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 two-person zero-sum game. ❋ Keiff, Laurent (2009)
This result is still not fully "finitary" because it deals with a sequence of finite structures, rather than with a single finite structure. ❋ Unknown (2009)
By Gödel's and Gentzen's result, already intuitionistic arithmetic contained principles that went beyond finitary reasoning. ❋ Unknown (2009)
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 ❋ Unknown (2009)
So when intuitionists deny that the Law of Excluded Middle holds in non-finitary 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). ❋ Berto, Francesco (2009)
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. ❋ Unknown (2009)
Gödel had, it seems, not thought of giving a consistency proof of arithmetic through the use of non-finitary but still constructive principles. ❋ Unknown (2009)
For Hilbert, the aims were a complete clarification of the foundational problems through finitary proofs of consistency, etc, aims in which proof theory failed. ❋ Unknown (2009)
Then, of course, the unexpected happened when Gödel 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. ❋ Unknown (2009)
An initial and rough answer to this last question is contained in our discussion so far: Dedekind's approach is set-theoretic and infinitary, while Kronecker's is constructivist and finitary. ❋ Reck, Erich (2008)
But a consequence of Meyer's construction was that within his arithmetic R# it was demonstrable by simple finitary means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations. ❋ Mortensen, Chris (2008)
In other words, the set of propositions is a (finitary) field or algebra A over a non-empty set W of possibilities: a set that contains W and is closed under complementation and finite intersection. ❋ Huber, Franz (2008)
It is mysterious why Hilbert wanted to prove directly the consistency of analysis by finitary methods. ❋ Kennedy, Juliette (2007)
Aumann (1976) gives an alternate finitary procedure for generating the common knowledge hierarchy in the special case in which the relevant number of possible worlds in Ω is finite and each agent's information system partitions Ω. ❋ Vanderschraaf, Peter (2007)
It is probably for this reason that Aumann is often credited with presenting the first finitary method of generating the common knowledge hierarchy (Aumann 1976). ❋ Vanderschraaf, Peter (2007)
Show, using finitary methods, that each proof in the extended system has a consistent interpretation. ❋ Avigad, Jeremy (2007)
Thus, checking the validity of a formula of L is equivalent to performing a finite number of finitary tests. ❋ Carnielli, Walter (2007)
The question addressed in the introduction to the paper is what abstract notions must be added to finitary mathematics in order to obtain a consistency proof for arithmetic. ❋ Kennedy, Juliette (2007)
In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the “attractive” features (e.g., compactness and completeness) of the latter. ❋ Bell, John L. (2006)
An interesting result on the finitary logics in the classes of fully selfextensional logics with conjunction and the ones with the deduction-detachment property for a single term is that their class of algebras ❋ Jansana, Ramon (2006)
