Shoenfield

In chapter 2 of Shoenfield's "Mathematical Logic", there is the following
exercise:

5. Let T be the theory with no nonlogical symbols and no nonlogical axioms.
a) Show that ~~(x = x) OR ~(x = x) is a theorem of T not provable without
propositional axioms. [Let f be a mapping from the set of formulas to the
set of truth values such that F(A) = T for A atomic; f(~A) = F; f(A OR B) =
f(B); f(Exists_x(A) = T. Show that if A is provable without propositional
axioms, then f(A) = T.]

I haven't the foggiest clue as to how to get started here. I don't
understand how the 'clue' helps to solve the problem. Heck, I don't even
understand the 'clue': I thought f(A) = T whether or not A is provable.
Could someone please explain how to 'show' the clue and why this proves that
the original formula is not provable without propositional axioms?

Many thanks.


.

Relevant Pages

  • Re: Phil phLIES again
    ... a natural that is preceded by infinitely many naturals, ... THIS IS *WHY* you have to use AXIOMS. ... infinity are MORE BASIC than sets and YOU NEVER ... Clue 2: There is no such thing as the axiom of specification. ...
    (sci.logic)
  • Re: Cantor Confusion
    ... Except that you offer not a shadow of a hint of a clue as to axioms for ... calculus that don't provide for infinite sets. ...
    (sci.math)
  • Re: Prove unprovable
    ... We have a first order theory with identity and no nonlogical symbols. ... or without using one of the rules of inference. ... The author provides us with a hint which consists of a truth valuation ... propositional axioms or A). ...
    (sci.logic)
  • Re: Shoenfield
    ... Let T be the theory with no nonlogical symbols and no nonlogical axioms. ... >> propositional axioms. ... Or does it mean non-logical axioms that just happen to be ...
    (sci.logic)