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Topics referred to by the same term
In mathematics, ω-logic can refer to: ω-logic, an infinitary extension of first-order logic Ω-logic, a deductive system in set theory developed by Hugh
Omega-logic
Fictional classification of mutant appearing in Marvel Comics
'logic puzzle' on which superhero stories thrives. In the Marvel Cinematic Universe film Deadpool & Wolverine, Cassandra Nova is referred as an Omega-level
Omega-level_mutants
Type of probabilistic logic
Subjective logic is a type of probabilistic logic that explicitly takes epistemic uncertainty and source trust into account. In general, subjective logic is suitable
Subjective_logic
Last letter of the Greek alphabet
of an elementary topos. In combinatory logic, the looping combinator, (S I I (S I I)). In group theory, the omega and agemo subgroups of a p-group, Ω(G)
Omega
Logic that allows infinitely long proofs
{L}}_{\omega _{1},\omega }} . Two infinitary logics stand out in their completeness. These are the logics of L ω , ω {\displaystyle L_{\omega ,\omega }} and
Infinitary_logic
Mathematical logic theory with exactly one countably infinite model up to isomorphism
In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is
Omega-categorical_theory
Subfield of mathematics
{\displaystyle L_{\omega _{1},\omega }} . In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have
Mathematical_logic
Mathematical logic concept
subjective logic expressed as: ( ω P | ~ Q A , ω P | ~ ¬ Q A ) = ( ω Q | P A , ω Q | ¬ P A ) ϕ ~ a P , {\displaystyle (\omega _{P{\tilde {|}}Q}^{A},\omega _{P{\tilde
Contraposition
Mathematical theory
In mathematical logic, an ω-consistent (or omega-consistent, or numerically segregative) theory is a theory (collection of sentences) that is not only
Ω-consistent_theory
Inference seeking the simplest and most likely explanation
{\displaystyle {\begin{aligned}\omega _{Y\,{\widetilde {\|}}\,X}&=\omega _{X\mid Y}\;{\widetilde {\circledcirc }}\;\omega _{X}\\&=(\omega _{X\mid Y}\;{\widetilde
Abductive_reasoning
Rule of logical inference
subjective logic expressed as: ω Q ‖ P A = ( ω Q | P A , ω Q | ¬ P A ) ⊚ ω P A , {\displaystyle \omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot
Modus_ponens
Function in logic
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and
Truth_function
Rule of logical inference
subjective logic expressed as: ω P ‖ ~ Q A = ( ω Q | P A , ω Q | ¬ P A ) ⊚ ~ ( a P , ω Q A ) , {\displaystyle \omega _{P{\tilde {\|}}Q}^{A}=(\omega _{Q|P}^{A}
Modus_tollens
Non-contradiction of a theory
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T {\displaystyle T} is consistent if there is no
Consistency
Infinite cardinal number
{\displaystyle \omega ,\omega _{\omega },\omega _{\omega _{\omega }},\cdots } which is sometimes denoted ω ω ⋱ {\textstyle \omega _{\omega _{\ddots }}}
Aleph_number
Whether a decision problem has an effective method to derive the answer
1007/978-3-642-30870-3_9. ISBN 978-3-642-30870-3. S2CID 8998263. "Lo.logic – Checkmate in $\omega$ moves?". Poonen, Bjorn (2014). "10. Undecidable Problems: A
Decidability_(logic)
Mathematical technique used in proof theory
{\displaystyle \omega _{1}^{\mathrm {CK} }} . In particular, the proof-theoretic ordinal of an inconsistent theory is equal to ω 1 C K {\displaystyle \omega _{1}^{\mathrm
Ordinal_analysis
Axiomatic set theory devised by W.V.O. Quine
In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification
New_Foundations
Elementary toposes can be used as models of intuitionistic higher-order logic. An elementary topos (hereafter just topos) can be pictured as an alternate
Elementary_topos
Axiomatic set theories based on the principles of mathematical constructivism
classical logic. E C S T {\displaystyle {\mathsf {ECST}}} has infinite sets, in particular the set of natural numbers ω {\displaystyle \omega } , but has
