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Axiom is a literate program. [9] The source code is becoming available in a set of volumes which are available on the axiom-developer.org website. These volumes contain the actual source code of the system.
Pastebin.com is a text storage site. It was created on September 3, 2002 by Paul Dixon, and reached 1 million active pastes (excluding spam and expired pastes) eight years later, in 2010. [3] It features syntax highlighting for a variety of programming and markup languages, as well as view counters for pastes and user profiles.
The most famous pastebin is the eponymous pastebin.com. [citation needed] Other sites with the same functionality have appeared, and several open source pastebin scripts are available. Pastebins may allow commenting where readers can post feedback directly on the page. GitHub Gists are a type of pastebin with version control. [citation needed]
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...
The case n = 2 is the axiom of pairing with A = A 1 and B = A 2. The cases n > 2 can be proved using the axiom of pairing and the axiom of union multiple times. For example, to prove the case n = 3, use the axiom of pairing three times, to produce the pair {A 1,A 2}, the singleton {A 3}, and then the pair {{A 1,A 2},{A 3}}.
Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive. [11] One example is the Banach–Tarski paradox , which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into ...
Antecedent of Playfair's axiom: a line and a point not on the line Consequent of Playfair's axiom: a second line, parallel to the first, passing through the point. In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):
If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.