Mathematicians identify two new types of infinities

How is “infinity” treated from mathematics? Let’s imagine that there is a hotel with infinite rooms, let’s call it Hotel Hilbert. The accommodation is at maximum capacity: there are infinite guests, each settling into a room. Then something unexpected happens: a person arrives urgently requesting a room. Although the hotel is full, the receptionist is not discouraged and asks the infinite guests to go to the room with the number next to the one they are occupying, and places the new one in number 1. Shortly after another problem arises, countless people arrive asking to stay, now the receptionist asks the tenants to move to a room with double the number they were occupying, and the newcomers settle into the rooms with the odd numbers that have just become vacant. With a little imagination and repeating “infinity” constantly, it can be shown that there will always be room for everyone at the Hilbert Hotel, even if infinite buses arrive, each with infinite tourists.

This thought experiment was created by German mathematician David Hilbert to illustrate the sometimes counterintuitive properties of the concept of infinity, which can be expanded arbitrarily without ever exhausting the available space. However, there is a complicating factor: there is no “single infinity.” Strangely enough, this is so, and we have known it for a long time. In 1878, mathematician George Cantor showed that “some infinities were more infinite than others.” He observed that the infinite set of natural numbers (1,2,3, etc.) is somehow “less infinite” than the set of real numbers: integers, rational and irrational; It is as if the first were an infinite “less dense” than the second.

The current landscape could change this; A group of mathematicians from the Vienna University of Technology discovered two types of infinity that could alter Cantor’s rules. [que dicen de algún modo que siempre hay más números decimales que enteros]. The new infinities: “cardinal exact” and “ultra exact”.

A house with infinite houses inside

“Historically, mathematicians have come up with increasingly complex notions of infinity, which have then been put into a kind of hierarchy,” he explains to the magazine. New Scientist Juan Aguilera, one of the authors of the study. The novelty is that Aguilera and his colleagues identified two new infinite dimensions, called “cardinal exact” and “ultra exact”, that do not fit into the existing hierarchy.and in fact, “they interact in a very strange way with the other notions of infinity.”

The authors define these new sets by predicting that they both contain exact mathematical copies of themselves, such as a house containing multiple full-size models of the same house, and “reduced” mathematical copies of larger sets, as if the house also contained models of the same house. neighborhood or city. In addition, ultra-exact ones also contain the mathematical rules that define how to do it: these “strange rules”, but totally legitimate when it comes to infinity, are the ones that cause the “unhinging” of current hierarchies.

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The problem of choice

The hierarchy of infinities is based on a set of basic rules or “axioms” synthesized in the so-called “Zermelo-Fraenkel Axiomatic Theory”. One of them, the “axiom of choice”, reinforces that it is always possible to construct a new set of numbers by choosing digits from another set, although it does not indicate how to select said digits. According to some mathematicians, this axiom would not work in the case of infinite sets, precisely because it leaves the problem of choice undefined.

However, the axiom is now accepted even for infinite sets, and is used to “organize” infinities into three regions of increasing complexity and depth. The problem is that the exact cardinals and the ultra exact ones cannot be placed in any of these three regions: “It is a complete chaos, it is not clear if they are in an “intermediate region” or if they are in a new fourth region, with rules and different axioms,” highlights Aguilera. It is abstract but it is also interesting because there are much larger and more important questions linked to the solution of the problem, in particular the so-called “Hodge Conjecture”, a yet unproven hypothesis about the validity of the axiom of choice for particularly “dense” sets of infinities. “If the existence of exact cardinals were accepted by the mathematical community, it would mean that the Hodge conjecture could be false and, therefore, chaos would reign”concludes the University of Berkeley expert, Gabriel Goldberg.

Article originally published in WIRED Italy. Adapted by Alondra Flores.