For finite sets X, the axiom of choice follows from the other axioms of set theory.

While Aquinas's theory follows generally the model of Aristotle, he develops a singular aesthetics which incorporates elements unique to his thought.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of Boltzmann's constant as, and the fourth equality follows from Stokes ' formula for the mobility.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

This theory states that an animal's embryonic development follows exactly the same sequence as the sequence of its evolutionary ancestors.

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

In it, he outlined his theory as follows ;

The key points where the theory of Karma in Jainism differs from the other religions such as theistic traditions of Hinduism, can be stated as follows:

" After tracing the history of the various opinions that have been and are held on the eternal fate of unbaptized infants, including that connected with the theory of the Limbo of Infants, and after examining the theological arguments, the document stated its conclusion as follows:

In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin.

One modern academic theory of religion, social constructionism, says that religion is a modern concept that suggests all spiritual practice and worship follows a model similar to the Abrahamic religions as an orientation system that helps to interpret reality and define human beings, and thus religion, as a concept, has been applied inappropriately to non-Western cultures that are not based upon such systems, or in which these systems are a substantially simpler construct.

Another way of stating Rice's theorem that is more useful in computability theory follows.

As a theory of knowledge, reliabilism can be roughly stated as follows:

As a theory of justified belief, reliabilism can be formulated roughly as follows:

Darwin's theory of evolution is based on key facts and the inferences drawn from them, which biologist Ernst Mayr summarised as follows:

In the sociology of Émile Durkheim, sui generis is used to illustrate his theory of an independence in social existence, as follows.

Paul Davidson follows Keynes closely in placing time and uncertainty at the centre of theory, from which flow the nature of money and of a monetary economy.

The theory is that a stimulus follows its individual dedicated neural path in the early stages of visual processing, and that intense or repetitive activity in that or interaction with active adjoining channels cause a physiological imbalance that alters perception.

It follows immediately from the fact that satisfies the guiding equation that also the configuration satisfies a guiding equation identical to the one presented in the formulation of the theory, with the universal wave function replaced with the conditional wave function.

The de Broglie – Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles ; the orientation of the apparatus affects the wavefunction.

Many theoretical perspectives attempt to explain development ; among the most prominent are: Jean Piaget's Stage Theory, Lev Vygotsky's Social constructivism ( and its heirs, the Cultural Theory of Development of Michael Cole, and the Ecological Systems Theory of Urie Bronfenbrenner ), Albert Bandura's Social learning theory, and the information processing framework employed by cognitive psychology.

For example, educational psychologists have conducted research on the instructional applicability of Jean Piaget's theory of development, according to which children mature through four stages of cognitive capability.

Piaget's views of moral development were elaborated by Kohlberg into a stage theory of moral development.

This is the reason why Piaget's theory of cognitive development was so influential for education, especially mathematics and science education.

Educational psychologists distinguish individual ( or psychological ) constructivism, identified with Piaget's theory of cognitive development, from social constructivism.

Ellis argues that Webster anticipated some of the insights currently associated with Jean Piaget's theory of cognitive development.

Piaget's theory of cognitive development was the first complete theory of reasoning development.

Piaget's theory of Developmental Psychology tackled cognitive development from infancy to adulthood.

A close follower of Jean Piaget's theory of cognitive development, Kohlberg's work reflected and extended his predecessor's ideas, at the same time creating a new field within psychology: " moral development ".

Piaget's theory of cognitive development -

Traill ( 1978 / 2006 ) suggests that this Ashby-hierarchy probably coincides with Piaget's well-known theory of developmental stages.

The second phase in Piaget's theory of moral development is referred to as the Autonomous Phase.

# REDIRECT Piaget's theory of cognitive development # Sensorimotor stage

In Piaget's theory of cognitive development infants develop this understanding by the end of the " sensorimotor stage ," which lasts from birth to about 2 years of age.

One criticism of Piaget's theory is that culture and education exert stronger influences on a child's development than Piaget maintained.

* Piaget's theory of cognitive development, a theory about the nature and development of human intelligence

Optimal human development generally occurs in identifiable stages ( e. g., Jean Piaget's theory of cognitive development, Erik Erikson's stages of psychosocial development, John Bowlby's attachment framework, and other developmental stage theories ).

Some researchers have linked the theory of fluid and crystallized intelligence to Piaget's conception of operative intelligence and learning.

# REDIRECT Piaget's theory of cognitive development # Preoperational Period

* The ' sensorimotor stage ' in Piaget's theory of cognitive development

# REDIRECT Piaget's theory of cognitive development

# REDIRECT Piaget's theory of cognitive development # Concrete operational stage