Austrian and Spanish mathematicians have recently discovered two new types of infinity, termed exacting and ultra-exacting cardinals. These infinities possess unusual properties, seemingly contradicting established mathematical principles like the Axiom of Choice and the Hereditarily Ordinal Definable theory. Their discovery challenges the existing hierarchy of infinities and suggests a more complex structure than previously understood. The findings, currently published on a preprint server, await wider acceptance within the mathematics community.

PS: This episode is from February 2025.

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The newly discovered infinities, called exacting and ultra-exacting cardinals, challenge existing mathematical frameworks because they do not fit into the established hierarchy of infinities and interact in unusual ways with other notions of infinity. Here are some of the ways that these new infinities challenge existing frameworks:

* They don't fit in the standard hierarchy: Mathematicians have long known that there are many kinds of infinities, forming a hierarchy of ever-greater complexity. However, the exacting and ultra-exacting infinities do not quite follow this standard ladder due to their unusual properties.

* They interact strangely with other infinities: These new infinities interact very, very strangely with other notions of infinity.

* They challenge the Axiom of Choice: The Axiom of Choice states that you can create a new set of numbers by picking numbers from other sets. Infinities are typically categorized into three groups: those that adhere to this axiom, infinities so large they are essentially chaos mathematics, and those that exist somewhere in between. While researchers initially thought these new infinities would fit in the in-between category, they couldn't definitively place them.

* They may contradict the Hereditarily Ordinal Definable theory: This theory suggests that as infinities get larger, the Axiom of Choice imposes order instead of contradiction. However, the new cardinals appear to defy that assumption.

* Their properties are unusual: Exacting cardinals are so large that they contain copies of themselves. Ultra-exacting cardinals additionally include mathematical rules on how to create them.

It is important to note that these new infinities have not yet been broadly accepted by the mathematics community. The researchers themselves are not sure if these new infinities form a fourth region beyond the chaotic region or if they are at the top of the middle region where axioms are compatible with set theory.

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