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A new type of infinity discovered by mathematicians appears to break the rules of behavior for extremely large numbers, and could redesign the way the mathematical universe is ordered.
We think that “infinity is infinity and that’s the end of it”, but it’s not quite like that.
Mathematicians have long known that there is more than one type of infinity.
Em 1878, Georg Cantor showed for the first time that the infinite set of real numbers, which includes negatives and decimals, is in fact larger than the infinite set of natural numbers, or integers. Proving this fact involves a careful comparison between the two sets, rather than attempting the impossible task of counting them all.
Mathematicians quickly realized that they could construct sets ever-increasing infinitiescreating a hierarchical ladder of sets that is itself infinite.
However, a group of mathematicians at the University of Vienna now propose two new infinity sizescalled exact and ultra-exact cardinalswho do not obey the rules.
These two new sizes of infinity, explains the research leader to . John Aguilera“do not fit well into this linear hierarchy. They interact in a very, very strange way with other notions of infinity.”
In a study, the results of which were recently published in , researchers defined these sets by making them so large that they must contain mathematically exact copies of its entire structure. Ultra-exact cardinals have an additional rule, which says that these sets must also contain the mathematical rules on how to do them.
As the aforementioned magazine explains, these unusual properties are what causes these sets to fall down the ladder of infinity, as they cause damage to some of the most profound rules of mathematics.
Axiom of choice
At the beginning of the 20th century, an attempt was made to establish a rigorous basis for mathematics, creating a set of fundamental axioms. Here came the Zermelo-Fraenkel set theory. This theory included a controversial rule called axiom of choice.
Some mathematicians felt that this axiom did not work when considering infinite sets, because it would require asserting the existence of mathematical objects without offering a way to prove them. However, over time, they ended up accepting the rule and it is now used as a fundamental measuring instrument to organize the infinite staircase, dividing it into three large regions.
These infinities are organized into three main regions: the first, containing infinities defined by the axioms of sets, such as real and natural numbers; the third, where the axioms, including that of choice, fail; and a second intermediate region, with infinities whose position is still uncertain.
Initially, the exact and ultra-exact cardinals seemed to fit into the second region, but when the team tried to identify them… they discovered it wasn’t possible.
A fourth mysterious region
“They may be at the top of this intermediate region, where the axioms are still compatible with all the other axioms of set theory, or they may be forming a fourth region that is more or less next to the chaotic regionbut on top of the previous ones”, says Aguilera.
If these exact cardinals are accepted by the mathematical community at large, then it “strongly suggests that chaos reigns,” giving life to a amazing infinityexplains Philip Gluckmember of the research team, for New Scientist.
