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Published online by Cambridge University Press: 10 May 2010
In the first part of the book, there were several assertions about which we said that they cannot be proved without the use of certain axioms. We now describe how a result concerning the unprovability of an assertion can be formulated precisely, and we will present two general methods that can be used to establish such results. According to the well-known Godel Incompleteness Theorem, the consistency (i.e., of the state of being contradiction free) of ZF cannot be proved inside ZF. On the other hand, it is known that, in an axiom system with sufficient expressive power, it is possible to formulate its own consistency. This formula is denoted by Con(Σ) for the axiom system Σ. As it is not possible to prove Con(ZF), it is possible to imagine a state of affairs such that one can prove, e.g., the formula 0 = 1 in ZF. For this reason we need the following definition.
Definition A8.1.Let L0 ⊂ L, Σ ⊂ Wff(L), and let φ ∈ Wff(L). We say that the formula is relatively consistent with σ if the formula Con(Σ) implies Con(Σ ∪ {φ}). The proof of such an implication is called a relative consistency proof.
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