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106-453: In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic . Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only

212-454: A {\displaystyle a} and b {\displaystyle b} . Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established

318-417: A first-order logic whose atomic formulas were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement . Appending this schema, as well as the axiom of regularity (first proposed by John von Neumann ), to Zermelo set theory yields the theory denoted by ZF . Adding to ZF either the axiom of choice (AC) or

424-434: A least element under the order R {\displaystyle R} . Given axioms 1  –  8 , many statements are provably equivalent to axiom 9 . The most common of these goes as follows. Let X {\displaystyle X} be a set whose members are all nonempty. Then there exists a function f {\displaystyle f} from X {\displaystyle X} to

530-786: A set of symbols . Let Φ {\displaystyle \Phi } be a maximally consistent set of S {\displaystyle S} -formulas containing witnesses . Define an equivalence relation ∼ {\displaystyle \sim } on the set of S {\displaystyle S} -terms by t 0 ∼ t 1 {\displaystyle t_{0}\sim t_{1}} if t 0 ≡ t 1 ∈ Φ {\displaystyle \;t_{0}\equiv t_{1}\in \Phi } , where ≡ {\displaystyle \equiv } denotes equality . Let t ¯ {\displaystyle {\overline {t}}} denote

636-418: A complete proof, and several ongoing projects hope to shorten and simplify this proof. Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by a layman. In mathematical logic , a formal theory is a set of sentences within a formal language . A sentence

742-400: 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. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which

848-467: A completely symbolic form—with the presumption that a formal statement can be derived from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and

954-583: A finite number of nodes. There are many equivalent formulations of the ZFC axioms. The following particular axiom set is from Kunen (1980) . The axioms in order below are expressed in a mixture of first order logic and high-level abbreviations. Axioms 1–8 form ZF, while the axiom 9 turns ZF into ZFC. Following Kunen (1980) , we use the equivalent well-ordering theorem in place of the axiom of choice for axiom 9. All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts

1060-461: A formal language is useful within proof theory , which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory , which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through interpretation . Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in

1166-405: A formal symbolic proof can in principle be constructed. In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, one might even be able to substantiate

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1272-411: A formal theorem is fundamentally syntactic, in contrast to the notion of a true proposition, which introduces semantics . Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. belief , justification or other modalities ). The soundness of a formal system depends on whether or not all of its theorems are also validities . A validity

1378-572: A hierarchy by assigning to each set the first stage at which that set was added to V . Inconsistent In classical , 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 formula φ {\displaystyle \varphi } such that both φ {\displaystyle \varphi } and its negation ¬ φ {\displaystyle \lnot \varphi } are elements of

1484-488: A natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 have the Mertens property, and the smallest number that does not have this property is only known to be less than the exponential of 1.59 × 10, which is approximately 10 to the power 4.3 × 10. Since the number of particles in the universe is generally considered less than 10 to

1590-535: A particular deductive logic , the logic is called complete . The completeness of the propositional calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic , are not complete. A consistency proof

1696-528: A property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension , thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes. There are many equivalent formulations of

1802-537: A set z {\displaystyle z} is a subset of a set x {\displaystyle x} if and only if every element of z {\displaystyle z} is also an element of x {\displaystyle x} : The Axiom of power set states that for any set x {\displaystyle x} , there is a set y {\displaystyle y} that contains every subset of x {\displaystyle x} : The axiom schema of specification

1908-498: A set X having infinitely many members. (It must be established, however, that these members are all different because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω which can also be thought of as the set of natural numbers N . {\displaystyle \mathbb {N} .} By definition,

2014-464: A set with exactly these two elements. The axiom of pairing is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement if we are given a set with at least two elements. The existence of a set with at least two elements is assured by either the axiom of infinity , or by the axiom schema of specification and the axiom of the power set applied twice to any set. The union over

2120-636: A single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs. For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 10. The Riemann hypothesis has been verified to hold for

2226-419: A statement that is equivalent to it yields ZFC. Formally, ZFC is a one-sorted theory in first-order logic . The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. The signature has a single predicate symbol, usually denoted ∈ {\displaystyle \in } , which

