Job 38:4
"It may sound strange, but mathematicians have created an entire
For centuries, mathematicians have categorized infinities into a kind of ladder. The infinite set of natural numbers — 1, 2, 3, and so on — sits on one rung. On a higher rung, the infinite set of real numbers, which includes decimals and negatives, dwarfs it. And from there, infinities cascade upward, forming an endless hierarchy.
Recently, researchers from Vienna University of Technology and the University of Barcelona uncovered two new layers of this vastness, and they don’t quite play by the usual rules.
These new types of infinities are called exacting and ultra-exacting cardinals. Unlike their predecessors, these cardinals refuse to slot neatly into the established hierarchy of infinities.
---Their discovery forces mathematicians to reconsider what infinity really means — and whether chaos might lurk at its core.
Mathematicians have long categorized infinities into a hierarchy where some infinities are larger than others. For example, the infinity of counting numbers (1, 2, 3, …) is smaller than the infinity of real numbers, which includes an infinity of decimals between 0 and 1 (and beyond).
Mathematicians use “large cardinal axioms” to describe these layers, defining specific types of infinite numbers with unique and powerful properties. At the base of the ladder is the infinity of natural numbers, ℵ₀ (aleph-null). Climbing higher reveals infinities of increasing size and complexity: measurable cardinals, supercompact cardinals, and even so-called “huge” cardinals.
These axioms followed a predictable, linear progression. Each new “rung” of the ladder built on the one before it, creating a stable structure.
But as these infinities grow, they stretch the foundational rules of
mathematics to their limits. Large cardinals, for instance, exist outside ZFC — the Zermelo-Fraenkel set theory with the Axiom of Choice, the framework underpinning nearly all modern mathematics.
Exacting cardinals are stronger (or “bigger”) in their properties than many previously known large cardinals, meaning they can interact with the mathematical universe in new and unexpected ways.
Exacting cardinals are stronger (or “bigger”) in their properties than many previously known large cardinals, meaning they can interact with the mathematical universe in new and unexpected ways.
Ultraexacting cardinals are an even more powerful and restrictive version of exacting cardinals. Think of them as exacting cardinals with additional “superpowers” that make them interact with infinity in a way that amplifies their effect on the mathematical universe.
HOD, or Hereditary Ordinal Definability, proposes that infinitely
large sets can be defined by “counting up to” them. If true, it would bring order to the mathematical universe, aligning the Axiom of Choice with the largest infinities.
But these new cardinals muddy the waters.
HOD, or Hereditary Ordinal Definability, proposes that infinitely
But these new cardinals muddy the waters.
If these new cardinals are accepted, they could provide strong evidence against the HOD Conjecture. “It could mean that the structure of infinity is more intricate than we thought,” Aguilera said.
Q: Can we ever fully understand the universe if infinity keeps surprising us?"
ZimeScience
