Philosophy Dictionary of Arguments

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Set Theory: set theory is the system of rules and axioms, which regulates the formation of sets. The elements are exclusively numbers. Sets contain individual objects, that is, numbers as elements. Furthermore, sets contain sub-sets, that is, again sets of elements. The set of all sub-sets of a set is called the power set. Each set contains the empty set as a subset, but not as an element. The size of sets is called the cardinality. Sets containing the same elements are identical. See also comprehension, comprehension axiom, selection axiom, infinity axiom, couple set axiom, extensionality principle.

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

Christian Thiel on Set Theory - Dictionary of Arguments

Thiel I 308
Set Theory: Bourbaki never talks about logicism, only about set theory. Sets are genuinely mathematical objects, not reducible to others (logic: classes). The set concept is an essential tool for the unification of mathematics.
I 308/309
Set Theory: as a fundamental discipline of mathematics: Basic concepts such as relation and function are traced back to the concept of set by explicit definition.
Relation as symmetrical or asymmetrical pair formation. Two-digit relation. Sometimes we need means to express the order. Ordered pairs.
Def I 310.
Functions: Def: right unambiguous relations.
If one presupposes the traceability of all higher types of numbers to the natural numbers once, one can also win these still set-theoretically.
I 311
The real question is a philosophical one and concerns the justification of the reductionist program behind everything.
Thiel: whether even numbers as mathematical entities turn out to be sets still appears today to be one of the most important philosophical questions, despite all the logical dead ends into which the classical logizistic approach has fallen.

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.

Chr. Thiel
Philosophie und Mathematik Darmstadt 1995

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Ed. Martin Schulz, access date 2022-09-25
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