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2332-399: A theorem by using a picture as its proof. Because theorems lie at the core of mathematics, they are also central to its aesthetics . Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood,

2438-601: A theorem if proven true. Until the end of the 19th century and the foundational crisis of mathematics , all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every natural number has a successor, and that there is exactly one line that passes through two given distinct points. These basic properties that were considered as absolutely evident were called postulates or axioms ; for example Euclid's postulates . All theorems were proved by using implicitly or explicitly these basic properties, and, because of

2544-556: A theorem of the ambient theory, although they can be proved in a wider theory. An example is Goodstein's theorem , which can be stated in Peano arithmetic , but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as Zermelo–Fraenkel set theory . Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises . In light of

2650-432: A theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. Logically , many theorems are of the form of an indicative conditional : If A, then B . Such a theorem does not assert B — only that B

2756-566: A variable assignment β Φ {\displaystyle \beta _{\Phi }} by β Φ ( x ) := x ¯ {\displaystyle \beta _{\Phi }(x):={\bar {x}}} for each variable x {\displaystyle x} . Let I Φ := ( T Φ , β Φ ) {\displaystyle {\mathfrak {I}}_{\Phi }:=({\mathfrak {T}}_{\Phi },\beta _{\Phi })} be

2862-449: Is Fermat's Last Theorem , and there are many other examples of simple yet deep theorems in number theory and combinatorics , among other areas. Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture . Both of these theorems are only known to be true by reducing them to a computational search that

2968-448: Is consistent when there is no formula φ {\displaystyle \varphi } such that φ ∈ ⟨ A ⟩ {\displaystyle \varphi \in \langle A\rangle } and ¬ φ ∈ ⟨ A ⟩ {\displaystyle \lnot \varphi \in \langle A\rangle } . A trivial theory (i.e., one which proves every sentence in

3074-420: Is a finite set is easily proved from axioms 1–8 , AC only matters for certain infinite sets . AC is characterized as nonconstructive because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed". One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann . In this viewpoint, the universe of set theory

3180-494: Is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program . Hilbert's program was strongly impacted by the incompleteness theorems , which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent). Although consistency can be proved using model theory, it

3286-437: Is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of logical consequence . Some accounts define a theory to be closed under the semantic consequence relation ( ⊨ {\displaystyle \models } ), while others define it to be closed under

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3392-537: 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. The classification of finite simple groups is regarded by some to be the longest proof of a theorem. It comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give

3498-408: Is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are tautologies ). A formal system is considered semantically complete when all of its theorems are also tautologies. Zermelo%E2%80%93Fraenkel set theory In set theory , Zermelo–Fraenkel set theory , named after mathematicians Ernst Zermelo and Abraham Fraenkel ,

3604-464: Is a member of b {\displaystyle b} ). There are different ways to formulate the formal language. Some authors may choose a different set of connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known as functional completeness . This section attempts to strike a balance between simplicity and intuitiveness. The language's alphabet consists of: With this alphabet,

3710-645: Is a member of X and, whenever a set y is a member of X then S ( y ) {\displaystyle S(y)} is also a member of X . or in modern notation: ∃ X [ ∅ ∈ X ∧ ∀ y ( y ∈ X ⇒ S ( y ) ∈ X ) ] . {\displaystyle \exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X)\right].} More colloquially, there exists

3816-419: Is a necessary consequence of A . 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. The two together (without the proof) are called the proposition or statement of the theorem (e.g. " If A, then B " is the proposition ). Alternatively, A and B can be also termed the antecedent and

3922-393: Is a predicate symbol of arity 2 (a binary relation symbol). This symbol symbolizes a set membership relation. For example, the formula a ∈ b {\displaystyle a\in b} means that a {\displaystyle a} is an element of the set b {\displaystyle b} (also read as a {\displaystyle a}

4028-499: Is a set A {\displaystyle A} containing every element that is a member of some member of F {\displaystyle {\mathcal {F}}} : Although this formula doesn't directly assert the existence of ∪ F {\displaystyle \cup {\mathcal {F}}} , the set ∪ F {\displaystyle \cup {\mathcal {F}}} can be constructed from A {\displaystyle A} in

