# 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,  ℵ0.  ℵ is the Hebrew letter aleph. The number of points on a line segment, a bigger infinity, is denoted by ℵ1.  We are immediately prompted to ask if there is an even bigger infinity? The answer is yes. ℵcan 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 2­M 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  ℵis the set of all subsets of ℵ.  ℵn+1 is equal to 2 raised to the power of  ℵn.

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)

## 3 Replies to “Robert J. Marks: Some infinities are bigger than others but there’s no biggest one”

1. 1
Silver Asiatic says:

Thus the immaterial human mind is capable of creating things that don’t exist in material reality.

I often ask myself, “Am I being fair by immediately thinking that this or that particular scientist, scholar or research is mentally ill? That the person is demented or otherwise profoundly stupid?” Usually, I’ll answer “yes” – it’s not a fair assessment. They might have had a bad day or maybe they’re a lot more intelligent than it seems.

Researcher Almira Osmanovic Thunström, a Swedish researcher at Gothenburg University, sought to get a whole research paper out of GPT-3 and publish it in a peer-reviewed academic journal.
After GPT-3 completed its scientific paper in just 2 hours, Thunström began the process of submitting the work and had to ask the algorithm if it consented to being published.
“It answered: Yes,” Thunström wrote. “Slightly sweaty and relieved (if it had said no, my conscience could not have allowed me to go on further), I checked the box for ‘Yes.'”

But in this case, I’m on solid ground. She’s insane.
With “sweaty palms” she asked the software. Is she going to ask GPT-3 to take her out on a date?
Her guilty conscience would have afflicted her if GPT-3 had said “No”.
Atheism is a form of idolatry. People will end up venerating and making appeals to some kind of god or another.

2. 2
Querius says:

To the OP, yes. some infinities are larger when dividing by smaller values of zero. (smirk)

-Q

3. 3
Querius says:

Or to put it another way, some series get larger, smaller, expand, or contract faster than others. Certainly rates of change to a series can differ this way, but any mathematical infinity is infinity as in unbounded.

The way Cantor and other mathematicians argue deals with the size of sets and 1:1 associations between different sets–Hilbert’s hotel comes to mind. But the concept of a bounded set (bounded by definition, not by number of elements) is where the paradox originates. One can also look at it this way: {any function} compared to {any other function}.

This perspective is the mathematical equivalent of dividing by zero: so is 6/0 larger than 3/0? And this is where L’Hôpital’s genius emerges.

-Q