Constructive_set_theory
Concerned with the notion of stability in model theory
cardinality restriction, a strong form of stability now called ω {\displaystyle \omega } -stability, and he made significant use of this equivalence. In the course
Stable_theory
Describes approximate behavior of a function
x\neq \Omega _{-}(1)\quad } as x → ∞ . {\displaystyle \quad x\to \infty ~.} For understanding the formal definitions, consult the list of logic symbols
Big_O_notation
Deductive system in set theory
In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of
Ω-logic
Hydra game in mathematical logic
In mathematics, especially mathematical logic, graph theory and number theory, the Buchholz hydra game is a type of hydra game, which is a single-player
Buchholz_hydra
sets Omega logic Ω-logic is a form of logic introduced by Hugh Woodin On The class of all ordinals order type A concept in set theory and logic that categorizes
Glossary_of_set_theory
1989 video game
simulation combat game by Nemesys RoboSport Logic simulation Scisco, Peter; Ferrell, Keith (October 1989). "Omega". Compute!. p. 100. Retrieved 11 November
Omega_(video_game)
Computational problem with high complexity
, {\displaystyle {\mathsf {F}}_{\omega },{\mathsf {F}}_{\omega ^{\omega }},{\mathsf {F}}_{\omega ^{\omega ^{\omega }}},} or F ϵ 0 {\displaystyle {\mathsf
Nonelementary_problem
following logics have the zero-one law: First-order logic (as seen above) First-order logic L ∞ ω ω {\displaystyle L_{\infty \omega }^{\omega }} with infinite
Zero–one_law_(logic)
Generalization of "n-th" to infinite cases
{\displaystyle \omega } (omega) to be the least element that is greater than every natural number, along with ordinal numbers ω + 1 {\displaystyle \omega +1}
Ordinal_number
Form of logic that allows quantification over predicates
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic
Second-order_logic
Family of mathematical functions
{\displaystyle g_{\omega }(n)=n} g ω + m ( n ) = n + m {\displaystyle g_{\omega +m}(n)=n+m} g ω m ( n ) = m n {\displaystyle g_{\omega m}(n)=mn} g ω m (
Slow-growing_hierarchy
Mathematical theory for handling uncertainty
{\displaystyle 2^{\Omega }} to [0, 1] such that: Axiom 1: Π ( ∅ ) = 0 {\displaystyle \Pi (\varnothing )=0} Axiom 2: Π ( Ω ) = 1 {\displaystyle \Pi (\Omega )=1} Axiom
Possibility_theory
Type of cardinal number in mathematics
{\displaystyle \omega _{\omega }} is the limit of the sequence ω {\displaystyle \omega } , ω 1 {\displaystyle \omega _{1}} , ω 2 {\displaystyle \omega _{2}} ,
Regular_cardinal
Subdiscipline of proof theory
_{n}\vdash \Delta _{n}\mid \Box \Sigma \vdash \Box \Pi \mid \Omega \vdash \Theta }}} for modal logic S5, where ◻ Σ {\displaystyle \Box \Sigma } means that every
Structural_proof_theory
Impossible task in computing
Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using
Entscheidungsproblem
this vague notion of a logic being "nice", in this sense, mathematically precise. The Leibniz operator Ω {\displaystyle \Omega } is the operator that
Leibniz_operator
Large countably-infinite ordinal number
_{0}=\psi _{0}(\Omega )} , B H O = ψ 0 ( Ω 2 ) {\displaystyle {\mathsf {BHO}}=\psi _{0}(\Omega _{2})} , ψ 0 ( Ω 3 ) {\displaystyle \psi _{0}(\Omega _{3})} ,
Buchholz's_ordinal
Theorem in mathematical logic
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model
Compactness_theorem
Paradox in set theory
numbers less than Ω {\displaystyle \Omega } is Ω {\displaystyle \Omega } itself. But this means that Ω {\displaystyle \Omega } , being the order type of a proper
Burali-Forti_paradox
Israeli mathematician
mathematical logic, in particular set theory. He served as president of the Hebrew University of Jerusalem, was president of the Association for Symbolic Logic from
Menachem_Magidor
Swiss mathematician (1930–2026)
