Some observations on intuitionistically elementary properties of linear orderings
The paper is about intuitionistic order structures of similarity type \(<R^ n,<_ n>\), where R is the intuitionistic continuum, \(<_ 1=<\) intuitionistic (linear) order and \(<_ n\) for \(n>1\) its lexicographic extension. Two such structures are intuitionistically elementary equivalent (IEE) if they have the same true first order sentences, and one is an intuitionistic elementary substructure (IES) of a second one if the set of the first is a subset of that of the second, and for every first order formula \(\phi\) any instance \(\phi [a_ 1,...,a_ n]\) is true in the first if and only if it is true in the second. Two subsets X and Y of R are IEE if their order structures are IEE, and order- elementary different (IED) if there is a first order sentence which is true in one and not in the other. It is shown that the non-isomorphic structures \(<(0,1)(1,2),<>\) and \(<(0,2),<>\) are order-IEE. There is a sequence of sentences \(\phi_ n\) and a function A assigning subsets A(\(\alpha)\) of R to elements \(\alpha\) of the binary spread \(^{\omega}2\) such that all A(\(\alpha)\) are dense in R, \(\neg \phi_ n\) holds in \(<A(\alpha),<>\) if \(\alpha (n)=0\), and \(\phi_ n\) holds in \(<A(\alpha),<>\) if \(\alpha (n)=1\). From this it can be deduced that there is an uncountable collection of subsets of the rationals which are mutually order-IED. A similar result holds for the lexicographical extensions: for all m and n such that \(0<m<n\) there is a sentence \(\phi\) such that \(\phi\) is true in \(<R^ m,<_ m>\), while \(\neg \phi\) is true in \(<R^ n,<_ n>\). This result is extended from \(R^ n\) to \(R^{\lambda}\), where \(\lambda \in \omega^{\omega}\), the ordered set of polynomials \(\omega^{m_ 1}\cdot n_ 1+...+\omega^{m_ k}\cdot n_ k\). It also can be proved that for \(0<\lambda <\mu\) in \(\omega^{\omega}\) the structures \(<2^{\lambda},<_{\lambda}>\) and \(<2^{\mu},<_{\mu}>\) are IED.
intuitionistically elementary equivalent
scientific article; zbMATH DE number 4130381