Anagram & Information om | Engelska ordet ZFC

Exempel på hur man kan använda ZFC i en mening

• While the title states that the set theory presented is 'naive', which is usually taken to mean without formal axioms, the book does introduce a system of axioms equivalent to that of ZFC set theory except the Axiom of foundation.
• In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.
• In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic.
• This is a more abstracted framework which can be grounded in ZFC (that is, both axioms can be proved within ZFC as well as most other formal foundations), but is closer to how most mathematicians use equality.
• The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.
• That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory.
• Note that if the axiom of the empty set is not taken as a part of this system (since it can be derived from ZF + Infinity), then the empty domain also satisfies ZFC − Infinity + ¬Infinity, as all of its axioms are universally quantified, and thus trivially satisfied if no set exists.
• Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom.
• The axioms I0, I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen's inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent.
• In the modern ZFC system, the "propositional function" referred to in the axiom of separation is interpreted as "any property definable by a first-order formula with parameters", so the separation axiom is replaced by an axiom schema.
• With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the continuum hypothesis), there are still highly nontrivial ZFC theorems about cardinal exponentiation.
• Some cardinalities cannot be proven to be equal to any particular aleph, for instance the cardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see Easton's theorem).
• Hatcher (1982) derives Peano's axioms from several foundational systems, including ZFC and category theory, and from the system of Frege's Grundgesetze der Arithmetik using modern notation and natural deduction.
• On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language.
• This was caused by several factors, including the extra work required to handle forcing in NBG, Cohen's 1966 presentation of forcing, which used ZF, and the proof that NBG is a conservative extension of ZFC.
• An alternative approach, developed by Edward Nelson, finds infinitesimals on the ordinary real line itself, and involves a modification of the foundational setting by extending ZFC through the introduction of a new unary predicate "standard".
• A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC.
• Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as an axiom for set theory in the same way as the ZFC axioms.
• Czesław Ryll-Nardzewski proved that Peano arithmetic cannot be finitely axiomatized, and Richard Montague proved that ZFC cannot be finitely axiomatized.
• Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "standard" satisfies three additional axioms I, S, and T.

Förberedelsen av sidan tog: 388,32 ms.