An axiomatic system is a finite sequence of propositions a_1,a_2..,a_N which are called axioms 56:23
In the whole lectures, two kind of logics are introduced:
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Proposition: A variable which is either true or false. It is also remarked, you can force a proposition to be either true or false.
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Predicate: A proposition valued function of some variable.
In the whole lecture, I see that proposition logic is used to define both a predicate and an axiom system. My question is why is it that we is that propositions are used over predicates in the definition in an axiomatic system? Would there be any issues in doing mathematics if we use predicates instead of proposition for the definition of axiom set?
Excuse me if the question is very stupid.
Best Answer
See also the post linked above referring to Schuller's lectures; what the author calls proposition (in the formal sense) is a formula of the language without free variable, like e.g. $\forall x \forall y (x \in y)$ in the language of set theory.
In the Lectures, the author starts with propositional calculus where a proposition is symbolized by a propositional variable $p_i$.
Then he introduces predicate logic, with predicates symbols: $P(x)$.
See 45:25 where the author states that a quantifier turns a predicate $P(x)$ of a single variable into a proposition.
Usually, we call this a sentence, i.e. a formula with no free variables; the issue is that it is like a proposition of propositional calculus because - having no free variables - it has a definite truth value.
No, because we want that axioms have a definite truth value.
In light of this, the statement above can be rewritten as:
See examples regarding arithmetic and geometry.