4134-615: Is a set for every x ∈ A , {\displaystyle x\in A,} then the range of f {\displaystyle f} is a subset of some set B {\displaystyle B} . The form stated here, in which B {\displaystyle B} may be larger than strictly necessary, is sometimes called the axiom schema of collection . Let S ( w ) {\displaystyle S(w)} abbreviate w ∪ { w } , {\displaystyle w\cup \{w\},} where w {\displaystyle w}

4240-399: Is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC, there are only sets, the interpretation of this logical theorem in the context of ZFC is that some set exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called free logic , in which it

4346-407: Is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox . Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics . Zermelo–Fraenkel set theory with

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4452-407: Is any existing set, the empty set can be constructed as Thus, the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on w {\displaystyle w} ). It is common to make a definitional extension that adds the symbol " ∅ {\displaystyle \varnothing } " to

4558-407: Is any infinite set and P {\displaystyle {\mathcal {P}}} is the power set operation. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in

4664-446: Is built up in stages, with one stage for each ordinal number . At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. The collection of all sets that are obtained in this way, over all the stages, is known as V . The sets in V can be arranged into

4770-466: Is consistent. If both A and ¬ A are consistent with T , then A is said to be independent of T . In the following context of mathematical logic , the turnstile symbol ⊢ {\displaystyle \vdash } means "provable from". That is, a ⊢ b {\displaystyle a\vdash b} reads: b is provable from a (in some specified formal system). Let S {\displaystyle S} be

4876-415: Is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion relative consistency is interesting in set theory (and in other sufficiently expressive axiomatic systems). If T is a theory and A is an additional axiom , T + A is said to be consistent relative to T (or simply that A is consistent with T ) if it can be proved that if T is consistent then T + A

4982-577: Is not free in φ {\displaystyle \varphi } . Then: (The unique existential quantifier ∃ ! {\displaystyle \exists !} denotes the existence of exactly one element such that it follows a given statement.) In other words, if the relation φ {\displaystyle \varphi } represents a definable function f {\displaystyle f} , A {\displaystyle A} represents its domain , and f ( x ) {\displaystyle f(x)}

5088-498: Is not provable from logic alone that something exists, the axiom of infinity asserts that an infinite set exists. This implies that a set exists, and so, once again, it is superfluous to include an axiom asserting as much. Two sets are equal (are the same set) if they have the same elements. The converse of this axiom follows from the substitution property of equality . ZFC is constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If

5194-426: Is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general. In theories of arithmetic, such as Peano arithmetic , there is an intricate relationship between

5300-438: Is some set. (We can see that { w } {\displaystyle \{w\}} is a valid set by applying the axiom of pairing with x = y = w {\displaystyle x=y=w} so that the set z is { w } {\displaystyle \{w\}} ). Then there exists a set X such that the empty set ∅ {\displaystyle \varnothing } , defined axiomatically,

5406-443: Is that it is falsifiable , that is, it makes predictions about the natural world that are testable by experiments . Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in

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5512-409: Is the case, so y {\displaystyle y} stands in a separate position from which it can't refer to or comprehend itself; therefore, in a certain sense, this axiom schema is saying that in order to build a y {\displaystyle y} on the basis of a formula φ ( x ) {\displaystyle \varphi (x)} , we need to previously restrict

5618-502: Is then used to define the power set P ( x ) {\displaystyle {\mathcal {P}}(x)} as the subset of such a y {\displaystyle y} containing the subsets of x {\displaystyle x} exactly: Axioms 1–8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed in Jech (2003) . Some ZF axiomatizations include an axiom asserting that

5724-632: Is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. Theorems in mathematics and theories in science are fundamentally different in their epistemology . A scientific theory cannot be proved; its key attribute

5830-433: The consequent , respectively. The theorem "If n is an even natural number , then n /2 is a natural number" is a typical example in which the hypothesis is " n is an even natural number", and the conclusion is " n /2 is also a natural number". In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in

5936-563: The axiom of choice , is presented here as a property about well-orders , as in Kunen (1980) . For any set X {\displaystyle X} , there exists a binary relation R {\displaystyle R} which well-orders X {\displaystyle X} . This means R {\displaystyle R} is a linear order on X {\displaystyle X} such that every nonempty subset of X {\displaystyle X} has

