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...

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...

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

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...

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

Continuing in this manner, it is possible to define a cardinal number ℵα for every ordinal number α, as described below. The concept and notation are due to...

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....

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...

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

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...

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

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

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...

and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim theorem (later reproved and strengthened by Thoralf Skolem...

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...

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...

then there is a minimal standard model (see Constructible universe). The Löwenheim–Skolem theorem can be used to show that this minimal model is countable...

its models are isomorphic. It follows from the definition above and the Löwenheim–Skolem theorem that any first-order theory with a model of infinite cardinality...

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...

{\displaystyle T} has a model. Another version, with connections to the Löwenheim–Skolem theorem, says: Every syntactically consistent, countable first-order...

this section T is a countable complete theory and κ is a cardinal. The Löwenheim–Skolem theorem shows that if I(T,κ) is nonzero for one infinite cardinal...

= | B | {\displaystyle |A|=|B|} ; however, if referring to the cardinal number of an individual set A {\displaystyle A} , it is simply denoted | A | {\displaystyle...

Skolemizations of formulas in prenex form as part of his proof of the Löwenheim–Skolem theorem (Skolem 1920). Herbrand worked with this dual notion of...

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

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...

geometrical shapes, variables, or even other sets. A set may have a finite number of elements or be an infinite set. There is a unique set with no elements...