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# Infinity and Intensionality: Towards a New Synthesis

**Alternative title:** Uendelighet og intensjonalitet: En ny syntese

#### Awarded: NOK 10.8 mill.

Source:

Project Manager:

Project Number:

314435

Application Type:

Project Period:

2021 - 2025

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We aim to undertake an integrated investigation of two clusters of questions, which have so far only been studied in isolation. The first cluster of questions concerns the theory of infinite sets, which is the standard foundation for nearly all of today?s mathematics. This foundation is threatened by various internal tensions and paradoxes, which have attracted much attention. The second cluster concerns intensional notion such as propositions, properties, and relations. Propositions, which represent the contents of beliefs and desires, are a central concern in linguistics, psychology, and philosophy.
Our overarching hypothesis is that real progress can only be made by means of a new synthesis, where the two clusters of questions are tackled in a unified way, thus bringing concepts and theories from each cluster to bear on the other one. To address the first cluster, we need to supplement standard set theory with intensional notions of collection, number, and generality. E.g., we can define a collection in terms of its membership condition rather than directly in terms of its members. Conversely, to develop satisfactory theories of intensional notions, we need to draw on concepts and ideas developed by some prominent critics of infinitary set theory (especially Poincaré, Brouwer, and Weyl). In short, real progress can only be made by an integrated investigation of the two clusters of questions.
In this way, we aim to develop (1) a new and distinctive approach to mathematics, which retains the theory of infinite sets but supplements it with intensional notions of collection, number, and generality; and (2) new theories of the intensional notions of proposition, property, and relation, of great significance for the foundations and philosophy of semantics and psychology.

We aim to undertake an integrated investigation of two clusters of questions—concerned with infinity and intensionality—which have so far only been studied in isolation. Our overarching hypothesis is that real progress can only be made by means of a new synthesis, where the two clusters of questions are tackled in a unified way, thus bringing concepts and theories from each cluster to bear on the other one.
The first cluster of questions concerns the theory of infinite sets, which is the standard foundation for nearly all of today’s mathematics. This foundation is threatened by various internal tensions and paradoxes, which have attracted much philosophical attention. The second cluster concerns intensional notion such as propositions, properties, and relations. Propositions, representing the contents of beliefs and desires, are a central concern in linguistics, psychology, and philosophy. To address the first cluster, we need to supplement standard set theory with intensional notions of collection, number, and generality. But to develop satisfactory theories of intensional notions, we need to draw on concepts and theories developed by critics of infinitary set theory. Thus, real progress can only be made by an integrated investigation of the two clusters of questions.
Our project will therefore develop a new synthesis in our theorizing about infinity and intensionality. Our objectives are to develop
- a novel and distinctive foundation and philosophy of mathematics, which retains the theory of infinite sets but supplements it with intensional notions of collection, number, and generality
- new theories of the intensional notions of proposition, property, and relation, of great significance for the foundations and philosophy of semantics and psychology
- a novel and surprising connection between areas so far only investigated separately
In research areas known for their male dominance, our interdisciplinary team has, very unusually, a perfect gender balance.

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1.7BILL. NOKtotal funding in the programme period751PROJECTShave received funding in the programme period1SOURCEhas financed the programme

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