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Large infinities and definable sets
J P Aguilera1, J Bagaria2,3, P Lücke4
1Institut für diskrete Mathematik und Geometrie, Technische Universität Wien, Vienna 1040, Austria.
Summary
Large cardinal axioms introduce very large infinite sets to mathematics, aiming to address Gödel incompleteness. New infinities reveal tensions with mathematical simplicity and the Axiom of Choice, posing new questions.
Area of Science:
- Set Theory
- Mathematical Logic
Background:
- Large cardinal axioms postulate the existence of very large infinite sets.
- Research in mathematical logic seeks stronger axioms to mitigate Gödel incompleteness.
- A tension exists between large cardinal axioms and principles of global mathematical simplicity, including the Axiom of Choice.
Purpose of the Study:
- To explore the implications of newly identified kinds of infinity.
- To investigate the tension between large cardinal axioms and global simplicity principles.
- To raise mathematical and philosophical questions arising from these new infinities.
Main Methods:
- Axiomatic Set Theory
- Exploration of large cardinal hierarchies
- Analysis of set-theoretic principles
Main Results:
- New kinds of infinity have been identified.
- These new infinities illuminate the tension between large cardinals and global simplicity.
- The identified infinities raise significant mathematical and philosophical questions.
Conclusions:
- The study highlights the complex interplay between large cardinal axioms and foundational principles in mathematics.
- New infinities offer novel perspectives on the structure of the mathematical universe.
- Further research is warranted to address the mathematical and philosophical implications.
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