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In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers. ==Main ideas== Like other strict finitists, ultrafinitists deny the existence of the infinite set N of natural numbers, on the grounds that it can never be completed. In addition, some ultrafinitists are concerned with acceptance of objects in mathematics that no one can construct in practice because of physical restrictions in constructing large finite mathematical objects. Thus some ultrafinitists will deny or refrain from accepting the existence of large numbers, for example, the floor of the first Skewes' number, which is a huge number defined using the exponential function as exp(exp(exp(79))), or : The reason is that nobody has yet calculated what natural number is the floor of this real number, and it may not even be physically possible to do so. Similarly, (in Knuth's up-arrow notation) would be considered only a formal expression which does not correspond to a natural number. The brand of ultrafinitism concerned with physical realizability of mathematics is often called ''actualism''. Edward Nelson criticized the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as 0 and numbers obtained by the iterative applications of the successor function to 0. But the concept of natural number is already assumed for the iteration. In other words, to obtain a number like one needs to perform the successor function iteratively, in fact exactly times to 0. Some versions of ultrafinitism are forms of constructivism, but most constructivists view the philosophy as unworkably extreme. The logical foundation of ultrafinitism is unclear; in his comprehensive survey ''Constructivism in Mathematics'' (1988), the constructive logician A. S. Troelstra dismissed it by saying "no satisfactory development exists at present." This was not so much a philosophical objection as it was an admission that, in a rigorous work of mathematical logic, there was simply nothing precise enough to include. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ultrafinitism」の詳細全文を読む スポンサード リンク
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