# Inner Models, Combinatorics, and Determinacy

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of North Texas (TX) · $100,000

## Abstract

The Large Cardinal Axioms (LCAs) are extensions of the standard axioms of set theory, Zermelo-Fraenkel set theory with the axiom of choice (ZFC). LCAs are designed to settle all natural theories that are independent of ZFC. This is Gödel's program in set theory. How can one test for “correctness” of an LCA? The inner model program, a major program in modern set theory, justifies correctness by constructing canonical models for LCAs much like the natural numbers are the canonical model for the Peano Axioms of arithmetic (PA). The canonicity of the models justifies the correctness of the LCAs much like the canonicity of the natural numbers justifies the correctness of PA. This research project contributes to the inner model program by advancing the state of the current knowledge regarding canonical models for LCAs and their relationship with other foundational frameworks of set theory. The project provides research opportunities for graduate students. 

This project builds on and expands the PI’s previous work on computing consistency lower bounds for the Proper Forcing Axiom (PFA), on studying canonical models of AD^+ up to the minimal model of the Largest Suslin Axiom (LSA), and on the Sealing phenomenon concerning universally Baire sets. These are important theories in our set-theoretic landscape. The PI plans to explore further various aspects of Sealing (particularly a weak form of Tower Sealing and its implications) and develop further techniques of the core model indu

## Key facts

- **NSF award ID:** 2449780
- **Awardee organization:** University of North Texas (TX)
- **SAM.gov UEI:** G47WN1XZNWX9
- **PI:** Nam Trang
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $100,000
- **Funds obligated:** $100,000
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2027

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2449780

## Citation

> US National Science Foundation, Award 2449780, Inner Models, Combinatorics, and Determinacy. Retrieved via AI Analytics 2026-06-06 from https://api.ai-analytics.org/grant/nsf/2449780. Licensed CC0.

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