Infinity contains everything, including multiple kinds of infinities.
Mathematicians have long known that there are many kinds of infinities (technically, there are an infinity of them).
Mathematicians revealed two new kinds of infinity—called exacting and ultra-exacting—infinities that appear to contradict foundational mathematics.
The concept of “infinity” appears simple at first glance, but becomes increasing more complex the more you think about it. Infinity means a never-ending sequence of numbers trailing off into, well, infinity. But that also necessitates that there’s technically an infinite number of infinities forming a hierarchy of ever-greater complexity.
Scientists and mathematicians have spent decades debating the nature of infinity, and have known for more than a century that there’s more than one kind. For example, one infinity—the one most people are familiar with—is an infinite set of natural numbers: 1, 2, 3, and so on. However, there’s also an infinite set of real numbers, which includes negatives and decimals. Keep following this train of thought, and you create an infinite set of infinities.
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Recently, mathematicians from the Vienna University of Technology in Austria and the University of Barcelona have discovered two new kinds of infinities, known as exacting and ultra-exacting cardinals. These infinities don’t quite follow the standard ladder of infinities due to their unusual properties. The researchers detailed these new infinite sets in a paper published on the non-peer reviewed preprint server arXiv.
“People have been coming up with larger and larger notions of infinity,” Juan Aguilera, a co-author of the paper from Vienna University of Technology, told New Scientist. “You can look at all the previous ones that people have come up with and you can fit them in a hierarchy […] they don’t quite fit in this linear hierarchy. They interact very, very strangely with other notions of infinity.”
New Scientist described exacting cardinals as being so large that they contain copies of themselves—sort of like a house with many full-scale copies of itself inside. Ultra-exacting copies additionally include mathematical rules on how to create them “as if the nested house was also wallpapered with blueprints of itself.”
However, things get wonky when comparing these new infinities with a foundational concept of mathematics known as the Axiom of Choice, which says that you can make a new set of numbers by picking out numbers from other sets. This puts infinities into three categories: infinities that adhere to this set theory (natural number sets, integer sets, etc...), infinities so large they’re essentially chaos mathematics, and infinities that exist somewhere in between. Although the researchers thought exacting and ultra-exacting cardinals fit in this in-between region, they couldn’t quite pin them down.
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“It’s not quite clear if they are at the top of this middle region, where the axioms are still compatible with all the other axioms of set theory, or whether they are forming a fourth region that is kind of to the side of the chaotic region, but on top of the previous ones,” Aguilera told New Scientist.
This could contradict an idea known as Hereditarily Ordinal Definable, which theorizes that infinities could get so large that the Axiom of Choice imposes order instead of contradiction. But these cardinals currently appear to defy that assumption.
Of course, these infinities have yet to be accepted by the broad mathematics community. But regardless of their potential confirmation, it looks like pondering infinity won’t be slowing down any time soon.
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Darren lives in Portland, has a cat, and writes/edits about sci-fi and how our world works. You can find his previous stuff at Gizmodo and Paste if you look hard enough.