6042-465: The axiom schema of replacement and the axiom of the empty set . On the other hand, the axiom schema of specification can be used to prove the existence of the empty set , denoted ∅ {\displaystyle \varnothing } , once at least one set is known to exist. One way to do this is to use a property φ {\displaystyle \varphi } which no set has. For example, if w {\displaystyle w}

6148-485: The division algorithm , Euler's formula , and the Banach–Tarski paradox . A theorem and its proof are typically laid out as follows: The end of the proof may be signaled by the letters Q.E.D. ( quod erat demonstrandum ) or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by Paul Halmos following their use in magazines to mark the end of an article. The exact style depends on

6254-400: The empty set exists . The axioms of pairing, union, replacement, and power set are often stated so that the members of the set x {\displaystyle x} whose existence is being asserted are just those sets which the axiom asserts x {\displaystyle x} must contain. The following axiom is added to turn ZF into ZFC: The last axiom, commonly known as

6360-449: The equivalence class of terms containing t {\displaystyle t} ; and let T Φ := { t ¯ ∣ t ∈ T S } {\displaystyle T_{\Phi }:=\{\;{\overline {t}}\mid t\in T^{S}\}} where T S {\displaystyle T^{S}} is the set of terms based on

6466-475: The logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem . The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory , such as Russell's paradox , led to

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6572-406: The set of all sets cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is inconsistent , and every well-formed assertion, as well as its negation, is a theorem. In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even

6678-430: The syntactic consequence , or derivability relation ( ⊢ {\displaystyle \vdash } ). For a theory to be closed under a derivability relation, it must be associated with a deductive system that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context. The closure of the empty set under the relation of logical consequence yields

6784-469: The term interpretation associated with Φ {\displaystyle \Phi } . Then for each S {\displaystyle S} -formula φ {\displaystyle \varphi } : There are several things to verify. First, that ∼ {\displaystyle \sim } is in fact an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of

6890-548: The above using the axiom schema of specification: The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set. Formally, let φ {\displaystyle \varphi } be any formula in the language of ZFC whose free variables are among x , y , A , w 1 , … , w n , {\displaystyle x,y,A,w_{1},\dotsc ,w_{n},} so that in particular B {\displaystyle B}

6996-400: The author or publication. Many publications provide instructions or macros for typesetting in the house style . It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in

7102-635: The axiom of choice included is abbreviated ZFC , where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set , so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by

7208-840: The axiom of extensionality can be reformulated as which says that if x {\displaystyle x} and y {\displaystyle y} have the same elements, then they belong to the same sets. Every non-empty set x {\displaystyle x} contains a member y {\displaystyle y} such that x {\displaystyle x} and y {\displaystyle y} are disjoint sets . or in modern notation: ∀ x ( x ≠ ∅ ⇒ ∃ y ( y ∈ x ∧ y ∩ x = ∅ ) ) . {\displaystyle \forall x\,(x\neq \varnothing \Rightarrow \exists y(y\in x\land y\cap x=\varnothing )).} This (along with

7314-397: The axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets a {\displaystyle a} and b {\displaystyle b} there is a new set { a , b } {\displaystyle \{a,b\}} containing exactly

7420-513: The axioms of pairing and union) implies, for example, that no set is an element of itself and that every set has an ordinal rank . Subsets are commonly constructed using set builder notation . For example, the even integers can be constructed as the subset of the integers Z {\displaystyle \mathbb {Z} } satisfying the congruence modulo predicate x ≡ 0 ( mod 2 ) {\displaystyle x\equiv 0{\pmod {2}}} : In general,

7526-410: The axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to proof theory , which allows proving general theorems about theorems and proofs. In particular, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of

7632-483: The consistency of the theory and its completeness . A theory is complete if, for every formula φ in its language, at least one of φ or ¬φ is a logical consequence of the theory. Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete. Gödel's incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to

