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** Well-ordering theorem: Every set can be well-ordered.

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## Some Related Sentences

** and theorem

__**__Tarski's

__theorem__

**:**For every infinite

**set**A, there is a bijective map between the sets A and A × A

**.**

__**__König's

__theorem__

**:**Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals

**.**

__**__If S is a

**set**of sentences of first-order logic and B is a consistent subset of S, then B is included in a

**set**that is maximal among consistent subsets of S

**.**The special case where S is the

**set**of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal

__theorem__; see the section " Weaker forms " below

**.**

__**__The Vitali

__theorem__on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable

**.**

__**__The Baire category

__theorem__about complete metric spaces, and its consequences, such as the open mapping

__theorem__and the closed graph

__theorem__

**.**

__**__Gödel's completeness

__theorem__for first-order logic

**:**every consistent

**set**of first-order sentences has a completion

**.**

__**__The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass

__theorem__); hence they are transcendental

**.**

__**__More generally, Rademacher's

__theorem__extends the differentiability result to Lipschitz mappings between Euclidean spaces

**:**a Lipschitz map ƒ

**:**U → R < sup > m </ sup >, where U is an open

**set**in R < sup > n </ sup >, is almost everywhere differentiable

**.**

__**__Artin reciprocity law, a general

__theorem__in number theory that provided a partial solution to Hilbert's ninth problem

** and Every

__**__Zorn's lemma

**:**

__Every__non-empty partially ordered

**set**in which every chain ( i

**.**e

**.**totally ordered subset ) has an upper bound contains at least one maximal element

**.**

__**__Tukey's lemma

**:**

__Every__non-empty collection of finite character has a maximal element with respect to inclusion

**.**

__**__

__Every__voiced consonant is written with a deep letter similar in shape to the corresponding voiceless consonant which in turn is written with a tall letter

**.**

__**__The Fugitive Guy

**:**

__Every__so often, Letterman would introduce " Roger Campbell " ( Elliott, wearing an extremely bad toupée ), a new member of the Late Night crew

**.**

__**__Luther Vandross and Marcus Miller for " Little Miracles ( Happen

__Every__Day )" performed by Luther Vandross

__**__1664-The Concession and Agreement of the Lords Proprietors of the Province of New Caesarea, or New Jersey, to and With All and

__Every__the Adventurers and All Such as Shall Settle or Plant There

__**__

__Every__item entry must

**be**within one level of its predecessor, such that each item must

**be**a sibling or child of the preceding item ( thus, no item

**can**

**be**a grandchild of the preceding item ).

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