conditions to omega-categoricity, an important concept in model theory. He returned to Switzerland in 1972, where he served as a professor of logic and computer
Erwin_Engeler
Basic circuit in quantum computing
computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. Quantum logic gates are the building
Quantum_logic_gate
\Theta \Rightarrow \Omega }}} Hypersequent calculi have been used to treat modal logics, intermediate logics, and substructural logics. Hypersequents usually
Hypersequent
Mathematical model of computation
{\displaystyle \omega } is the output function. If the output function depends on the state and input symbol ( ω : S × Σ → Γ {\displaystyle \omega :S\times \Sigma
Finite-state_machine
First letter of the Greek alphabet
thing. The New Testament has God declaring himself to be the "Alpha and Omega, the beginning and the end, the first and the last." (Revelation 22:13,
Alpha
Large cardinal number
every function f : [ κ ] < ω → { 0 , 1 } {\displaystyle f:[\kappa ]^{<\omega }\to \{0,1\}} , there is a set of order type α {\displaystyle \alpha } that
Erdős_cardinal
Ordinals in mathematics and set theory
_{0}+1,\qquad \omega ^{\varepsilon _{0}+1}=\varepsilon _{0}\cdot \omega ,\qquad \omega ^{\omega ^{\varepsilon _{0}+1}}=(\varepsilon _{0})^{\omega },\qquad {\text{etc
Large_countable_ordinal
Type of mathematical function
\gamma <\omega ^{\omega ^{\beta +1}}\land n<\omega \implies \omega ^{\omega ^{\beta +1}}\cdot \alpha +\omega ^{\omega ^{\beta }}\cdot (\omega \cdot \gamma
Ordinal_notation
Concept in model checking (computer science)
P=\{A_{0}A_{1}....\in (2^{AP})^{\omega }\mid \forall j\in \mathbb {N} :A_{j}\ {\text{satisfies}}\ \Phi \}} for some propositional logic formula Φ {\displaystyle
Linear_time_property
or permit fundamental sequences of length ω 1 {\displaystyle \mathrm {\omega } _{1}} . The n th {\displaystyle n^{\text{th}}} element of the fundamental
Fundamental sequence (set theory)
Fundamental_sequence_(set_theory)
generalized to other logics, like L κ , ω {\displaystyle L_{\kappa ,\omega }} , or L ω 1 , ω ( Q ) {\displaystyle L_{\omega _{1},\omega }(Q)} , where Q {\displaystyle
Abstract_elementary_class
Method for constructing models of set theory
consideration is that, typically, it is necessary that ω 1 {\displaystyle \omega _{1}} is not collapsed. This is often accomplished by the use of a preservation
Iterated_forcing
Branch of mathematics concerning probability
sense, denoted by Ω {\displaystyle \Omega } . It is then assumed that for each element x ∈ Ω {\displaystyle x\in \Omega \,} , an intrinsic "probability"
Probability_theory
Mathematical conjecture
ω 2 , ω 1 ) ↠ ( ω 1 , ω ) {\displaystyle (\omega _{2},\omega _{1})\twoheadrightarrow (\omega _{1},\omega )} . The axiom of constructibility implies that
Chang's_conjecture
Order type of the set of all recursive ordinals
{\displaystyle \omega } . The notation ω 1 C K {\displaystyle \omega _{1}^{\mathsf {CK}}} is in reference to ω 1 {\displaystyle \omega _{1}} , the first
Nonrecursive_ordinal
Undertale character
souls of the six fallen humans, he evolves into a grotesque form known as "Omega Flowey", alternatively known as "Photoshop Flowey". In the "True Pacifist"
Flowey
Certain large countable ordinal
ω ( 0 ) {\displaystyle \theta _{\Omega ^{\omega }}(0)} or ψ ( Ω Ω ω ) {\displaystyle \psi (\Omega ^{\Omega ^{\omega }})} is the limit of ordinals that
Small_Veblen_ordinal
∈ M {\displaystyle g\in M} mapping ω {\displaystyle \omega } to ω ∖ { 0 } {\displaystyle \omega \setminus \{0\}} such that g {\displaystyle g} diverges
Sacks_property
Analog of Grothendieck topology
object A with classifier χ s : A → Ω {\displaystyle \chi _{s}:A\rightarrow \Omega } , then the composition j ∘ χ s {\displaystyle j\circ \chi _{s}} defines
Lawvere–Tierney_topology
Typed lambda calculus
isomorphism, System F corresponds to second-order propositional intuitionistic logic. System F can be seen as part of the lambda cube, together with even more
