|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. See also axiom systems, systems, strength of theories, proofs, provability._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
Saul A. Kripke on Axioms - Philosophy Dictionary of Arguments
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?"> rules/Kripke).
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 rule)
KripkeVsWallace: same problems apply to the >referential quantification._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Naming and Necessity, Dordrecht/Boston 1972
Name und Notwendigkeit Frankfurt 1981
Saul A. Kripke
"Speaker’s Reference and Semantic Reference", in: Midwest Studies in Philosophy 2 (1977) 255-276
Eigennamen, Ursula Wolf, Frankfurt/M. 1993
Saul A. Kripke
Is there a problem with substitutional quantification?
Truth and Meaning, G. Evans/J McDowell, Oxford 1976
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