A theorem is a statement that can be logically or mathematically proven or has been proved. A theorem’s proof is a logical argument that makes use of the inference rules of a deductive system in order to establish that it is a logical consequence of the axioms made for this example and previously proven theorems.
In mathematical logic, theorems and proofs have been formalized as a means of allowing mathematical reasoning about them. From within the logical bounds of this statement, it is clear that statements become well-formed formulae of formal scientific language. A theory consists of statements called axioms that form its basis, as well as some rules that can help deduce them. These are sometimes included in the axioms themselves. The theorems of that theory are the statements that can be derived from the axioms by the above-mentioned rules. This formalization is what has led to proof theory, which allows proving general theorems about proofs and theorems. Gödel’s incompleteness theorems, in particular, show that every consistent theory that contains natural numbers has true statements on the same that are not theorems of the theory.
The validity of a theorem depends solely on the correctness of its proof. A theorem is independent of the truth, or even the significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but rather that the validity of a theorem is independent of the significance of the axioms. This independence is useful because it allows the use of results of certain areas of mathematics in apparently unrelated areas.
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