System_F
Mathematician and philosopher (1906–1978)
Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building
Kurt_Gödel
Linear circuit that produces an output voltage that is a fraction of its input voltage
}}}={\frac {\frac {1}{\mathrm {j} \omega C}}{{\frac {1}{\mathrm {j} \omega C}}+R}}={\frac {1}{1+\mathrm {j} \omega RC}}\ .} The product τ (tau) = RC is
Voltage_divider
Area of mathematical logic
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing
Model_theory
Base set of symbols with which a language is formed
binary string. Infinite sequences of symbols may be considered as well (see Omega language). Strings are often written as the concatenation of their symbols
Alphabet_(formal_languages)
given an infinite subtree of the binary tree 2 < ω {\displaystyle 2^{<\omega }} , it is possible to find an infinite path through the tree with particular
Low_basis_theorem
Mathematical function characterizing set membership
the function that indicates membership in a set. In fuzzy logic and modern many-valued logic, predicates are the characteristic functions of a probability
Indicator_function
Extension of first-order logic with atoms expressing variable dependencies
t_{n-1}} . Dependence logic is a logic of imperfect information, like branching quantifier logic or independence-friendly logic (IF logic): in other words
Dependence_logic
Theory of relational databases
S)\cup ((R-\pi _{r_{1},r_{2},\dots ,r_{n}}(R\bowtie S))\times \{(\omega ,\dots ,\omega )\})} The right outer join (⟖) behaves almost identically to the
Relational_algebra
Mathematical object in category theory
category theory, a subobject classifier is a special object Ω {\displaystyle \Omega } of a category such that, informally, the subobjects of any object X {\displaystyle
Subobject_classifier
Mathematical concept
numbers up to ω ω {\displaystyle \omega ^{\omega }} . Each turn of the spiral represents one power of ω {\displaystyle \omega } . Transfinite induction requires
Transfinite_induction
Set theory concept
[\omega ]^{\omega }} the set of all infinite subsets of ω {\displaystyle \omega } . For any a , b ∈ [ ω ] ω {\displaystyle a,b\in [\omega ]^{\omega }}
Cardinal characteristic of the continuum
Cardinal_characteristic_of_the_continuum
Topics referred to by the same term
BCE in the lowlands of eastern Cilicia An informal term for a member of Omega Psi Phi fraternity, a historically African American Greek-letter fraternity
Que
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
System of arithmetic in proof theory
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic
Elementary function arithmetic
Elementary_function_arithmetic
Countable ordinal that is the order type of a computable well-ordering of natural numbers
{\displaystyle \omega } . Since there exists a Turing machine that decides x < y {\displaystyle x<y} , this means that ω {\displaystyle \omega } is a
Computable_ordinal
Polish computer scientist (born 1981)
studies at RWTH Aachen University under Erich Grädel. His dissertation, Logic and Games on Automatic Structures, studied games and logical definability
Lukasz_Kaiser
Proposition in mathematical logic
{\displaystyle \aleph _{\omega }} or ℵ ω 1 + ω {\displaystyle \aleph _{\omega _{1}+\omega }} or any cardinal with cofinality ω {\displaystyle \omega } . The continuum
Continuum_hypothesis
Computer science and logic conference
on Logic in Computer Science (LICS) is an annual academic conference on the theory and practice of computer science in relation to mathematical logic. Extended
Symposium on Logic in Computer Science
Symposium_on_Logic_in_Computer_Science
2^{\aleph _{\omega }}<\aleph _{\omega _{1}}} holds. The analogous bound 2 ℵ ω 1 < ℵ ω 2 {\displaystyle 2^{\aleph _{\omega _{1}}}<\aleph _{\omega _{2}}} follows
Pcf_theory
Number measuring the chance an event occurs
