I have an idea about the uniformity of infinity between $aleph_0$ and $aleph_1$, motivated by directly disputing Cantor’s proof which I learned of in my discrete mathematics class. I’m going to skip all the formalization of this proof that I’ve written out for brevity’s sake, but I’m rather convinced of it. Surely, I cannot be right, so I’m looking for some criticism on where it is wrong.

The formalized version states that if we prove $(0,1)$ is countably infinite, then $(-infty,infty)$ is. I can explain my reasoning for this if you’d like.

Now this proof relies on Byers’ demonstration I learned about in grade-school that a repeating radix-point number can be represented in two equivalent ways:

$$

x=0.overline1_2text{; definition}\

10_2x=1.overline1_2text{; multiply both sides by radix}\

10_2x=1_2+0.overline1_2text{; decompose into sum substituents}\

10_2x=1_2+xtext{; substitute}\

x=1_2text{; subtract both sides by }x\

$$

The generation of the sequence $(0,1)$ we’ll use to run against Cantor’s diagonal proof is written out with three parts here in a brief, not particularly rigorous manner. Hopefully it’s still rigorous enough to be convincing — if you’re not convinced with the first two steps, just start at step 3, since the first two steps are just contextual. It’ll be done in radix-two, or binary, for consistency with the above demonstration.

Step 1: Enumerate 0 and all positive integers in ascending order — prepend leading zeros for uniformity’s sake. Use each value as the row index.

- $…00000$
- $…00001$
- $…00010$
- $…00011$
- $…00100$

Step 2: Treat each number like a string of symbols are reverse it.

- $00000…$
- $10000…$
- $01000…$
- $11000…$
- $00100…$

Step 3: Prepend bicimal point. Recall that trailing zeros are insignificant.

- $0.00000…$
- $0.10000…$
- $0.01000…$
- $0.11000…$
- $0.00100…$

Run this through Cantor’s diagonal test to generate a unique number (ignoring everything before the bicimal point). Note that the last digit in this sequence is $0.overline1_2$, which is $1$. There are two numbers you can arrive at: $0.overline10$ and $0.overline1$, depending on how you want to view “reaching the end”. For the former, you can say that Cantor’s proof relies on their being an infinitesimal in order to generate a unique number, which means the series is countable in the first place. For the latter, that number is the final number in our sequence by definition: $1$, so it’s not unique.

A caveat of this method is that irrational and transcendental numbers cannot currently be found in the series in a finite number of steps due to their nature of not being perfectly expressible in a radix-point number. I’m not particularly supporting the idea that there really is only one form of infinity, but more disputing Cantor’s proof stating that there is. The call in this is that Cantor’s proof isn’t rigorous, and that proving different infinities requires a definition for the set of irrational and transcendental numbers that is not based on other sets.

Despite the lack of rigorousness in this post, I still would like all criticism, including that of the lack of rigor, and ways to better articulate what I’m saying. Feel free to submit edits or comments and I’ll revise the post.

Most importantly, is there anything in the literature that discusses this idea already?