# Infrafiltration Theorem and Some Inductive Sequence of Models of Generalized Second-Order Dedekind Theory of Real Numbers With Exponentially Increasing Powers

## TL;DR

This paper constructs a sequence of non-isomorphic models for a generalized second-order Dedekind theory of real numbers, utilizing infraproducts and a generalized infrafiltration theorem to extend classical model theory results.

## Contribution

It introduces a new construction of models using infraproducts and generalized equalities, expanding the understanding of models in second-order Dedekind theory.

## Key findings

- Models are non-isomorphic despite sharing the same theory.
- Develops a generalized infrafiltration theorem.
- Establishes a generalized compactness theorem for the theory.

## Abstract

The paper is devoted to construction of some closed inductive sequence of models of the generalized second-order Dedekind theory of real numbers with exponentially increasing powers. These models are not isomorphic whereas all models of the standard second-order Dedekind theory are. The main idea in passing to generalized models is to consider instead of superstructures with the single common set-theoretical equality and the single common set-theoretical belonging superstructures with several generalized equalities and several generalized belongings for first and second orders. The basic tools for the presented construction are the infraproduct of collection of mathematical systems different from the factorized Lo\'s ultraproduct and the corresponding generalized infrafiltration theorem. As its auxiliary corollary we obtain the generalized compactness theorem for the generalized second-order language.