What is it about?
In the 19th century, German mathematician Georg Cantor proved that infinities come in different sizes. Investigating large infinities is central in our quest to understanding the limits of mathematics, its axioms, and the phenomenon of incompleteness. This article illustrates that there is a possible tension between the idea that very large infinities exist and the idea that the universe of mathematics is globally well-behaved and/or orderly. Together, both ideas lead to inconsistencies.
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Why is it important?
The article raises the possibility that some of our long-held conceptions about infinity might need to be revised. As infinity plays a key role in our understanding of the very fundaments of mathematics (the "axioms"), these revisions could have rippling effect throughout the field.
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This page is a summary of: Large infinities and definable sets, Proceedings of the National Academy of Sciences, April 2026, Proceedings of the National Academy of Sciences,
DOI: 10.1073/pnas.2528175123.
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