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XVII. CHAPTER SEVENTEEN: AXIOM SYSTEMS
A. True or False
1. There are complete and consistent axiom systems for both sentential and predicate
logic.
2. There is a complete and consistent axiom system for arithmetic.
3. It is possible to construct an inconsistent axiom system for arithmetic in which all
truths of arithmetic would be derivable.
4. There are decision procedures both for sentential and predicate logic.
5. Truth table analysis constitutes a decision procedure for sentential logic.
6. There can be no decision procedure for any consistent formulation of arithmetic.
7. An axiom system for predicate logic is complete if either every well formed
formula or its negation is derivable as a theorem.
8. An axiom system for arithmetic is complete if either every well formed formula
or its negation is derivable as a theorem.
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A. Answers
B. Theory
1. If an axiom system is not complete, can we say whether it is consistent? (Defend
your answer.)
2. Can we say that an axiom system for predicate logic is complete if every formula
or its negation is derivable as a theorem? If so, why? If not, why not?
3. a. Roughly, how does one prove that a given consistent axiom system is
consistent?
b. Roughly, how does one prove that a given inconsistent axiom system is
inconsistent?
B. Answers
1. If an axiom system is not complete then there are formulas that we desire to prove