corollary

WHATISTHEDIFFERENCEBETWEENATHEOREM,ALEMMA,

ANDACOROLLARY?

CHESON

(1)Definition—apreciandunambiguousdescriptionofthemeaningofamathe-

acterizesthemeaningofawordbygivingalltheproperties

andonlythopropertiesthatmustbetrue.

(2)Theorem—amathematicalstatementthatisprovedusingrigorousmathemat-

hematicalpaper,thetermtheoremisoftenrervedfor

themostimportantresults.

(3)Lemma—

casionallylemmascan

takeonalifeoftheirown(Zorn’slemma,Urysohn’slemma,Burnside’slemma,

Sperner’slemma).

(4)Corollary—aresultinwhichthe(usuallyshort)proofreliesheavilyonagiven

theorem(weoftensaythat“thisisacorollaryofTheoremA”).

(5)Proposition—aprovedandofteninterestingresult,butgenerallylessimportant

thanatheorem.

(6)Conjecture—astatementthatisunproved,butisbelievedtobetrue(Collatz

conjecture,Goldbachconjecture,twinprimeconjecture).

(7)Claim—tenudlikeaninformallemma.

(8)Axiom/Postulate—astatementthatisassumedtobetruewithoutproof.

Thearethebasicbuildingblocksfromwhichalltheoremsareproved(Eu-

clid’sfivepostulates,Zermelo-Frankelaxioms,Peanoaxioms).

(9)Identity—amathematicalexpressiongivingtheequalityoftwo(oftenvariable)

quantities(trigonometricidentities,Euler’sidentity).

(10)Paradox—astatementthatcanbeshown,usingagiventofaxiomsand

definitions,xesareoftenudtoshowthe

inconsistenciesinaflawedtheory(Rusll’sparadox).Thetermparadoxisoften

udinformallytodescribeasurprisingorcounterintuitiveresultthatfollows

fromagiventofrules(Banach-Tarskiparadox,Alabamaparadox,Gabriel’s

horn).

定理

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