infinitary

The former created in 1951 an infinitary sequent calculus to present consistency proofs in a perspicuous way, the latter instead used a more traditional Gentzen-style calculus.

Chores 2009

On the other hand, the consistency of an atomistic theory is typically guaranteed by the trivial one-element model (with ˜P™ interpreted as identity), though we can also have models of atomistic theories that allow for infinitary decomposition.

Wild Dreams Of Reality, 3 2009

The theorem fails badly for nearly all infinitary languages.

First-order Model Theory Hodges, Wilfrid 2009

For there is a sense in which (P. 16ψ) might tought to be redundant in the presence of infinitary sum principles such as (P. 15ψ) and the like.

Wild Dreams Of Reality, 3 2009

(It appears, in an infinitary form, as Mackey's axiom V; a related but stronger condition appears in the definition of a partial Boolean algebra in the work of Kochen and Specker [1965].)

Puppet X: 1 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.

Wild Dreams Of Reality, 3 2009

There are similar but more complicated theorems for uncountable first-order languages; some of these can be paraphrased as omitting types theorems for infinitary languages.

First-order Model Theory Hodges, Wilfrid 2009

We can get even stronger composition principles by considering infinitary bounds and sums.

Wild Dreams Of Reality, 3 2009

Thus, intuitively, each of the infinitary sum principles above should have a substitution instance that yields

Wild Dreams Of Reality, 3 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.

Dedekind's Contributions to the Foundations of Mathematics Reck, Erich 2008