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Encyclopedia > Finitism

In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. (Most constructivists, in contrast, allow a countably infinite number of steps.) The most famous proponent of finitism was Leopold Kronecker, who said:

"God created the natural numbers, all else is the work of man."

Although most modern constructivists take a weaker view, they can trace the origins of constructivism back to Kronecker's finitist work.

Even stronger than finitism is ultrafinitism (also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin.

Results from FactBites:

 Finite - Wikipedia, the free encyclopedia (175 words) In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2,..., n} where n is a natural number. All finite sets are countable, but not all countable sets are finite. It is a theorem (assuming the axiom of choice) that a set is finite if and only if there exists no bijection between the set and any of its proper subsets.
 Finite state machine - Wikipedia, the free encyclopedia (1085 words) A finite state machine (FSM) or finite automaton is a model of behaviour composed of states, transitions and actions. Finite state machines are one type of the automata studied in automata theory and the theory of computation. In computer science, finite state machines are widely used in modelling of application behaviour, design of hardware digital systems, software engineering, compilers, study of computation and languages.
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