Undecidability of the Real‐Algebraic Structure of Scott's Model

Abstract

We show that true first-order arithmetic of the positive integers is interpretable over the real-algebraic structure of Scott's topological model for intuitionistic analysis. From this the undecidability of the structure follows. Mathematics Subject Classification: 03D45, 03F55.

Encoding true second‐order arithmetic in the real‐algebraic structure of models of intuitionistic elementary analysis

“…For the encoding of sets of natural numbers we shall generalize the following construction used in Scott's model in [3]. Let denote the set of positive natural numbers.…”

“…There were results by Smoryński (in [15]) and by Cherlin (in [1]) suggesting that this might be the case. Indeed, building on Cherlin's work, in [3] we were able to interpret true first order arithmetic in the L ‐theory of Scott's model. Then, in [4] we extended this result to the classes of models defined by Scowcroft [13, 14] and Krol [7].…”

Undecidability of the real-algebraic structure of models of intuitionistic elementary analysis

“…Let L denote the language of ordered rings. In [5], using a coding similar to the one used by Cherlin in [1], we showed that the L-theory of Scott's topological model for intuitionistic analysis is undecidable by encoding true first-order arithmetic in that structure. (Scott defined the model in question in [9].)…”