Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.01935

Intuitionistic Mathematics and Logic

Joan R. Moschovakis·Garyfallia Vafeiadou
View original source ↗Read source document ↗Export metadata ↓

Abstract

The first seeds of mathematical intuitionism germinated in Europe over a century ago in the constructive tendencies of Borel, Baire, Lebesque, Poincaré, Kronecker and others. The flowering was the work of one man, Luitzen Egbertus Jan Brouwer, who taught mathematics at the University of Amsterdam from 1909 until 1951. By proving powerful theorems on topological invariants and fixed points of continuous mappings, Brouwer quickly build a mathematical reputation strong enough to support his revolutionary ideas about the nature of mathematical activity. These ideas influenced Hilbert and Gödel and established intuitionistic logic and mathematics as subjects worthy of independent study. Our aim is to describe the development of Brouwer's intuitionism, from his rejection of the classical law of excluded middle to his controversial theory of the continuum, with fundamental consequences for logic and mathematics. We borrow Kleene's formal axiomatic systems (incorporating earlier attempts by Kolmogorov, Glivenko, Heyting and Peano) for intuitionistic logic and arithmetic as subtheories of the corresponding classical theories, and sketch his use of gödel numbers of recursive functions to realize sentences of intuitionistic arithmetic including a form of Church's Thesis. Finally, we present Kleene and Vesley's axiomatic treatment of Brouwer's continuum, with the function-realizability interpretation which establishes its consistency.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joan R. Moschovakis, Garyfallia Vafeiadou. 2020-03-04. Intuitionistic Mathematics and Logic. https://arxiv.org/abs/2003.01935

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On n-dependent groups and fields III. Multilinear forms and invariant connected components

We develop some model theory of multilinear forms, generalizing Granger's work in the bilinear case. In particular, after proving a quantifier elimination result, we show that for an NIP field $K$, the theory of infinite-dimensional non-degenerate alternating $n$-linear spaces over $K$ is strictly $n$-dependent, and is NSOP$_1$ if $K$ is. These results rely on a new Composition Lemma for functions of arbitrary arity and NIP relations (which in turn relies on certain higher-arity generalizations of the Sauer--Shelah lemma). We also study the invariant connected components $G^{\infty}$ in $n$-dependent groups, demonstrating their relative absoluteness.

math.LO↗
arXiv2026

Loops, Inverse Limits and Non-Determinism

We introduce an operator on problems in Weihrauch complexity, which we call the infinite loop or inverse limit, and which corresponds to an infinite compositional product. This operation arises naturally whenever one implements algorithms that produce a sequence of results in an infinite loop, using some fixed subroutine. We prove that the corresponding operator is monotone with respect to (strong) Weihrauch reducibility but that it is not a closure operator. One of our findings is that weak Kőnig's lemma is closed under infinite loops, which implies that the class of non-deterministically computable problems is also closed under this operation. Consequently, this class allows for a high degree of flexibility in programming. As our main technical tools, we present an injective version of the recursion theorem and an infinitary version of the so-called independent choice theorem. We also show that, in general, the infinite loop operator is more powerful than the composition of the diamond operator followed by the parallelization operator. However, in many practical scenarios, these compositions yield a result, which coincides with the application of the infinite loop operator. Finally, we discuss the special situation of loops for single-valued problems and for problems on Turing degrees.

math.LO↗
arXiv2026

A vector logic for intensional formal semantics

Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model-theoretic structures; the latter as vectors in high-dimensional spaces shaped by usage. This paper establishes which part of intensional formal semantics admits a linear vector-space encoding. Kripke-style intensional models, with any finite collection of index sorts collected in a compound index space, embed injectively into vector spaces: primitive domains go to free carriers; intensions and other functions go to linear operators. Semantic functions lift to unique multilinear maps on the free carriers, and composition is preserved. The operator encoding of a function domain compresses its free carrier, and we characterize the functionals: a Boolean-valued functional of a power set acts linearly on operator encodings exactly when it is constant, an ultrafilter indicator, or the complement of one, so that on finite domains the nonconstant ones are Montague's individuals and their negations. Determiners over a restrictor of two or more elements, modal operators over two or more accessible indices, and attitude operators over two or more alternatives are outside the linear regime, and take, instead, the form of a linear accumulation followed by a decision. Modality is defined uniformly over measure frames, in which counting measure recovers Kripke semantics at every cardinality, and continuous measures make necessity truth almost everywhere; we give the correspondence conditions for the axioms D, T, B, 4, and 5 under measures, the Kronecker factorization of accessibility over compound indices, and the reading of measure-based modality as graded modality.

math.LO↗