sample space of the experiment, sometimes denoted as Ω {\displaystyle \Omega } . The power set of the sample space is formed by considering all different
Probability
Theoretical computer science concept
the new ω-language KL. Omega (infinite iteration) As the notation hints, the operation ( ⋅ ) ω {\displaystyle (\cdot )^{\omega }} is the infinite version
Omega_language
Mathematical result on infinite trees
branching subtree of ω < ω {\displaystyle \omega ^{<\omega }} has an infinite path. Here ω {\displaystyle \omega } denotes the set of natural numbers (thought
Kőnig's_lemma
Interpretation of quantum mechanics
using Einstein's light quanta hypothesis: E = ℏ ω {\displaystyle E=\hbar \omega } and de Broglie's hypothesis: p = ℏ k {\displaystyle \mathbf {p} =\hbar
De_Broglie–Bohm_theory
Branch of logic
more generally for sentences in the infinitary logic L ∞ ω ω {\displaystyle L_{\infty \omega }^{\omega }} , which allows for potentially arbitrarily long
Finite_model_theory
Variation of a finite automaton that runs on infinite input
accepting states of A {\textstyle A} , formally a subset of Q ω {\textstyle Q^{\omega }} . An input for A {\textstyle A} is an infinite string over the alphabet
Ω-automaton
with P preserves stationary subsets of [ λ ] ω {\displaystyle [\lambda ]^{\omega }} . The proper forcing axiom asserts that if P {\displaystyle P} is proper
Proper_forcing_axiom
Single-player iterative mathematical game played on a mathematical tree
{Hydra} (6)>f_{\underbrace {\omega ^{\cdots ^{\omega }}} _{6}}(5)} The Buchholz hydra game is a hydra game in mathematical logic, a single player game based
Hydra_game
∈ M {\displaystyle g\in M} mapping ω {\displaystyle \omega } to ω ∖ { 0 } {\displaystyle \omega \setminus \{0\}} such that g {\displaystyle g} diverges
Laver_property
American logician (1943–2011)
2011) was an American mathematician who worked in set theory, mathematical logic and foundations, and topology. Baumgartner was born in Wichita, Kansas,
James_Earl_Baumgartner
Mathematical system
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative
Second-order_arithmetic
Theory and paradigm of statistics
'}:=\{\omega \in \Omega \mid (\theta ',\omega )\in M\})} . Now, let A θ := { θ } × Ω {\displaystyle A_{\theta }:=\lbrace \theta \rbrace \times \Omega } and
Bayesian_statistics
Being equally consistent
In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In
Equiconsistency
Type of non-sinusoidal waveform
{4}{\pi }}\left(\sin(\omega t)+{\frac {1}{3}}\sin(3\omega t)+{\frac {1}{5}}\sin(5\omega t)+\ldots \right),&{\text{where }}\omega =2\pi f.\end{aligned}}}
Square_wave_(waveform)
Theorem about natural numbers
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein
Goodstein's_theorem
Concept in philosophy and set theory
Further ω < ω + 1 < ⋯ < ω + ω = ω ⋅ 2 {\displaystyle \omega <\omega +1<\dots <\omega +\omega =\omega \cdot 2} , and so on. Specifically, ordinal numbers
Absolute_infinite
Transforming a function in such a way that it only takes a single argument
product. The internal language of such categories is linear logic, a form of quantum logic; the corresponding type system is the linear type system. Such
Currying
Model theory concept
In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities
Spectrum_of_a_theory
Fifteenth letter of the Latin alphabet
Greek, a variation of the form later came to differentiate this long sound (omega, meaning "large O") from the short o (Omicron, meaning "small o"). The Greek
O
Hierarchy of complexity classes for formulas defining sets
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej
Arithmetical_hierarchy
1999 film by Dean Parisot
character as he confronts the Thermians' enemy, Sarris, who demands the "Omega 13", a secret super weapon with unknown capabilities mentioned but never
Galaxy_Quest
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