Editing Ultrafinitism (section)

==Main ideas==
Like other [[finitism|finitists]], ultrafinitists deny the existence of the [[infinite set]] <math>\N</math> of [[natural numbers]], on the basis that it can never be completed (i.e., there is a largest natural number).

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 function|floor]] of the first [[Skewes's number]], which is a huge number defined using the [[exponential function]] as exp(exp(exp(79))), or
: <math> e^{e^{e^{79}}}. </math>
The reason is that nobody has yet calculated what [[natural number]] is the [[Floor and ceiling functions|floor]] of this [[real number]], and it may not even be physically possible to do so. Similarly, <math>2\uparrow\uparrow\uparrow 6</math> (in [[Knuth's up-arrow notation]]) would be considered only a formal expression that 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 <math>2\uparrow\uparrow\uparrow 6</math> one needs to perform the successor function iteratively (in fact, exactly <math>2\uparrow\uparrow\uparrow 6</math> times) to 0.

Some versions of ultrafinitism are forms of [[constructivism (mathematics)|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.