7738-818: The construction of entities of the more general form: This restriction is necessary to avoid Russell's paradox (let y = { x : x ∉ x } {\displaystyle y=\{x:x\notin x\}} then y ∈ y ⇔ y ∉ y {\displaystyle y\in y\Leftrightarrow y\notin y} ) and its variants that accompany naive set theory with unrestricted comprehension (since under this restriction y {\displaystyle y} only refers to sets within z {\displaystyle z} that don't belong to themselves, and y ∈ z {\displaystyle y\in z} has not been established, even though y ⊆ z {\displaystyle y\subseteq z}

7844-409: The desire for a more rigorous form of set theory that was free of these paradoxes. In 1908, Ernst Zermelo proposed the first axiomatic set theory , Zermelo set theory . However, as first pointed out by Abraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of

7950-425: The elements of a set exists. For example, the union over the elements of the set { { 1 , 2 } , { 2 , 3 } } {\displaystyle \{\{1,2\},\{2,3\}\}} is { 1 , 2 , 3 } . {\displaystyle \{1,2,3\}.} The axiom of union states that for any set of sets F {\displaystyle {\mathcal {F}}} , there

8056-429: The evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the interior angles of a triangle equals 180°, and this was considered as an undoubtable fact. One aspect of the foundational crisis of mathematics was the discovery of non-Euclidean geometries that do not lead to any contradiction, although, in such geometries,

8162-528: The existence of a set, although he notes that he does so only "for emphasis". Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself, ∃ x ( x = x ) {\displaystyle \exists x(x=x)} . Consequently, it

8268-647: The fact that ∼ {\displaystyle \sim } is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of t 0 , … , t n − 1 {\displaystyle t_{0},\ldots ,t_{n-1}} class representatives. Finally, I Φ ⊨ φ {\displaystyle {\mathfrak {I}}_{\Phi }\vDash \varphi } can be verified by induction on formulas. In ZFC set theory with classical first-order logic , an inconsistent theory T {\displaystyle T}

8374-458: The first 10 trillion non-trivial zeroes of the zeta function . Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved. Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e.,

8480-495: The foundations of mathematics to make them more rigorous . In these new foundations, a theorem is a well-formed formula of a mathematical theory that can be proved from the axioms and inference rules of the theory. So, the above theorem on the sum of the angles of a triangle becomes: Under the axioms and inference rules of Euclidean geometry , the sum of the interior angles of a triangle equals 180° . Similarly, Russell's paradox disappears because, in an axiomatized set theory,

8586-484: The interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. 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 ). Although theorems can be written in

8692-414: The language of ZFC with all free variables among x , z , w 1 , … , w n {\displaystyle x,z,w_{1},\ldots ,w_{n}} ( y {\displaystyle y} is not free in φ {\displaystyle \varphi } ). Then: Note that the axiom schema of specification can only construct subsets and does not allow

8798-414: The language of ZFC. If x {\displaystyle x} and y {\displaystyle y} are sets, then there exists a set which contains x {\displaystyle x} and y {\displaystyle y} as elements, for example if x = {1,2} and y = {2,3} then z will be {{1,2},{2,3}} The axiom schema of specification must be used to reduce this to

8904-421: The language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability . A theory is satisfiable if it has a model , i.e., there exists an interpretation under which all axioms in

9010-455: The meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement about a formal system (as opposed to within a formal system) is called a metatheorem . Some important theorems in mathematical logic are: The concept of

9116-448: The most important results, and use the terms lemma , proposition and corollary for less important theorems. In mathematical logic , the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language . A theory consists of some basis statements called axioms , and some deducing rules (sometimes included in

9222-498: The physical axioms on which such "theorems" are based are themselves falsifiable. A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time. Other terms may also be used for historical or customary reasons, for example: A few well-known theorems have even more idiosyncratic names, for example,

9328-472: The power 100 (a googol ), there is no hope to find an explicit counterexample by exhaustive search . 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 ). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role;

9434-456: The proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem. It has been estimated that over a quarter of a million theorems are proved every year. The well-known aphorism , "A mathematician

9540-625: The recursive rules for forming well-formed formulae (wff) are as follows: A well-formed formula can be thought as a syntax tree. The leaf nodes are always atomic formulae. Nodes ∧ {\displaystyle \land } and ∨ {\displaystyle \lor } have exactly two child nodes, while nodes ¬ {\displaystyle \lnot } , ∀ x {\displaystyle \forall x} and ∃ x {\displaystyle \exists x} have exactly one. There are countably infinitely many wffs, however, each wff has

