Definition:Theorem
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General Definition
In all contexts, the definition of the term theorem is by and large the same.
That is, a theorem is a statement which has been proved to be true.
Logic
A theorem in logic is a statement which can be shown to be the conclusion of a logical argument which depends on no premises except axioms.
A sequent which denotes a theorem $\phi$ is written $\vdash \phi$, indicating that there are no premises.
In this context, $\vdash$ is read as:
- It is a theorem that ...
Formal Systems
Let $\mathcal F$ be a formal system consisting of a formal language with deductive apparatus $\mathcal D$.
A theorem is a well-formed word in $\mathcal F$ which can be deduced from the axioms and/or axiom schemata of $\mathcal D$ by means of its rules of inference.
As can be seen, this is equivalent to the definition in logic above if one defines the apparatus of logic as a formal system.
Mathematics
The term theorem is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch.
Note that statements which are taken as axioms in one branch of mathematics may be theorems in others.
It is possible (and this is the ultimate aim of ProofWiki) to justify basing the whole of mathematics on a handful of axioms.
Also see
Sources
- Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{I}: \S 5$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.2.1$: Definition $1.10$
