finitism

Temporal finitism is the idea that time is finite.

John Philoponus was probably the first to use the argument that infinite time is impossible, establishing temporal finitism.

Becker argued that Hilbert could not stick with finitism, but had to assume the potential infinite.

In medieval philosophy, there was much debate over whether the universe had a finite or infinite past (see Temporal finitism).

The most famous proponent of finitism was Leopold Kronecker, who said:

Reuben Goodstein is another proponent of finitism.

Although he denied it, much of Ludwig Wittgenstein's writing on mathematics has a strong affinity with finitism.

Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity.

Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics.

Philoponus' arguments for temporal finitism were severalfold.

The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic.

He published many works on finitism and the reconstruction of analysis from a finitistic viewpoint, for example "Constructive Formalism.

Ultrafinitism is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources.

In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism.

Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter.

Immanuel Kant's argument for temporal finitism, at least in one direction, from his First Antinomy, runs as follows:

In his book The Infinite, Moore offers a thorough discussion of the idea of infinity and its history, and a defence of finitism.

This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and intuitionism.

The primitive recursive functions are closely related to mathematical finitism, and are used in several contexts in mathematical logic where a particularly constructive system is desired.

The theory of temporal finitism was inspired by the doctrine of creation shared by the three Abrahamic religions: Judaism, Christianity and Islam.

The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics (for example, that includes real analysis).

Ultrafinitism (also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism and objects to the existence of finite mathematical objects if they are too large.

The essential distinction, similar to the concept of definable numbers, contrasts the finiteness of the domain of concepts that we can specify and discuss with the unbounded infinity of the set of numbers; compare finitism.

These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis.

Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarily finite sets constitutes a model of Zermelo-Fraenkel set theory with the axiom of infinity replaced by its negation.