Lexicon of Arguments

Philosophical and Scientific Issues in Dispute
 
[german]


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Sc. Camps
Theses I
Theses II

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III 389ff
Axioms/infinite/Kripke: not all Tarski sentences are derivable anymore. Proof/Kripke: 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 following/Kripke.
III 389
Infinitely many axioms/Kripke: one cannot derive Tarski sentences for any kind of f's, from an infinite number of truth sentences T(f) ↔ 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 derived from a finite number of axioms: P(m1)...P(mn). M: m is the 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. This is also applicable to finite systems.
III 390
Solution: we must allow an infinity rule (e.g.> omega rule)
III 391
KripkeVsWallace: the same problems apply to the >referential quantification.

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