Robert J. Marks: Some infinities are bigger than others but there’s no biggest one

Marks: Georg Cantor came up with an ingenious proof that infinities can differ in size
even though both remain infinite:

Cantor called the countable infinite, the infinity of all counting numbers,  ℵ_0.  ℵ is the Hebrew letter aleph. The number of points on a line segment, a bigger infinity, is denoted by ℵ_1.  We are immediately prompted to ask if there is an even bigger infinity? The answer is yes. ℵ₂ can be thought of as all of the set of points, squiggles, clumps of points, and combinations thereof that can be written in a square.

Cantor showed that a bigger infinity can always be constructed by taking the set of all subsets of a lower infinity. In general, there are 2­^M subsets of a set with M  elements. If there are 3 elements in a set, there should be 2­³ =2×2×2=8 subsets. The eight subsets are A, B, C, AB, AC, BC, ABC and the null set.   So a higher infinity than  ℵ_n is the set of all subsets of ℵ_n .  ℵ_n+1 is equal to 2 raised to the power of  ℵ_n.

So there is no biggest infinity! A larger infinity can always be constructed.

Robert J. Marks, “Some infinities are bigger than others but there’s no biggest one” at Mind Matters News (July 8, 2022)

Takehome: In Marks’s view, infinity is a beautiful — and provable — theory in math that can’t exist in reality without ludicrous consequences. (Thus the immaterial human mind is capable of creating things that don’t exist in material reality.)

Here are all five parts — and a bonus:

Part 1: Why infinity does not exist in reality. A few examples will show the absurd results that come from assuming that infinity exists in the world around us as it does in math. In a series of five posts, I explain the difference between what infinity means — and doesn’t mean — as a concept.

Part 2. Infinity illustrates that the universe has a beginning. The logical consequences of a literally infinite past are absurd, as a simple illustration will show. The absurdities that an infinite past time would create, while not a definitive mathematical proof, are solid evidence that our universe had a beginning.

Part 3. In infinity, lines and squares have an equal number of points Robert J. Marks: We can demonstrate this fact with simple diagram. This counterintuitive result, driven by Cantor’s theory of infinities is strange. Nevertheless, it is a valid property of the infinite.

Part 4. How almost any numbers can encode the Library of Congress. Robert J. Marks: That’s a weird, counterintuitive — but quite real — consequence of the concept of infinity in math. Math: Almost every number between zero and one, randomly chosen by coin flipping, will at some point contain the binary encoding of the Library of Congress.

and

Part 5: Some infinities are bigger than others but there’s no biggest one Georg Cantor came up with an ingenious proof that infinities can differ in size even though both remain infinite. In this short five-part series, we show that infinity is a beautiful — and provable — theory in math that can’t exist in reality without ludicrous consequences.

You may also wish to read: Yes, you can manipulate infinity in math. The hyperreals are bigger (and smaller) than your average number — and better! (Jonathan Bartlett)