Economics Dictionary of ArgumentsHome
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| Sets: a set is a summary of objects relating to a property. In the set theory, conditions are established for the formation of sets. In general, sets of numbers are considered. Everyday objects as elements of sets are special cases and are called primordial elements. Sets are, in contrast to e.g. sequences not ordered, i.e. no order is specified for the consideration of the elements. See also element relation, sub-sets, set theory, axioms._____________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. | |||
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Leon Henkin on Sets - Dictionary of Arguments
Quine IX 222 Set/Quine: the condition to be a set: "Ey(z ε y)". (6) {x:∀y[(0 ε y u S'' y ≤ y) › x ε y } ≠ {x: ∀y[(y e ϑ u 0 ε y u S''y ≤ y) › x ε y]}. Here the right side contains extras! 0,1,2 and their successors belong to both classes. An inequality should therefore discredit the right and not the left expression as a version of "N". IX 223 On the right, of course, there are no extras if there is a set of y whose elements are exactly 0,1,2 and their successors. Then, on the contrary, the right side will be exactly this class y. Conversely, if there is no such set y, then the right side contains extras. Because the formula proves to be stratified, so the class qualifies as a set, so it would itself be a set y, unless it contains extras. N.B.: can we in any case rely on the fact that the left side contains the scarcest of the two, exactly 0,1,2 and their successors? No! Henkin: simple proof that no definition of "N" of any kind enables us to prove that N contains just 0,1,2 and its successors without extras. As long as "0 ε N", "1 ε N", "2 ε N" etc. all apply, there can be no contradiction in the assumption that in addition an x ε N exists with (7) x ε N, x ≠ 0, x ≠ 1, x ≠ 2... ad infinitum Proof: ...since a proof can only use finitely many premises, any proof of a contradiction from (7) uses only finitely many of the premises (7), but any such finite set is true for a certain x. Classes/Quantification/Concepts/Quine: quantification via classes allows us to use concepts that would otherwise be out of our reach. (see above Section II) Example "and their successors". Example predecessor. Universe/Set Theory/Quine: is an unregulated matter that looks different from theory to theory. Concept/Set Theory: a similar relativity must be feared with the concepts. This was emphasized by Skolem (1922/23). Especially for the deceptively familiar "and its successors". >Consistency/Henkin. Def Omega-contradictory/(w)/Goedel: (Goedel 1931) is a system when there is a formula "Fx" such that any one of the statements "F0", "F1", "F2",... can be proved ad infinitum in the system, but also "Ex(x ε N and ~Fx)". >Proofs, >Provability._____________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. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Henkin I Leon Henkin Retracing elementary mathematics New York 1962 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 In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From 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 In Zur 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 |
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