an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage of an axiomatic...
4 KB (470 words) - 13:47, 21 November 2024
axiom schema of predicative separation, or of restricted, or Δ0 separation, is a schema of axioms which is a restriction of the usual axiom schema of...
3 KB (345 words) - 03:00, 6 July 2024
Constructive set theory (category Systems of set theory)
Intersection (which is related to the Axiom schema of predicative separation) and the Set Induction schema. Taken as axioms, the aforementioned principles constitute...
213 KB (35,228 words) - 09:33, 13 June 2025
Set theory (redirect from Axiom of set theory)
the axiom schema of replacement with that of separation; General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and...
54 KB (6,575 words) - 19:15, 10 June 2025
In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no set of all sets. The axiom of regularity together...
26 KB (3,222 words) - 15:49, 19 June 2025
φz^. The axiom of reducibility states that any truth function (i.e. propositional function) can be expressed by a formally equivalent predicative truth function...
39 KB (6,052 words) - 19:23, 13 February 2025
it gives the axiom of separation with P {\displaystyle P} . Mancini, Antonella; Montagna, Franco (Spring 1994). "A minimal predicative set theory". Notre...
4 KB (480 words) - 21:56, 17 March 2025
Kripke–Platek set theory (redirect from Kripke–Platek axioms of set theory)
containing precisely those elements x for which φ(x) holds. (This is an axiom schema.) Axiom of Δ0-collection: Given any Δ0 formula φ(x, y), if for every set x...
10 KB (1,586 words) - 11:54, 3 May 2025
Diaconescu's theorem (category Axiom of choice)
particular, in the axiom schema of predicative separation only sentences with set bound quantifiers may be used. The restricted form of excluded middle provable...
11 KB (1,926 words) - 21:56, 17 March 2025
Principia Mathematica (redirect from Whitehead-Russell axioms)
difference between predicative and non-predicative functions, so they introduced the axiom of reducibility, saying that for every non-predicative function there...
70 KB (9,476 words) - 10:31, 19 June 2025
set Axiom schema of predicative separation Axiom of separation for formulas whose quantifiers are bounded Axiom schema of replacement The image of a set...
91 KB (11,628 words) - 12:22, 21 March 2025
Heyting arithmetic (category Formal theories of arithmetic)
theories suffice: They shall adopt the Axiom of infinity, the Axiom schema of predicative separation to prove induction of arithmetical formulas in ω {\displaystyle...
37 KB (6,285 words) - 21:10, 9 March 2025
Morse–Kelley set theory (category Systems of set theory)
variables in NBG's axiom schema of Class Comprehension are restricted to sets; hence Class Comprehension in NBG must be predicative. (Separation with respect...
21 KB (3,186 words) - 12:28, 4 February 2025
Reverse mathematics (redirect from Arithmetical comprehension axiom)
program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described...
38 KB (4,782 words) - 10:20, 2 June 2025
Cantor's diagonal argument (category Theorems in the foundations of mathematics)
non-existence of a set of all sets also already follows from Predicative Separation. In a set theory, theories of mathematics are modeled. Weaker logical axioms mean...
27 KB (2,751 words) - 02:16, 12 April 2025
axiom schema to imply nonprojectibility, in fact there are transitive models of K P {\displaystyle KP} + Σ 1 {\displaystyle \Sigma _{1}} -separation of...
40 KB (5,519 words) - 11:55, 15 May 2025