Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective Of a function, taking a
finite number ofarguments to produce an output.
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Fully spelled out it means that for an entity to be a proposition there must exist a dialogue game associated with this entity, i.e., the proposition A, such that an individual play of the game where A occupies the initial position, i.e., a dialogue D (A) about A, reaches a final position with either win or loss after a finite number of moves according to definite rules: the dialogue game is defined as a finitary open twoperson zerosum game.

This result is still not fully "finitary" because it deals with a sequence of finite structures, rather than with a single finite structure.

By GÃ¶del's and Gentzen's result, already intuitionistic arithmetic contained principles that went beyond finitary reasoning.

The algebraic strength of GEM, and of its weaker finitary and infinitary variants, is worth emphasizing, but it also reflects substantive mereological postulates whose philosophical underpinnings leave room for controversy.

For Hilbert, the aims were a complete clarification of the foundational problems through finitary proofs of consistency, etc, aims in which proof theory failed.

It can be checked that all these generalized formulations include the corresponding finitary principles as special cases, taking ˜Ïw™ to be the formula ˜w = x

GÃ¶del had, it seems, not thought of giving a consistency proof of arithmetic through the use of nonfinitary but still constructive principles.

So when intuitionists deny that the Law of Excluded Middle holds in nonfinitary contexts, they are actually taking truth as provability; and when paraconsistentists claim that some formula can be true (in some weird circumstances) together with its negation, they are not talking of negation anymore (see e.g. Berto 2006).

Then, of course, the unexpected happened when GÃ¶del proved the impossibility of a complete formalization of elementary arithmetic, and, as it was soon interpreted, the impossibility of proving the consistency of arithmetic by finitary means, the only ones judged “absolutely reliable” by Hilbert.

An initial and rough answer to this last question is contained in our discussion so far: Dedekind's approach is settheoretic and infinitary, while Kronecker's is constructivist and finitary.
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