Infinity with mise en abyme

Infinity is more complex than you might think - From the New Scientist, April 2025.

Mathematicians have discovered a mind-blowing new kind of infinity

It may sound strange, but mathematicians have created an entire ladder of infinities, each larger than the next. Now a new kind of infinity threatens to upset that order, and perhaps redefine the structure of the mathematical universe

By Alex Wilkins

“People have been coming up with larger and larger notions of infinity,” says Juan Aguilera at the Vienna University of Technology in Austria. “You can look at all the previous ones that people have come up with and you can fit them in a hierarchy.”

But now, Aguilera and his colleagues have proposed two new sizes of infinity, called exacting and ultra-exacting cardinals, that don’t obey the rules. “They don’t quite fit in this linear hierarchy,” says Aguilera. “They interact very, very strangely with other notions of infinity.”

Aguilera and his team defined these sets by making them so large that they must both contain mathematically exact copies of their entire structure – a bit like a house that contains multiple, full-scale models of itself – and also contain small versions of larger sets, like adding models of the surrounding neighbourhood or city to the house. Ultra-exacting cardinals have one further rule, which says that these sets must also contain the mathematical rules of how to make them – as if the nested house was also wallpapered with blueprints of itself. (My emphasis)