# Set theory of the real line and its subsets

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Miami University (OH) · $80,000

## Abstract

Techniques from mathematical logic can be used to measure the complexity of mathematical concepts. Areas of mathematics with physical applications tend to appear in the low levels of the corresponding complexity hierarchy. Passing to higher levels of complexity enables mathematicians to make connections between different areas of mathematics, and to develop productive general theories. There is a corresponding division of the universe of mathematics into an absolute part, where natural questions tend to be resolved by the standard axioms, and a more abstract part where extensions of the standard axiom system are needed to resolve many fundamental questions. The main focus of PI's research is the relationship between these parts. The technical machinery involved in this project includes Cohen's forcing technique, axioms asserting the existence of winning strategies in infinite games, and axioms asserting the existence of infinite objects whose existence cannot be proved from the standard axioms for mathematics. These techniques originate in set theory, which serves as the most commonly accepted foundations for mathematics.  


The PI plans to work on a related collection of projects involving the complexity of sets of real numbers. Two of these projects aim to extend the work published in two of his recent books. The first of these is another book exposing part of W. Hugh Woodin's work on his axiom AD^+. The first part of this book is to be on methods for producing maxim

## Key facts

- **NSF award ID:** 2452139
- **Awardee organization:** Miami University (OH)
- **SAM.gov UEI:** T6J6AF3AM8M8
- **PI:** Paul B Larson
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $80,000
- **Funds obligated:** $80,000
- **Transaction type:** Standard Grant
- **Period:** 06/15/2025 → 05/31/2027

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2452139

## Citation

> US National Science Foundation, Award 2452139, Set theory of the real line and its subsets. Retrieved via AI Analytics 2026-06-06 from https://api.ai-analytics.org/grant/nsf/2452139. Licensed CC0.

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