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A Deep Math Dive into Why Some Infinities Are Bigger Than Others

The size of certain infinite sets has been a mystery. Now, it turns out, each one is different than the next, and they can all be ordered by size

Credit: Lidiia Moor Getty Images
Credit: Lidiia Moor Getty Images

Simple mathematical concepts such as counting appear to be firmly anchored in the natural process of thinking. Studies have shown that even very young children and animals possess such skills to a certain extent. This is hardly surprising because counting is extremely useful in terms of evolution. For example, it is required for even very simple forms of trading. And counting helps in estimating the size of a hostile group and, accordingly, whether it is better to attack or retreat.

Over the past millennia, humans have developed a remarkable notion of counting. Originally applied to a handful of objects, it was easily extended to vastly different orders of magnitude. Soon a mathematical framework emerged that could be used to describe huge quantities, such as the distance between galaxies or the number of elementary particles in the universe, as well as barely conceivable distances in the microcosm, between atoms or quarks.

We can even work with numbers that go beyond anything currently known to be relevant in describing the universe. For example, the number 10^10^100 (one followed by 10^100 zeros, with 10^100 representing one followed by 100 zeros) can be written down and used in all kinds of calculations. Writing this number in ordinary decimal notation, however, would require more elementary particles than are probably contained in the universe, even employing just one particle per digit. Physicists estimate that our cosmos contains fewer than 10100 particles.

Yet even such unimaginably large numbers are vanishingly small, compared with infinite sets, which have played an important role in mathematics for more than 100 years. Simply counting objects gives rise to the set of natural numbers, ℕ = {0, 1, 2, 3, …}, which many of us encounter in school. Yet even this seemingly simple concept poses a challenge: there is no largest natural number. If you keep counting, you will always be able to find a larger number.

Can there actually be such a thing as an infinite set? In the 19th century, this question was very controversial. In philosophy, this may still be the case. But in modern mathematics, the existence of infinite sets is simply assumed to be true—postulated as an axiom that does not require proof.

Set theory is about more than describing sets. Just as, in arithmetic, you learn to apply arithmetical operations to numbers—for example, addition or multiplication—you can also define set-theoretical operations that generate new sets from given ones. You can take unions—{1, 2} and {2, 3, 4} becomes {1, 2, 3, 4}—or intersections—{1, 2} and {2, 3, 4} becomes {2}. More excitingly, you can form power sets—the family of all subsets of a set.

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