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** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

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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

**.**

__**__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 sum

__**__The Lebesgue measure

**of**

**a**countable disjoint union

**of**measurable sets

**is**equal to

**the**

__sum__

**of**

**the**measures

**of**

**the**individual sets

**.**

__**__Atkinson independently discovered

**the**Midpoint circle algorithm for fast drawing

**of**circles by using

**the**

__sum__

**of**consecutive odd numbers

**.**

__**__Bézout domain

**,**an integral domain in which

**the**

__sum__

**of**two principal ideals

**is**again

**a**principal ideal

__**__Signal-to-noise and distortion ratio ( SNDR ) indicates in dB

**the**ratio between

**the**powers

**of**

**the**converted main signal and

**the**

__sum__

**of**

**the**noise and

**the**generated harmonic spurs

__**__it

**is**proved on oath

**,**or in such other manner as may be prescribed

**,**that he has been duly served with

**the**summons

**,**and that

**a**reasonable

__sum__has been paid or tendered to him for costs and expenses

**,**and

__**__Khaliq-O-Vision

**,**Ray Bardani ( engineers ), David Isaac ( producer ) & Marcus Miller ( producer & artist ) for M <

__sum__>

__**__Roger Frye used experimental mathematics techniques to find

**the**smallest counterexample to Euler's

__sum__

**of**powers conjecture

**.**

__**__July 10-He discovers that every positive integer

**is**representable as

**a**

__sum__

**of**at most three triangular numbers

**,**noting in his diary " Heureka!

__**__In

**a**variation on

**the**basic definition

**of**

**a**bioship

**,**

**the**Star Trek

**:**The Next Generation episode " Emergence ", features

**a**story in which

**the**Enterprise-D develops an artificial intelligence from

**the**

__sum__

**of**

**the**ship's experiences

**,**for

**the**sole purpose

**of**creating

**a**biological life form which

**is**released into space

**.**

__**__Consequently

**,**

**the**

__sum__

**of**distances d ( F < sub > 1 </ sub >, P ) + d ( F < sub > 2 </ sub >, P ) must be constant as P moves along

**the**curve because

**the**

__sum__

**of**distances d ( P < sub > 1 </ sub >, P ) + d ( P < sub > 2 </ sub >, P ) also remains constant

**.**

__**__Compute

**the**number

**of**forest descendants for this node

**,**by adding one to

**the**

__sum__

**of**its children's descendants

**.**

__**__

**the**

__sum__

**of**votes remaining in

**the**multi-seat constituencies after

**the**distribution

**of**

**the**seats

**,**plus

__**__

**the**

__sum__

**of**votes cast for losing candidates

**of**each party in

**the**first valid round

**of**each single-seat constituency ( similar to

**the**scorporo system ).

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