|Axiom: principle or rule for linking elements of a theory that is not proven within the theory. It is assumed that axioms are true and evident. Adding or eliminating axioms turns a system into another system. Accordingly, more or less statements can be constructed or derived in the new system. > System.|
|Kripke, Saul Aaron
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|Axioms||EMD II 389ff
Axioms/infinite/Kripke: then not all Tarski sentences are derivable anymore - proof/Kripke: only has a finite number of steps and cites only a finite number of axioms - otherwise rule (rule of evidence): "implicit definition" (Hilbert, "which axioms are valid?"> rule).
EMD II 389
Infinitely many axioms/Kripke: From an infinite number of truth sentences T(f) ↔ f the Tarski sentences cannot be deduced for any f - e.g. assuming we add a biconditional to a simple predicate P(x) and take P(0), P(1), P(2)... as number-theoretic axioms. These new axioms have the power that P(x) is valid for every number - does (x)P(x) still follow the normal rules of deduction? No - evidence cites only a finite number of axioms - reductio ad absurdum: if (x)P(x) was deducible (derivable), it would have to be derivable from a finite number of axioms: P(m1)...P(mn) - m: Number name in the formal language of the biconditional which denotes the number m - it is clear that it cannot be derived from a finite number of axioms - if we define P(x) as true of m1...mn, each of the finite axioms will be true, but (x)P(x) will be false - every instance is known but not the generalization - also applicable to finite systems.
Solution: we must allow an infinity rule (e.g. Omega rules).
KripkeVsWallace: same problems apply to the referential quantification.
Name und Notwendigkeit Frankfurt 1981
S. A. Kripke
Outline of a Theory of Truth (1975)
Recent Essays on Truth and the Liar Paradox, R. L. Martin (Hg), Oxford/NY 1984
G. Evans/J. McDowell
Truth and Meaning Oxford 1977
The Varieties of Reference (Clarendon Paperbacks) Oxford 1989