Liberal Arts Math- not for the faint-hearted.
I am a Math-phobic. This question has me stuck. Please help if you can.
(a) Explain why the deletion of a dependent axiom from an axiom system has no effect on which theorems can be proved or on which models the system has.
(b) if an axiom system has two dependent axioms, does it follow that both could be deleted with no substantive effect on the system. Please explain.
Re: Liberal Arts Math- not for the faint-hearted.
Welcome to the forum. Logic can be considered as a part of discrete mathematics (since traditionally there are just two truth values 0 and 1 and not the whole continuum between 0 and 1), so logic-related questions can be posted to the Discrete Math subforum.
(a) Suppose T is a set of axioms and A is another axioms derivable from T. Then everything derivable from T is also derivable from T with A, using the same derivation. Conversely, if a derivation from T plus A uses A at some point, then we can construct a derivation from T only, which first derives A and then uses it as necessary.
Suppose M is a model of T plus A; then M is a model of T only. Conversely, suppose M is a model of T. Since A is derivable from T, the soundness theorem says that every model of T is also a model of A, so M is a model of T together with A.
(b) Suppose that T consists of independent axioms and for some A ∈ T, let A' be A /\ (p -> p) (a conjunction of A and some tautology). Then A and A' are derivable from each other, i.e., they are dependent. If we remove both A and A' from T plus A', then we are left with T without A, which is weaker than T because axioms in T were independent (in particular, A was not derivable from other axioms).