not involving the functional symbol f {\displaystyle f} ). Shoenfield states the theorem in the form for a new function
Extension by new constant and function names
Extension_by_new_constant_and_function_names
Theorem in set theory
dimensions cannot be continuous Schröder–Bernstein theorem for measurable spaces Schröder–Bernstein theorems for operator algebras Schröder–Bernstein property
Schröder–Bernstein_theorem
Polyhedra are determined by surface distance
uniqueness theorems for convex polyhedra is Cauchy's theorem, which states that a convex polyhedron is uniquely determined by the shape and connectivity of its
Alexandrov's theorem on polyhedra
Alexandrov's_theorem_on_polyhedra
Non-contradiction of a theory
incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies
Consistency
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
Logical connective OR
logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as ∨ {\displaystyle \lor } and read aloud as "or". For
Logical_disjunction
Graphs of d-dimensional polytopes are d-connected
In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex
Balinski's_theorem
Category of mathematical proof
of impossibility, negative proofs, or negative results. Impossibility theorems often resolve decades or centuries of work spent looking for a solution
Proof_of_impossibility
Mathematical construction
include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization
Ultraproduct
Paradox in set theory
Cantor's diagonal argument – Proof in set theory Gödel's incompleteness theorems – Limitative results in mathematical logic Hilbert's first problem – Proposition
Russell's_paradox
Axiom of set theory
less common than the type that requires the axiom of choice to be true. Theorems of ZF hold true in any model of that theory, regardless of the truth or
Axiom_of_choice
Every set is smaller than its power set
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
Cantor's_theorem
Bound on eigenvalues
In mathematics, the Gershgorin circle theorem (also called Gershgorin Disk Theorem) may be used to bound the spectrum of a square matrix. It was first
Gershgorin_circle_theorem
Subfield of mathematics
mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary
Mathematical_logic
Logical connective AND
truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as ∧ {\displaystyle \wedge }
Logical_conjunction
Theorem in mathematical logic
In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the
Lindström's_theorem
Model of (first-order) Peano arithmetic that contains non-standard numbers
standard model, so a model where Goodstein's theorem fails must be non-standard. Gödel's incompleteness theorems also imply the existence of non-standard
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
Mathematical graph theorem
Diks & Stanczyk (2010). If G is a regular graph of degree d whose edge connectivity is at least d − 1, and G has an even number of vertices, then it has
Petersen's_theorem
Term in logic and deductive reasoning
Using the narrow definition of theorem, for sentences provable from no premises, weak soundness says that all theorems are tautologies. Strong soundness
Soundness
Standard system of axiomatic set theory
axiom turns the axioms of infinity, power set, and choice (7–9 above) into theorems. Many important statements are independent of ZFC. The independence is
Zermelo–Fraenkel_set_theory
functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even formal theories are considered as mathematical objects
Mathematical_object
Properties linking logical conjunction and disjunction
logic. The duality consists in these metalogical theorems: In classical propositional logic, the connectives for conjunction and disjunction can be defined
Conjunction/disjunction duality
Conjunction/disjunction_duality
Undecidability of equality of real numbers
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ln
Richardson's_theorem
Mathematical logic concept
arithmetic and that its consistency is therefore less controversial. Gentzen's theorem is concerned with first-order arithmetic: the theory of the natural numbers
Gentzen's_consistency_proof
Square matrix used to represent a graph or network
the maximum degree. This can be seen as result of the Perron–Frobenius theorem, but it can be proved easily. Let v be one eigenvector associated to λ
Adjacency_matrix
Mathematical proof at least partially generated by computer
proofs of mathematical theorems from the bottom up using automated reasoning techniques such as heuristic search. Such automated theorem provers have proved
Computer-assisted_proof
Branch of mathematics that studies sets
that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with Peano arithmetic. The result was
Set_theory
On linear-time algorithms for graph logic
Parker & Tovey (1992). It is considered the archetype of algorithmic meta-theorems. In one variation of monadic second-order graph logic known as MSO1, the
Courcelle's_theorem
Area of mathematical logic
