Proposition (mathematics)
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In mathematics, the term proposition is used for a proven statement that is of more than passing interest, but whose proof is neither profound nor difficult.
Comparison
The general term for proven mathematical statements is theorem, which also is used in a second, more particular sense, for a proven statement which required some effort, or is in some way a final result. In increasing order of difficulty, the names used for different levels of (general) theorems is approximately:
Technically, since a proposition is sometimes followed by a proof, it is a theorem in the general sense, but when the word proposition is used the proof is not challenging enough to call the result a theorem in the particular sense.
Propositions are minor building-blocks for major theorems, like lemmas. But the word lemma is used to describe proofs that establish statements that are stepping stones for further theorems when the proof is somewhat difficult. [dubious – discuss] The term proposition is used for statements that are easy consequences of earlier definitions, possibly presented without proof; when a proposition is a simple consequence of a previous theorem, the term is a synonym of corollary (which is preferred).
Alternate meanings
In mathematical logic, the term proposition is also used as an abbreviation for propositional formula. This use of the term does not imply or suggest that the formula is provable or true.
The word proposition is also sometimes used to name the section of a theorem that gives the statement of fact that is to be proven.
Decidability
Kurt Godel believed that every mathematical statement was decidable. In other words, he believed we were able to know whether a given statement was true or false. However, there has been a lot of philosophical discussion on it.
See also
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