ZF

See also: zf and zF

Translingual

Symbol

ZF

1. (metrology) Symbol for zettafarad, an SI unit of electrical capacitance equal to 10²¹ farads.

English

Noun

ZF (uncountable)

1. (set theory) Initialism of Zermelo-Fraenkel (set theory): a particular axiomatic formulation of set theory without the axiom of choice.
   • 1967, Carol Karp, “A Proof of the Relative Consistency of the Continuum Hypothesis”, in John N. Crossley, editor, Sets, Models, and Recursion Theory, North-Holland, page 7:

     Gödel's model is an example of a simple type of inner model that might be called a definable transitive inner model³), where the universe is replaced by a transitive subuniverse defined within ZF, the membership relation is just the original one restricted to the subuniverse, and where the axioms of ZF relativized to the subuniverse, are provable within ZF.

   • 1971, Ulrich Felgner, Models of ZF-Set Theory, Springer, Lecture Notes in Mathematics 223, page 21,

     1. Corollary: ZF is not finitely axiomatizable.

     2. Corollary: ZF is reflexive (i.e. the consistency of every finite subtheory of ZF can be proved within ZF).

   • 1991 [Kluwer Academic], Fred Landman, Structures for Semantics, 1991, Springer, Softcover, page 56,

     However, the problem with it,^{[the generalized continuum hypothesis]} and the reason why it is not part of ZF strictly (apart from the fact that it implies the axiom of choice) is that it is rather arbitrary.

Adjective

ZF (not comparable)

1. (telecommunications, signal processing) Initialism of zero forcing.
   • 2008, Byeong Gi Lee, Sunghyun Choi, Broadband Wireless Access and Local Networks:

     In the case of the ZF detector, we suppress the interference among the transmit streams by multiplying the received signal vector y with the Moore-Penrose pseudo-inverse of the channel matrix […]

   • 2010, Dayan Adionel Guimaraes, Digital Transmission, page 348:

     However, due to its simplicity, ZF equalization is a good choice when the channel is known and fixed, and also when there is no frequency notch in the channel frequency response […]

   • 2013, Academic Press Library in Signal Processing, page 308:

     Thus, the importance of the upper bound is in showing that even in the most optimistic case, the achievable rate cannot exceed the ZF rates by more than 10%.

See also

• (set theory): NBG, ZFC

Further reading

• Zermelo–Fraenkel set theory on Wikipedia.Wikipedia
• Zero-forcing precoding on Wikipedia.Wikipedia
• Zero forcing equalizer on Wikipedia.Wikipedia

Chinese

Noun

ZF

1. (Mainland China, Internet slang) Initialism of 政府 (zhèngfǔ).