9646-408: The same way such evidence is used to support scientific theories. Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find

9752-481: The set of consequences of T {\displaystyle T} . Let A {\displaystyle A} be a set of closed sentences (informally "axioms") and ⟨ A ⟩ {\displaystyle \langle A\rangle } the set of closed sentences provable from A {\displaystyle A} under some (specified, possibly implicitly) formal deductive system. The set of axioms A {\displaystyle A}

9858-436: The set of symbols S {\displaystyle S} . Define the S {\displaystyle S} - structure T Φ {\displaystyle {\mathfrak {T}}_{\Phi }} over T Φ {\displaystyle T_{\Phi }} , also called the term-structure corresponding to Φ {\displaystyle \Phi } , by: Define

9964-449: The set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative to a given semantics, or relative to the standard interpretation of the underlying language. A theory that is inconsistent has all sentences as theorems. The definition of theorems as sentences of

10070-407: The sets y {\displaystyle y} will regard within a set z {\displaystyle z} that leaves y {\displaystyle y} outside so y {\displaystyle y} can't refer to itself; or, in other words, sets shouldn't refer to themselves). In some other axiomatizations of ZF, this axiom is redundant in that it follows from

10176-635: The significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas. An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as mathematical objects , and to prove theorems about them. Examples are Gödel's incompleteness theorems . In particular, there are well-formed assertions than can be proved to not be

10282-525: The structure of proofs. Some theorems are " trivial ", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example

10388-543: The subset of a set z {\displaystyle z} obeying a formula φ ( x ) {\displaystyle \varphi (x)} with one free variable x {\displaystyle x} may be written as: The axiom schema of specification states that this subset always exists (it is an axiom schema because there is one axiom for each φ {\displaystyle \varphi } ). Formally, let φ {\displaystyle \varphi } be any formula in

10494-462: The sum of the angles of a triangle is different from 180°. So, the property "the sum of the angles of a triangle equals 180°" is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of sets leads to the contradiction of Russell's paradox . This has been resolved by elaborating the rules that are allowed for manipulating sets. This crisis has been resolved by revisiting

10600-463: The theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic . Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of

10706-409: The theory (that is they cannot be proved inside the theory). As the axioms are often abstractions of properties of the physical world , theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law , which is experimental , the justification of the truth of a theorem is purely deductive . A conjecture is a tentative proposition that may evolve to become

10812-408: The theory are true. This is what consistent meant in traditional Aristotelian logic , although in contemporary mathematical logic the term satisfiable is used instead. In a sound formal system , every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in

10918-529: The theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own Gödel sentence—provided that they are consistent, which

11024-629: The time, notably the cardinal number ℵ ω {\displaystyle \aleph _{\omega }} and the set { Z 0 , P ( Z 0 ) , P ( P ( Z 0 ) ) , P ( P ( P ( Z 0 ) ) ) , . . . } , {\displaystyle \{Z_{0},{\mathcal {P}}(Z_{0}),{\mathcal {P}}({\mathcal {P}}(Z_{0})),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(Z_{0}))),...\},} where Z 0 {\displaystyle Z_{0}}

11130-527: The union of the members of X {\displaystyle X} , called a " choice function ", such that for all Y ∈ X {\displaystyle Y\in X} one has f ( Y ) ∈ Y {\displaystyle f(Y)\in Y} . A third version of the axiom, also equivalent, is Zorn's lemma . Since the existence of a choice function when X {\displaystyle X}

11236-601: The variety of first-order logic in which you are constructing set theory does not include equality " = {\displaystyle =} ", x = y {\displaystyle x=y} may be defined as an abbreviation for the following formula: ∀ z [ z ∈ x ⇔ z ∈ y ] ∧ ∀ w [ x ∈ w ⇔ y ∈ w ] . {\displaystyle \forall z[z\in x\Leftrightarrow z\in y]\land \forall w[x\in w\Leftrightarrow y\in w].} In this case,

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