Argument
Axiom schema of specification
Overview
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of class construction, or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Some mathematicians refer to this axiom as the axiom schema of comprehension, although others reserve that term only for unrestricted comprehension; this axiom is a "restricted" version of unrestricted comprehension. Because restricting comprehension avoids Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Axiom schema of specification is.
This observation with regard to the nature of mathematics gives a hint as to the first and supreme condition of its possibility: it must be based on some pure percept in which it can exhibit all its concepts _in concreto_ and yet _à priori,_ or, as it is called, _construe_ them.' The phrase, and the emphasis on the doctrine, that perception must be taken as an auxiliary in mathematics,' belong specially to the…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Axiom schema of specification is.
CV. In forming axioms, we must invent a different form of induction from that hitherto in use; not only for the proof and discovery of principles (as they are called), but also of minor, intermediate, and, in short, every kind of axioms. The induction which proceeds by simple enumeration is puerile, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too…
Francis Bacon, Novum organum
States what Axiom schema of specification is.
Philosophers who discussed Axiom schema of specification
Related ideas
- If and only if
- Zermelo–Fraenkel set theory
- Kurt Gödel
- Intuitionistic logic
- Ernst Zermelo
- Logical conjunction
- Intersection (set theory)
- Set (mathematics)
- Set theory
- Subset
- Predicate (logic)
- Theorem
- Willard Van Orman Quine
- Second-order logic
- Type theory
- Well-formed formula
- Tautology (logic)
- Russell's paradox
- Naive set theory
- Axiom of extensionality
- Axiom schema of replacement
- Abraham Fraenkel
- Complement (set theory)
- Classical logic
Place in the tradition
Part of Analytic philosophy.
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