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If we have a set of sentences S in first order logic. We know that we can create a first order theory Th(S) from S, which is the "set S" union "the sentences which we can prove them from S" . Also as first order theory, this theory follow the rules (axioms and rules of inferences) of first order logic. Does it has sense to define the Th(S) as above but with an extra union with "axioms of first order logic" ? Also can i consider the Th(S) as formal system which has "intersection" with the formal system First Order Logic?

zaxsqwedc
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  • What sort of formal logic system are you imagining using? For example, with a natural deduction type system, first-order logic is implemented entirely in terms of deduction rules that specify how the various logical operators behave. Whereas with a Hilbert-style proof system, you do have a combination of deduction rules and logical axioms (and in fact, as I understand it, some versions of Hilbert-style proof systems have an infinite family of logical axioms generated according to some rules). – Daniel Schepler Nov 08 '23 at 20:29
  • “We can prove from $S$” means prove using a deduction system for first order logic. Any logical axioms included in such a system will be provable of course, so would already be included. As will everything in $S$ (so your first union is unnecessary as well). – spaceisdarkgreen Nov 08 '23 at 20:34
  • @spaceisdarkgreen Can i say that th(S)= the formal system which has: (S+axioms of FOL) as axioms+ rules of inference of FOL+ language of FOL ? your comment It helped me to understand that because i use the term of "proof" it means that i used all this pack "S+axioms of FOL+ rules of inference of FOL+ language of FOL" – zaxsqwedc Nov 08 '23 at 20:59
  • @chers Yes, mostly, though I don’t know if you need to include “language” in there. Language just tells you what formulas there are (as a function of the nonlogical symbols) and is kind of implicit in everything else. – spaceisdarkgreen Nov 08 '23 at 21:15
  • Usually, we have the "underlying logic": first order logic plus some specific mathematical axioms, like e.g. the axioms of set theory. The axioms (if any) and the inference rules of logic are common to every theory: thus, the mathematical theory is "defined" by its own specific axioms. – Mauro ALLEGRANZA Nov 09 '23 at 07:22
  • See The definition of First Order Theory in Mendelson – Mauro ALLEGRANZA Nov 09 '23 at 08:17

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