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Disputed term/author/ism | Author |
Entry |
Reference |
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Consistency | Quine | IX 209/10
Consistency/Set Theory/Quine: we have been able to prove it twice as we have a simple model in finite sets - which does not apply if we once have added an axiom of infinity. Consistency is questionable and more difficult and more urgent to prove. And the evidence is even less convincing - problem: the question of whether the methods themselves are consistent.
>Set Theory/Quine.II 178
The essence of the Corollary of Goedel's incompleteness theorem is that the internal consistency of a mathematical theory can usually only be proved by resorting to another theory based on further premises and it is therefore less reliable than the original one. This has a melancholic connotation.
But this helps us to prove that one theory is stronger than another: This is achieved by proving in one theory that the other is consistent. II 180
Goedel's third great discovery: the consistency of the continuum hypothesis and the axiom of choice. II 210
Possible Worlds/QuineVsKripke: possible worlds allow proof of consistency, but no clear interpretation: when are objects equal? For example Bishop Buttler said that any thing is this thing and "no other thing": Problem/QuineVsButler: Identity does not follow necessarily.
>Possible Worlds/Quine. IX 192
Set theory/Modern Type Theory/Consistency/Quine: we can prove the freedom of contradiction of this version of set theory with cumulative types:
Def Cumulative Types/Set Theory/Quine: Type 0: only L is of type 0. Type 1: L and {L} and nothing else. Type n: should generally include and only include the 2n sets belonging to type n -1. So each quantification interprets only finitely many cases. Every closed statement can be checked mechanically for truth. Such a simple proof will no longer work if the infinity axiom is added. IX 210
Infinite Classes/Consistency: the proof also becomes less convincing if we have to accept infinite amounts.
Problem: whether the methods themselves are consistent (only with infinite classes). The highest thing we can often strive for is that we prove that such a system is consistent when another, corresponding system is less distrusted. IX 239
If the consistency of one set theory can be proven in another, the latter is the stronger (unless both are contradictory). Zermelo's system is stronger than type theory. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference InFrom a Logical Point of View, Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz InZur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |