A Wisdom Archive on axiom of choice - quotes

In mathematics, the axiom of choice, or AC, is an axiom of set theory. It was formulated in 1904 by Ernst Zermelo. While it was originally controversial, it is now used without embarrassment by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, that either reject the axiom of choice, or even investigate consequences of axioms inconsistent with AC. Intuitively speaking, AC says that given a collection of bins, each containing at least one object, then exactly one ob ... Including:

• Axiom of choice - Statement
• Axiom of choice - Usage
• Axiom of choice - Independence of AC
• Axiom of choice - Weaker forms of AC
• Axiom of choice - Results requiring AC or weaker forms
• Axiom of choice - Results requiring ¬AC
• Axiom of choice - Results requiring choice in intuitionistic logic though not classically
• Axiom of choice - Quotes

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By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (ZF). This means that neither it nor its negation can be proven to be true in ZF. Consequently, assuming the axiom of choice, or its negation, will never lead to a contradiction that could not be obtained without that assumption. So the decision whether or not it is appropriate to make use of the axiom of choice in a proof cannot be made by appeal to other axioms of s ...

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Independence of AC

Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. Theory - Etymology. The word ‘theory’ derives from the Greek ‘theorein’, which means ‘to look at’. According to some sources, it was used frequently in terms of ‘looking at’ a theatre stage, which may explain why sometimes the word ‘theory’ is used as something provisional or not completely resembling real. The term ‘theoria’ (a noun) was already used by ... Including:

• Theory - Etymology
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Politics series Factions Minarchism Anarcho-capitalism Paleolibertarianism Neolibertarianism Left-libertarianism Influences Austrian School Classical liberalism Individualist anarchism Objectivism Mixed economy Ideas Liberty Free markets Capitalism Non-aggression Key issues Parties Economic views Vi ... Including:

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Schools Anarcho-capitalism Anarcho-communism Anarcho-primitivism Anarcho-syndicalism Christian anarchism Eco-anarchism Individualist anarchism Mutualism Anarchism in culture Anarchism and religion Anarchism and society Anarchism and the arts Anarcho-punk Anarchist theory Anarchism and capitalism Anarchism and Marxism Anarchist economics Anarchist law Anarchist symbolism Anarchism without adjectives P ... Including:

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Albert Einstein (March 14, 1879–April 18, 1955) was a Jewish theoretical physicist, born in Ulm, Germany, who is widely regarded as the greatest scientist of the 20th century. He proposed the theory of relativity and also made major contributions to the development of quantum mechanics, statistical mechanics, and cosmology. He was awarded the 1921 Nobel Prize for Physics for his explanation of the photoelectric effect in 1905 (his "miracle ... Including:

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Suppose P is any predicate in two variables that doesn't use the symbol B. Then in the formal language of the Zermelo-Fraenkel axioms, the axiom schema reads: or in words: If, given any set X, there is a unique set Y such that P holds for X and Y, then, given any set A, there is a set B such that, given any set C, C is a member of B if and only if there is a set D such that D is a member of A ...

Read more here: » Axiom schema of replacement: Encyclopedia II - Axiom schema of replacement - Statement

Despite the elegance and appeal of some aspects of non-standard analysis, there is a great deal of skepticism in the mathematical community about whether this machinery really adds anything that cannot just as easily be achieved by standard methods. One noted critic of non-standard analysis is the Fields Medalist Alain Connes, as evinced by the following quote The answer given by nonstandard analysis, a so-called nonstandard real, is equally deceiving. From every nonstandard real number one can construct canonically a sub ...

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In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it often does in other contexts. Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, or conjectures. A theory is in this context a set of hypotheses that are logically bound together (See also hypothetico-deductive method). Theories are typically ways of explaining why things happen, often, but not always after their occurrence is no longer in scientific di ...

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Albert Einstein was much respected for his kind and friendly demeanor rooted in his pacifism. He was modest about his abilities, and had distinctive attitudes and fashions—for example, he minimized his wardrobe so that he would not need to waste time in deciding on what to wear. He occasionally had a playful sense of humor, and enjoyed sailing and playing the violin. He was also the stereotypical "absent-minded professor"; he was often forgetful of everyday items, such as keys, and would focus so intently on solving physics problems that h ...

Read more here: » Albert Einstein: Encyclopedia II - Albert Einstein - Personality