A theory is a subset of a formal language together with a set of inference rules on that formal language in which the members of the theory need no premises to be true, right? So if I had a subset ##\mathcal{T}## of a formal language ##\mathcal{F}##, and a set of inference rules in which all members of ##\mathcal{T}## were true without any premises, that would make ##\mathcal{T}## a theory, right?(adsbygoogle = window.adsbygoogle || []).push({});

Now a formal system is an alphabet ##\Sigma## together with a subset ##F## of all words over ##\Sigma## whose members are well formed formulas, a set of inference rules on ##F## and another subset of ##F## that make up the axioms of the formal system.

In short:

- theory: formal language, inference rules, axioms.
- formal system: alphabet, wff, inference rules, axioms.

But isn't a subset of all words over an alphabet a formal language anyway? Making theories and formal systems the "same" concepts?

Or are theories more general than subsets of formal languages? So the premises and conclusions in the rules of inference of a theory need not be well formed formulas of a formal language. Is that the case?

Also, what field of mathematics should I look into to learn more about these concepts? Logic? Proof theory?

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# Difference between formal systems and theories

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