Sentence examples for weak theory from inspiring English sources

At the base one has the very weak theory Q and one climbs upward by adding "closure principles".

NFU + Infinity + Choice is a relatively weak theory: like Zermelo set theory it does not prove even that ℵω exists.

It is however a very weak theory, as it can not prove the existence of the addition function on the natural numbers (Rathjen 2008).

In fact, Q is such a weak theory, it is not clear that a semantically contentful statement of consistency can be formulated for Q at all.

When selection on codon bias is weak, theory shows that it has only small effects on the rate of synonymous substitutions even if the effects on codon usage can be significant [ 25].

(There are some very weak theories that are undecidable but not essentially undecidable).

Since any proponent of ZFC must accept that ZFC is consistent, these weak theories are enough to get several variants of Putnam's argument off the ground.

In particular, Button notes that even very weak theories can prove theorems like: "if ZFC is consistent, then ZFC has a countable model".

However, the axioms ∀A(T[TA] ↔ TA) and ∀A(T[¬A] ↔ ¬TA) are inconsistent over weak theories of syntax, so one of them has to be given up.

But because the concepts of Herbrand provability and provability separate in weak theories, the question whether there is an intensionally adequate version of the Second Incompleteness Theorem for Robinson's Q remains open.

However, the applicability of the incompleteness theorems can be dramatically extended outside the language of first-order arithmetic and its extensions, when it is noted that all that is needed is that weak theories such as Q, or PRA, can be interpreted in the system in question.