Nonetheless, it turns out that infinite sets do have a well-defined notion of size ( or more properly, of cardinality, which is the technical term for the number of elements in a set ), and not all infinite sets have the same cardinality.

While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both " an abomination " and " a cholera bacillus of mathematics ".

The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

One is then immediately led to the more subtle notion of κ-categoricity, which asks: for which cardinals κ is there exactly one model of cardinality κ of the given theory T up to isomorphism?

There is also a notion of stationary subset of, for a cardinal and a set such that, where is the set of subsets of of cardinality:.

His notion of abelian category is now the basic object of study in homological algebra.

In view of implacable demands by the political leadership of both Hindu and Islamic communities in British India for a separate Hindu and a separate Muslim homeland, Mountbatten conceded the notion of two nations consisting of a Hindu-majority India and a Muslim-majority Pakistan ( which incorporated East Pakistan, now Bangladesh ).

In it he stated another notion of " effective computability " with the introduction of his a-machines ( now known as the Turing machine abstract computational model ).

Parallel with this generational change, the articles and books on the doctrine and practice of fellowship with the main " Central " grouping now reject the notion itself of separate " fellowships " among those who recognise the same baptism as " schism ".

The notion of temporary insanity argues that a defendant was insane, but is now sane.

The actual titles are in question ( as is the notion that these were album-length recordings ), but the first title has been confirmed by a former head of the now defunct Smelly Tongues fan club.

But now another problem about the notion that God is a mind arises.

Adams, once a political visionary and now a man blind to the needs of his country, not only held fast to his notion of defending the monarchy but also made additional attempts to form similarly flawed Federation-like entities after that union's demise.

In 1966 Per Martin-Löf introduced a new notion which is now generally considered the most satisfactory notion of algorithmic randomness.

The fact that many immigrants settle in impoverished city centres led to the notion of the " peripheralisation of the core ", which simply describes that people who used to be at the periphery of the former empires now live right in the centre.

Though transformations continue to be important in Chomsky's current theories, he has now abandoned the original notion of Deep Structure and Surface Structure.

Scholars now reject the former notion, but note that there were certain traditions and practices used in both the Irish and British churches but not in the wider Christian world.

Nagel and Newman can now define the notion of tautologous: " a formula is a tautology if, and only if, it falls in the class K < sub > 1 </ sub > no matter in which of the two classes its elements are placed ".

To Be or Not To Be, now regarded as one of the best films of Lubitsch's, Benny's and Lombard's careers, was initially not well received by the public, many of whom could not understand the notion of making fun out of such a real threat as the Nazis.

Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path ( depending on the type of differential structure applied ).

This essay, first published in the Collected Works of 1894-1911, contains the first known appearance of what are now called linear algebra and the notion of a vector space.

The notion of " graded ring " now becomes the same thing as a N-graded ring, where N is the monoid of non-negative integers under addition.

The notion of a personal life, as now understood, is in part a creation of modern Western society.

However, this usage is now uncommon, as the notion of " just plain ASCII " ( though very attractive at first glance ), has turned out to have difficulties.

The notion of local hidden variables is now introduced by considering the following question:

It can now run atop TCP with a small adaptation layer that adds a packet length to each SMB message ; this is necessary because TCP only provides a byte-stream service with no notion of packet boundaries.

The U. S. government has tried to sell the notion that the war against Iraq is critical to the American " War on Terrorism ": " In the war on terror, Iraq is now the central front ..." President Bush said on December 14, 2005 ( Bush, George W. " President Discusses Iraqi Elections, Victory in the War on Terror.

In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory.

The Chinese had a somewhat different series of elements, namely Fire, Earth, Metal, Water and Wood, which were understood as different types of energy in a state of constant interaction and flux with one another, rather than the Western notion of different kinds of material.

The term Sileni-especially when juxtaposed with the character of Alcibiades-can therefore be understood as an evocation of the notion that something on the inside is more expressive of a person's character than what one sees on the outside.

It is perhaps best understood in relation to the notion of the ' habitus ' and ' field ', which describes the relationship between individual agents and the contextual environment.

This is understood by both mainstream Muslims and historians to have been stated by Muhammad to emphasize upon the notion of equality between all the prophets and the law of making no distinction between any of the messengers.

Central to the Pure Theory of Law is the notion of a ' basic norm ( Grundnorm )'— a hypothetical norm, presupposed by the jurist, from which in a hierarchy all ' lower ' norms in a legal system, beginning with constitutional law, are understood to derive their authority or ' bindingness '.

This argument resonates with a notion of the fine-tuned Universe, understood as an alternative to the anthropic principle.

The basic notion of being all-powerful can be understood well enough, it seems, at first glance: something is omnipotent, or all-powerful, if the being can do anything we can think of.

The notion that the tail of the vector coincides with the origin is implicit and easily understood.

Moses Maimonides wrote that " by virtue of the existence of the Creator everything exists " and argues in his 12th-century Guide for the Perplexed ( 2: 13 ) that " time itself is part of creation " and that therefore, " when God is described as existing before the creation of the universe, the notion of time should not be understood in its normal sense.

For the same reason the notion of " Aztec civilization " is best understood as a particular horizon of a general Mesoamerican civilization.

However, Benjamin Penny argues that Li's teachings are better understood in the context of a " Buddhist notion of the cycle of the Dharma or the Buddhist law ".

As harmonics are universal to all physical sounds, the notion of authenticity is best understood in terms of musical quality.

" Gore's attorneys therefore understood that they could fight on, and could petition the Florida Supreme Court to repudiate the notion that December 12 was final under Florida law.

" It was more or less a compendium of theological concepts, as articulated by Armstrong, which included the notion that God deliberately coded the bible " so that it would not be understood until our modern time " -.

The notion of selling a product into a specific thermal envelope was typically not understood by the mass of reviewers, who tended to compare Efficeon to the gamut of x86 microprocessors, regardless of power consumption or application.

While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita who first codified the notion of a tensor.

Formalist theories differ according to how the notion of ' form ' is understood.

The very notion of a " thing in itself " should be understood as an option of a set of functions for an operating mind, such that we consider something that appears without respect to the specific manner in which it appears.

The methodology employed by this Yeshiva places emphasis on the notion that the initial assumptions of the Talmud must be highly rigorous, and that the movement between the initial thought process ( known as the hava aminah ) of the Talmudic sugya to the final thought process must be fully unfolded and understood.

This notion is often understood as a simple historical narrative: primitive communism had developed into slave states.

He understood that the interconnectedness of varying forces in the universe is so vast and innumerable that the notion of causality, as presently understood, is wasted.