Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Theorem
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{short description|In mathematics, a statement that has been proven}} {{distinguish|Teorema (disambiguation){{!}}Teorema|Theorema (disambiguation){{!}}Theorema|Theory (disambiguation){{!}}Theory}} [[File:Pythagorean Proof (3).PNG|thumb|200px|right|The [[Pythagorean theorem]] has at least 370 known proofs.<ref name='Loomis'>{{cite web|url=http://www.eric.ed.gov/PDFS/ED037335.pdf|author=Elisha Scott Loomis |title=The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs |access-date=2010-09-26 |work=[[Education Resources Information Center]] |publisher=[[Institute of Education Sciences]] (IES) of the [[U.S. Department of Education]] }} Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics.</ref>]] In [[mathematics]] and [[formal logic]], a '''theorem''' is a [[statement (logic)|statement]] that has been [[Mathematical proof|proven]], or can be proven.{{efn|In general, the distinction is weak, as the standard way to prove that a statement is provable consists of proving it. However, in mathematical logic, one considers often the set of all theorems of a theory, although one cannot prove them individually.}}<ref>{{Cite Merriam-Webster|Theorem|access-date=1 December 2024}}</ref><ref>{{Cite web|url=https://www.lexico.com/en/definition/theorem|archive-url=https://web.archive.org/web/20191102041621/https://www.lexico.com/en/definition/theorem|url-status=dead|archive-date=November 2, 2019|title=Theorem {{!}} Definition of Theorem by Lexico|website=Lexico Dictionaries {{!}} English|language=en|access-date=2019-11-02}}</ref> The ''proof'' of a theorem is a [[logical argument]] that uses the inference rules of a [[deductive system]] to establish that the theorem is a [[logical consequence]] of the [[axiom]]s and previously proved theorems. 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]].{{efn|An exception is the original [[Wiles's proof of Fermat's Last Theorem]], which relies implicitly on [[Grothendieck universe]]s, whose existence requires the addition of a new axiom to set theory.<ref>{{cite journal| title=What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory| first=Colin|last= McLarty| journal=The Review of Symbolic Logic| volume=13| number=3| pages=359–377| year=2010| publisher=Cambridge University Press| doi=10.2178/bsl/1286284558| s2cid=13475845}} </ref> This reliance on a new axiom of set theory has since been removed.<ref> {{cite journal| title=The large structures of Grothendieck founded on finite order arithmetic| first=Colin|last= McLarty| journal=Bulletin of Symbolic Logic| volume=16| number=2| pages=296–325| year=2020| publisher=Cambridge University Press| doi=10.1017/S1755020319000340| arxiv=1102.1773| s2cid=118395028}}</ref> Nevertheless, it is rather astonishing that the first proof of a statement expressed in elementary [[arithmetic]] involves the existence of very large infinite sets.}} 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 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 [[formal system|formalized]] in order to allow mathematical reasoning about them. In this context, statements become [[well-formed formula]]s of some [[formal language]]. A [[Theory (mathematical logic)|theory]] consists of some basis statements called ''axioms'', and some ''deducing rules'' (sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules.{{efn|A theory is often identified with the set of its theorems. This is avoided here for clarity, and also for not depending on [[set theory]].}} This formalization led to [[proof theory]], which allows proving general theorems about theorems and proofs. In particular, [[Gödel's incompleteness theorems]] show that every [[consistency|consistent]] theory containing the natural numbers has true statements on natural numbers that are not theorems of 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]].<ref name=":0">{{Citation|last=Markie|first=Peter|title=Rationalism vs. Empiricism|date=2017|url=https://plato.stanford.edu/archives/fall2017/entries/rationalism-empiricism/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2017|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-11-02}}</ref>{{efn|However, both theorems and scientific law are the result of investigations. See {{harvnb|Heath|1897|p=clxxxii |loc=Introduction, The terminology of [[Archimedes]]}}: "theorem (θεὼρνμα) from θεωρεἳν to investigate"}} A ''[[conjecture]]'' is a tentative proposition that may evolve to become a theorem if proven true.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)