Josie Miller
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Is There a Bigger Infinity Than Infinity?

— Josie Miller

We often think of infinity as a simple concept: a value with no limits that goes on forever. But that isn’t the full picture. In fact, infinite sets of numbers are not equal in size—some are bigger than others. 

The amount of numbers in an infinite set is known as its ‘cardinality.’ The smallest infinite set is the set of natural numbers (1, 2, 3, …), since there are only so many countable whole numbers. If an infinite set is equal to the set of natural numbers, then the amount of numbers in this set is the smallest infinite ‘cardinal,’ denoted ℵ0 (aleph null). In other words, ℵ0 is how many natural numbers there are. 

If the set of natural numbers were to have the smallest infinity, then what would have the largest? The set of real numbers, including integers and decimals, is considered the biggest infinite set. This set has a cardinality of , or ‘continuum.’ 

In the 1870s, a famous German logician Georg Cantor wondered if the cardinality of the real set of numbers was the smallest cardinal above ℵ0. That is, whether ℵ1 = continuum. His conjecture that the size of all real numbers is the smallest uncountable cardinal is known as the continuum hypothesis.

Another helpful metric for measuring the different sizes of infinite sets is ordinals. Think of ordinals as positions of something (i.e. 1st, 2nd, 3rd) and cardinals as how many of something there are (i.e. 1, 2, 3). The ordinal number that describes the first infinite position is ω (omega). In other words, instead of saying an ordinal of an infinite number is the ∞th number, you would say that an ordinal of an infinite number is in the ω range. 

Following this logic, the ordinality of the cardinal number ℵ0 is ω, as both ℵ0 and ω describe the first infinity of their respective measures. So, with both cardinals and ordinals, you may have a bigger infinite amount or position of a set of numbers (i.e. ℵ1, ω1). With ℵ and ω ranges, we can explore how these sizes and positions exist in different infinite values, and how infinity may not be as simple of a concept as it may seem!


References

Goldstern, M., & Kellner, J. (2021, August 16). A deep math dive into why some infinities are bigger than others. Scientific American. https://www.scientificamerican.com/article/a-deep-math-dive-into-why-some-infinities-are-bigger-than-others/

Jach, T. (2002, July 11). Set Theory. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/Archives/Win2004/entries/set-theory/

lhfgraphics. (n.d.). Doodle style infinity math symbol illustration. 123RF. https://www.123rf.com/photo_11790093_doodle-style-infinity-math-symbol-illustration-suitable-for-web-print-or-advertising-use.html