Editing Finitism (section)

== History ==
The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase when [[Georg Cantor]] in 1874 introduced what is now called [[naive set theory]] and used it as a base for his work on [[transfinite number]]s. When paradoxes such as [[Russell's paradox]], [[Berry's paradox]] and the [[Burali-Forti paradox]] were discovered in Cantor's naive set theory, the issue became a heated topic among mathematicians.

There were various positions taken by mathematicians. All agreed about finite mathematical objects such as natural numbers. However there were disagreements regarding infinite mathematical objects. 
One position was the [[intuitionistic mathematics]] that was advocated by [[L. E. J. Brouwer]], which rejected the existence of infinite objects until they are constructed.

Another position was endorsed by [[David Hilbert]]: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to [[Hilbert's program]] of proving both [[consistency]] and [[completeness (logic)|completeness]] of set theory using finitistic means as this would imply that adding ideal mathematical objects is [[conservative extension|conservative]] over the finitistic part. Hilbert's views are also associated with the [[Formalism (mathematics)|formalist philosophy of mathematics]]. Hilbert's goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due to [[Kurt Gödel]]'s [[incompleteness theorems]]. However, [[Harvey Friedman (mathematician)|Harvey Friedman]]'s [[Friedman's grand conjecture|grand conjecture]] would imply that most mathematical results are provable using finitistic means.

Hilbert did not give a rigorous explanation of what he considered finitistic and referred to as elementary. However, based on his work with [[Paul Bernays]] some experts such as {{harvtxt|Tait|1981}} have argued that [[primitive recursive arithmetic]] can be considered an upper bound on what Hilbert considered finitistic mathematics.{{sfn|Schirn|Niebergall|2005}}

As a result of Gödel's theorems, as it became clear that there is no hope of proving both the consistency and completeness of mathematics, and with the development of seemingly consistent [[axiomatic set theory|axiomatic set theories]] such as [[Zermelo–Fraenkel set theory]], most modern mathematicians do not focus on this topic.