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Full Idea
A 'theory' is any logically closed set of propositions, ..and since any proposition has infinitely many consequences, including all the logical truths, so that theories have infinitely many premisses.
Gist of Idea
A theory is logically closed, which means infinite premisses
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.43
A Reaction
Read is introducing this as the essential preliminary to an account of the Compactness Theorem, which relates these infinite premisses to the finite.
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| 16906 | The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Frege, by Jeshion] |
| 16865 | 'Theorems' are both proved, and used in proofs [Frege] |
| 17807 | To study formal systems, look at the whole thing, and not just how it is constructed in steps [Curry] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 14620 | Theories in logic are sentences closed under consequence, but in truth discussions theories have axioms [Fine,K] |
| 10973 | A theory is logically closed, which means infinite premisses [Read] |
| 15328 | A theory is 'non-conservative' if it facilitates new mathematical proofs [Horsten] |
| 16310 | A theory is some formulae and all of their consequences [Halbach] |