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

Abstract

We show that true first-order arithmetic is interpretable over the real-algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras. From this the undecidability of the structures follows. We also show that Scott's model is equivalent to true second-order arithmetic. In the appendix we argue that undecidability on the language of ordered rings follows from intuitionistically plausible properties of the real numbers.

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

“…Remark The results in [4] were true for a third class of Heyting algebras, the algebras of the coperfect open sets of . In particular true first order arithmetic can be interpreted in the corresponding real algebras (see below).…”

“…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]. In [4] we interpreted true second‐order arithmetic in the L ‐theory of Scott's model, and also showed that Moschovakis' model for second‐order arithmetic can be encoded in true second‐order arithmetic giving the exact complexity of 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]. In [4] we interpreted true second‐order arithmetic in the L ‐theory of Scott's model, and also showed that Moschovakis' model for second‐order arithmetic can be encoded in true second‐order arithmetic giving the exact complexity of the L ‐theory of Scott's model. Here we prove the following theorem extending the results in [4].…”

“…In [4] we interpreted true second‐order arithmetic in the L ‐theory of Scott's model, and also showed that Moschovakis' model for second‐order arithmetic can be encoded in true second‐order arithmetic giving the exact complexity of the L ‐theory of Scott's model. Here we prove the following theorem extending the results in [4]. Theorem True second‐order arithmetic can be interpreted in the real‐algebraic structure of models of intuitionistic analysis built on elements of a class of Heyting algebras (nice Heyting algebras of height ω, to be defined below in § 4).…”

The Creating Subject, the Brouwer–Kripke Schema, and infinite proofs

“…When Brouwer in his dissertation writes that 'strictly speaking the construction of intuitive mathematics in itself is an act and not a science 10, p.99n; trl. 45, p.61n1, modified] 58 he is thinking of constructions in the first sense; and it is clear that constructions in the other two senses presuppose for their existence a construction process in the first sense. Constructions in the first sense are ontologically prior to the others.…”