(6) {x:∀y[(0 ε y u S'' y ≤ y) ">

Psychology Dictionary of Arguments

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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.
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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.

 
Author Concept Summary/Quotes Sources

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.

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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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