Theorem

theoremspropositionconversemathematical theoremFormal theorempropositionsstatementclassical resultscorollaryhypothesis
In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms, or on the basis previously established statements such as other theorems.wikipedia
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Mathematical proof

proofproofsprove
In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms, or on the basis previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system.
The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference.

Axiom

axiomspostulateaxiomatic
In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms, or on the basis previously established statements such as other theorems.
All other assertions (theorems, in the case of mathematics) must be proven with the aid of these basic assumptions.

Propositional calculus

propositional logicpropositionalsentential logic
Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability.
These derived formulas are called theorems and may be interpreted to be true propositions.

Mathematical beauty

mathematical eleganceelegantAesthetics
Because theorems lie at the core of mathematics, they are also central to its aesthetics.

Mathematics

mathematicalmathmathematician
In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms, or on the basis previously established statements such as other theorems.
According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."

Kepler conjecture

Kepler's conjectureFlyspeck 1Flyspeck proof
The most prominent examples are the four color theorem and the Kepler conjecture.
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space.

Pythagorean theorem

Pythagoras' theoremPythagorasPythagoras's theorem
The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.
This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

Conjecture

conjecturalconjecturesconjectured
In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture), and B the conclusion of the theorem.
It was the first major theorem to be [[computer-assisted proof#List of theorems proved with the help of computer programs|proved using a computer]].

Banach–Tarski paradox

Banach-Tarski paradoxBanach-TarskiBanach-Tarski decomposition
The Banach–Tarski paradox is a theorem in measure theory that is paradoxical in the sense that it contradicts common intuitions about volume in three-dimensional space.
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

Formal system

logical systemdeductive systemsystem of logic
A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic).
A deductive system, also called a deductive apparatus, consists of the axioms (or axiom schemata) and rules of inference that can be used to derive theorems of the system.

Bézout's identity

Bézout's lemmaBézout identityAryabhata equation
Bézout's identity is a theorem asserting that the greatest common divisor of two numbers may be written as a linear combination of these numbers.
In elementary number theory, Bézout's identity (also called Bézout's lemma) is the following theorem:

Euclid's Elements

ElementsEuclid's ''ElementsEuclid
It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions.

Lemma (mathematics)

lemmalemmasLemma 1
In mathematics, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result.

Corollary

corollariesconsequenceCorollarial
In mathematics, a corollary is a theorem connected by a short proof to an existing theorem.

Scientific law

laws of physicsphysical lawlaws of nature
In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.
As with other kinds of scientific knowledge, laws do not have absolute certainty (as mathematical theorems or identities do), and it is always possible for a law to be contradicted, restricted, or extended by future observations.

Law of large numbers

strong law of large numbersweak law of large numbersBernoulli's Golden Theorem
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.

Proof theory

proof-theoreticProof-theoreticallyderive
The field of mathematics known as proof theory studies formal languages, axioms and the structure of proofs. Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language.
Its defining method can be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms.

Four color theorem

four-color theoremfour colour theoremfour color problem
Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read.
It was the first major theorem to be [[computer-assisted proof#List of theorems proved with the help of computer programs|proved using a computer]].

Non-classical logic

non-Aristotelian logicnon-classicalnon-classical" logic
However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic).

Gödel's incompleteness theorems

Gödel's incompleteness theoremincompleteness theoremincompleteness theorems
The most famous result is Gödel's incompleteness theorems; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.
Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic.

Mathematical theory

theoretical frameworkmathematical theoriesmathematical theorizing
The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory (see mathematical theory).

Formula

formulaeformulasmathematical formulae
Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language.
with the most important ones being mathematical theorems.

Paul Erdős

ErdősPál ErdősP. Erdős
The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.
His colleague Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities (this quotation is often attributed incorrectly to Erdős, but Erdős himself ascribed it to Rényi ).

Q.E.D.

quod erat demonstrandumQED
The end of the proof may be signaled by the letters Q.E.D. (quod erat demonstrandum) or by one of the tombstone marks "□" or "∎" meaning "End of Proof", introduced by Paul Halmos following their usage in magazine articles.
The style and system of the book are, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions.

Theory (mathematical logic)

theorytheoriesformal theories
A set of formal theorems may be referred to as a formal theory.
In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a theory T is then called a theorem of the theory.