Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

Area can be defined as a function from a collection M of special kind of plane figures ( termed measurable sets ) to the set of real numbers which satisfies the following properties:

By symmetry, any algebra which satisfies any two of:

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

A B *- algebra is a Banach *- algebra in which the involution satisfies the following further property:

which satisfies the Cauchy – Riemann equations everywhere, but fails to be continuous at z = 0.

In number theory, a Carmichael number is a composite positive integer which satisfies the congruence

The veteran gets a voucher good only for educational expense and he is completely free to choose the school at which he uses it, provided that it satisfies certain standards.

The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties.

This formula is indeed equivalent to φ because it states that for every x, if there is a y which satisfies Q '( x, y ), then ( something ) holds, and furthermore, we know that there is such a y, because for every x ', there is a y ' which satisfies Q '( x ', y ').

Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.

In other words, is the infimum of the set of numbers such that there is some ( indexed ) collection of balls covering with for each which satisfies < math >

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

A function which satisfies these properties is called a linear function ( or linear operator, or more generally a linear map ).

A Lie algebra is a vector space over some field F together with a binary operation called the Lie bracket, which satisfies the following axioms:

This leads to the web design principle of the " Rule of Least Power ", which advocates using the least ( computationally ) powerful language that satisfies a task to facilitate such manipulation and reuse.

After introducing the moral law, Lewis argues that thirst reflects the fact that people naturally need water, and there is no other substance which satisfies that need.

A field which satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.

The first reason would never apply to a reorganization transfer which meets the conditions of section 381(a), which is the only type presently under discussion.

But for some fool reason he couldn't remember which men he'd put on the transfer detail.

A third concept was proposed in 1923 by Gilbert N. Lewis which includes reactions with acid-base characteristics that do not involve a proton transfer.

The higher value for alpha radiation is generally attributable to the high linear energy transfer ( LET ) coefficient, which is about one ionization of a chemical bond for every angstrom of travel by the alpha particle.

The justification for attributing life to objects was stated by David Hume in his Natural History of Religion ( Section III ): " There is a universal tendency among mankind to conceive all beings like themselves, and to transfer to every object those qualities with which they are familiarly acquainted, and of which they are intimately conscious.

The 68000 has a 16-bit external data bus so must transfer 32 bits of data in two consecutive steps, a technique called multiplexing: all this is transparent to the software, which was 32-bit from the beginning.

The U. S. Department of Defense investigation contract, which is ongoing, will also evaluate the effectiveness of copper alloy touch surfaces to prevent the transfer of microbes to patients and the transfer of microbes from patients to touch surfaces, as well as the potential efficacy of copper-alloy based components to improve indoor air quality.

* more recently, compare entire genomes, which permits the study of more complex evolutionary events, such as gene duplication, horizontal gene transfer, and the prediction of factors important in bacterial speciation,

Pairing is the mechanism by which codons on messenger RNA molecules are recognized by anticodons on transfer RNA during protein translation.

The CDC categorizes these agents ( A, B or C ) and administers the Select Agent Program, which regulates the laboratories which may possess, use, or transfer select agents within the United States.

Options include a minimally invasive procedure known as platelet injection fat transfer, which transfers fat cells from a woman's thighs to her smaller breast.

After mastectomy ( the removal of a diseased breast ), the reconstruction of the breast or breasts is done with breast implants or autologous tissue transfer, using fat and tissues from the abdomen, which is performed with a TRAM flap or with a back ( latissiumus muscle flap ).

Brooks's institute has charged varying fees to prospective clients who wished to learn how to live without food, which have ranged from US $ 100, 000 with an initial deposit of $ 10, 000 to one billion dollars, to be paid via bank wire transfer with a preliminary deposit of $ 100, 000, for a session called " Immortality workshop ".

Well-known communications protocols are Ethernet, a hardware and Link Layer standard that is ubiquitous in local area networks, and the Internet Protocol Suite, which defines a set of protocols for internetworking, i. e. for data communication between multiple networks, as well as host-to-host data transfer, and application-specific data transmission formats.

Upon the transfer of the colony to British control, the British government issued the Royal Proclamation of 1763, which imposed English common law on the colony, under the principle set out in Blackstone relating to captured colonies.

A dye-sublimation printer ( or dye-sub printer ) is a printer which employs a printing process that uses heat to transfer dye to a medium such as a plastic card, paper or canvas.

The other part of a nation's money supply consists of bank deposits ( sometimes called deposit money ), ownership of which can be transferred by means of cheques, debit cards, or other forms of money transfer.

Harvard Square was originally the northwestern terminus of the Red Line and a major transfer point to streetcars that also operated in a short tunnel — which is still a major bus terminal, although the area under the Square was reconfigured dramatically in the 1980s when the Red Line was extended.

The main purpose of the bridge on a classical guitar is to transfer the vibration from the strings to the soundboard, which vibrates the air inside of the guitar, thereby amplifying the sound produced by the strings.

FADH < sub > 2 </ sub >, therefore, facilitates transfer of electrons to coenzyme Q, which is the final electron acceptor of the reaction catalyzed by the Succinate: ubiquinone oxidoreductase complex, also acting as an intermediate in the electron transport chain.

Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.