It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems) that a theory has a model if and only if it
Model_theory
Set of sentences in a formal language
definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference
Theory_(mathematical_logic)
Logical operation
a negation is called a negand or negatum. Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions
Negation
Logical connective
over the set of connectives { → , ⊥ } {\displaystyle \{\to ,\bot \}} are called f-implicational. In classical logic the other connectives, such as ¬ {\displaystyle
Material_conditional
Sequence of words formed by specific rules
by its theorems. Two formal systems F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have all the same theorems and yet
Formal_language
Summary of a mathematical proof
gives a sketch of a proof of the first of Gödel's incompleteness theorems. This theorem applies to any formal theory that satisfies certain technical hypotheses
Proof sketch for Gödel's first incompleteness theorem
Proof_sketch_for_Gödel's_first_incompleteness_theorem
Partition of a graph whose components are reachable from all vertices
are strongly connected themselves. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time
Strongly_connected_component
Type of theory in mathematical logic
countable language, the downward and upward Löwenheim–Skolem theorems, plus the completeness theorem, implies that I ( T , κ ) ≥ 1 {\displaystyle I(T,\kappa
Categorical_theory
Function in mathematical logic
Kurt Gödel developed the concept for the proof of his incompleteness theorems. A Gödel numbering can be interpreted as an encoding in which a number
Gödel_numbering
System of mathematical set theory
not requiring infinite sets, and is the weakest known set theory whose theorems include the first-order Peano axioms. The ontology of GST is identical
General_set_theory
Theorem equivalent to the Axiom of Choice
American Mathematical Society, 53 (2): 209 Tarski, A. (1924), "Sur quelques theorems qui equivalent a l'axiome du choix", Fundamenta Mathematicae, 5: 147–154
Tarski's_theorem_about_choice
Relation between deterministic and nondeterministic space complexity
space O ( f ( n ) 2 ) {\displaystyle O(f(n)^{2})} . Thus by deciding connectivity in a graph representing nondeterministic Turing machine configurations
Savitch's_theorem
Model in mathematical logic not isomorphic to the standard model
is infinite and the language is first-order, then the Löwenheim–Skolem theorems guarantee the existence of non-standard models. The non-standard models
Non-standard_model
Yes-or-no question that cannot ever be solved by a computer
undecidable for Turing machines. The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs
Undecidable_problem
System of formal deduction in logic
other logics as well. It is defined as a deductive system that generates theorems from axioms and inference rules, especially if the only postulated inference
Hilbert_system
Mathematical proposition equivalent to the axiom of choice
the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space
Zorn's_lemma
Method of deriving conclusions
constructs. To establish theorems, mathematicians apply rules of inference to these axioms, aiming to demonstrate that the theorems are logical consequences
Rule_of_inference
sets Boolean algebra (structure) Boolean algebra Field of sets Logical connective Propositional calculus Ampheck Analysis of Boolean functions Balanced
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Proof that only uses basic techniques
once thought that certain theorems, like the prime number theorem, could only be proved by invoking "higher" mathematical theorems or techniques. However
Elementary_proof
Symbol representing a property or relation in logic
properties can be derived from these, and they are sufficient for proving theorems in mathematics. Similarly, set membership can be understood solely through
Predicate_(logic)
1956 computer program written by Allen Newell, Herbert A. Simon and Cliff Shaw
the first 52 theorems in chapter two of Whitehead and Bertrand Russell's Principia Mathematica, and found a new and shorter proof for Theorem 2.85. In 1955
Logic_Theorist
Generalizations in graph theory
theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Possessing negative truth value
the " ⊥ {\displaystyle \bot } " connective is defined to be consistent, if and only if the false is not among its theorems. In the absence of propositional
False_(logic)
In logic, a statement which is always true
binary connectives ∨ {\displaystyle \lor } and ∧ {\displaystyle \land } representing disjunction and conjunction respectively, and the unary connective ¬ {\displaystyle
Tautology_(logic)
Symbol representing a mathematical concept
for example, in the context of proving metalogical theorems (such as Gödel's incompleteness theorems), where one doesn't want to allow the introduction
Function_symbol
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS
CONNECTIVITY THEOREMS