mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds....

mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem...

the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists. The Löwenheim number of first-order...

possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the...

Lindenbaum–Tarski algebra Abstract model theory Löwenheim number – Smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds Lindström's theorem –...

Extension (predicate logic) Herbrandization List of logic symbols Lojban Löwenheim number Nonfirstorderizability Prenex normal form Prior Analytics Prolog Relational...

structures. Philosophy portal First-order logic Higher-order logic Löwenheim number Omega language Second-order propositional logic Monadic second-order...

numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number....

Leopold Löwenheim gave the first proof of what Skolem would prove more generally in 1920 and 1922, the Löwenheim–Skolem theorem. Löwenheim showed that...

possible to define an infinite cardinal number ℵ α {\displaystyle \aleph _{\alpha }} for every ordinal number α , {\displaystyle \alpha ,} as described...

greatly simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim–Skolem theorem, which states that if a countable...

model theory was a special case of the downward Löwenheim–Skolem theorem, published by Leopold Löwenheim in 1915. The compactness theorem was implicit in...

compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order...

cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality...

that the axioms of the theory are satisfied within this structure. The Löwenheim–Skolem theorem shows that any model of set theory in first-order logic...

injection of the infinite product of N into the ultraproduct. However, by the Löwenheim–Skolem theorem there must exist countable non-standard models of arithmetic...

first-order arithmetic, the induction principles, bounding principles, and least number principles are three related families of first-order principles, which may...

intended model is infinite and the language is first-order, then the Löwenheim–Skolem theorems guarantee the existence of non-standard models. The non-standard...

non-isomorphic, elementarily equivalent models, which can be obtained via the Löwenheim–Skolem theorem. Thus, for example, there are non-standard models of Peano...

establishing independence results in set theory. Leopold Löwenheim and Thoralf Skolem obtained the Löwenheim–Skolem theorem, which says that first-order logic...

{\displaystyle \Rightarrow } Löwenheim–Skolem theorem" — that is, D C {\displaystyle {\mathsf {DC}}} implies the Löwenheim–Skolem theorem. See table Moore...

certain rules, the Gödel number of a proof can be defined. Now, for every statement p, one may ask whether a number x is the Gödel number of its proof. The relation...

Thoralf Skolem simplified a previous result by Leopold Löwenheim, leading to the Löwenheim–Skolem theorem and, in 1930, to the notion of a Herbrand...

nonstandard elements cannot be excluded in first-order logic. The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite...

reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specifically, a line was not considered as the set of...

possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite...

article the word "number" refers to a natural number (including 0). The key property these numbers possess is that any natural number can be obtained by...

problematic. One approach to the problem might be to run the program for some number of steps and check if it halts. However, as long as the program is running...

of a single set). Based upon work of the German mathematician Leopold Löwenheim (1915) the Norwegian logician Thoralf Skolem showed in 1922 that every...

the Löwenheim–Skolem number of K {\displaystyle K} . Note that we usually do not care about the models of size less than the Löwenheim–Skolem number and...