In mathematics, the law of **trichotomy** states that every real number is either positive, negative, or zero.^{ [1] }

More generally, a binary relation *R* on a set *X* is **trichotomous** if for all *x* and *y* in *X*, exactly one of *xRy*, *yRx* and *x* = *y* holds. Writing *R* as <, this is stated in formal logic as:

- A relation is trichotomous if, and only if, it is asymmetric and connected.
- If a trichotomous relation is also transitive, then it is a strict total order; this is a special case of a strict weak order.
^{ [2] }^{ [3] }

- On the set
*X*= {*a*,*b*,*c*}, the relation*R*= { (*a*,*b*), (*a*,*c*), (*b*,*c*) } is transitive and trichotomous, and hence a strict total order. - On the same set, the cyclic relation
*R*= { (*a*,*b*), (*b*,*c*), (*c*,*a*) } is trichotomous, but not transitive; it is even antitransitive.

A **law of trichotomy** on some set *X* of numbers usually expresses that some tacitly given ordering relation on *X* is a trichotomous one. An example is the law "For arbitrary real numbers *x* and *y*, exactly one of *x* < *y*, *y* < *x*, or *x* = *y* applies"; some authors even fix *y* to be zero,^{ [1] } relying on the real number's additive linearly ordered group structure. The latter is a group equipped with a trichotomous order.

In classical logic, this **axiom of trichotomy** holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers.^{[ clarification needed ]} The law does not hold in general in intuitionistic logic.^{[ citation needed ]}

In Zermelo–Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderable sets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).^{ [4] }

*Begriffsschrift*contains an early formulation of the law of trichotomy- Dichotomy
- Law of noncontradiction
- Law of excluded middle
- Three-way comparison

An **axiom**, **postulate** or **assumption** is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek *axíōma* (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'

**First-order logic**—also known as **predicate logic**, **quantificational logic**, and **first-order predicate calculus**—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists*"* is a quantifier, while *x* is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

In mathematical logic, the **Peano axioms**, also known as the **Dedekind–Peano axioms** or the **Peano postulates**, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

In mathematical logic, a **universal quantification** is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

In set theory, **Zermelo–Fraenkel set theory**, named after mathematicians Ernst Zermelo and Abraham Fraenkel, 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 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.

**Relevance logic**, also called **relevant logic**, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics.

In axiomatic set theory and the branches of mathematics and philosophy that use it, the **axiom of infinity** is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.

In mathematical logic, **sequent calculus** is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional theorems in a first-order language rather than conditional tautologies.

In mathematics, a binary relation *R* is called **well-founded** on a class *X* if every non-empty subset *S* ⊆ *X* has a minimal element with respect to *R*, that is, an element *m* not related by *sRm* for any *s* ∈ *S*. In other words, a relation is well founded if

In mathematics, two sets or classes *A* and *B* are **equinumerous** if there exists a one-to-one correspondence between them, that is, if there exists a function from *A* to *B* such that for every element *y* of *B*, there is exactly one element *x* of *A* with *f*(*x*) = *y*. Equinumerous sets are said to have the same cardinality. The study of cardinality is often called **equinumerosity** (*equalness-of-number*). The terms **equipollence** (*equalness-of-strength*) and **equipotence** (*equalness-of-power*) are sometimes used instead.

In the foundations of mathematics, **von Neumann–Bernays–Gödel set theory** (**NBG**) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel-Choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

In mathematics, **intransitivity** is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of **antitransitivity**, which describes a relation that is never transitive.

In mathematics, an **asymmetric relation** is a binary relation on a set where for all if is related to then is *not* related to

**Tarski's axioms**, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory. Other modern axiomizations of Euclidean geometry are Hilbert's axioms and Birkhoff's axioms.

In the foundations of mathematics, **Morse–Kelley set theory** (**MK**), **Kelley–Morse set theory** (**KM**), **Morse–Tarski set theory** (**MT**), **Quine–Morse set theory** (**QM**) or the **system of Quine and Morse** is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over proper classes as well as sets, as first suggested by Quine in 1940 for his system ML.

In Mathematical logic, a **tautology** is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". A less abstract example is "either the ball is green, or the ball is not green". This would be true regardless of the color of the ball.

Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.

In mathematical logic, the **ancestral relation** of a binary relation *R* is its transitive closure, however defined in a different way, see below.

In constructive mathematics, a **pseudo-order** is a constructive generalisation of a linear order to the continuous case. The usual trichotomy law does not hold in the constructive continuum because of its indecomposability, so this condition is weakened.

- 1 2 Trichotomy Law at MathWorld
- ↑ Jerrold E. Marsden & Michael J. Hoffman (1993)
*Elementary Classical Analysis*, page 27, W. H. Freeman and Company ISBN 0-7167-2105-8 - ↑ H.S. Bear (1997)
*An Introduction to Mathematical Analysis*, page 11, Academic Press ISBN 0-12-083940-7 - ↑ Bernays, Paul (1991).
*Axiomatic Set Theory*. Dover Publications. ISBN 0-486-66637-9.

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