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Why infinity is unreachable
Infinity is defined as a number that is not reachable in terms of finite numbers. This is true even if we only consider the smallest infinity, aleph-0. However, it always feels as though if you go “just a little further” each time, you will reach it. You will not.
Imagine a number line where aleph-0 is a finite distance along the line to the right. Now consider any point. If it is even a minute distance to the right of 0, aleph-0 is a finite multiple of it. Hence that number is infinite. If all numbers to the right of 0 are infinite, where do the positive finite numbers go? They must naturally all be crammed into where 0 is. Thus, no matter how large your number is, be it 1, 2, Graham’s Number, or TREE (3), you might as well stay at 0. (This argument was due to Sbiis Saibian)
Now imagine an “aleph-0-beast”. It lives in a space where aleph-0 is finite, like on the number line in the previous example. What would it think of us and our natural numbers? Here’s what it might say. “So let me contemplate what 10 billion is. Let me add it to a basic unit, aleph-0. I still get aleph-0. On the contrary, adding aleph-0 to aleph-0 gives a new, bigger number. [It is actually not if we consider it in terms of real numbers, but it will soon be apparent that even though 1≠2, to the beast they are the same]. So really 10 billion is 0! So is this 1 and 2. Now 2 = 0 and 5 = 0. Also, 0+0 = 0. Therefore, 2+2=5.” Actually, that makes perfect sense. To imagine it, think of the reals in this case not as we think of reals, but the way we think of infinitesimals. Consider the following proof that 0.9 recurring = 1.
- Let 1 - 0.9999... = x.
- Multiply both sides by 10 to get 10-9.9999... = 10x.
- Add 9.9999... - 10x to both sides to get 10 - 10x = 9.9999...
- Subtract 9 from both sides to get 1 - 10x = 0.9999...
- Now replace 0.9999... with 1 - x to get 1 - 10x = 1 - x.
- Multiply by - 1 and add 1 to get 10x = x.
- Subtract x so 9x = 0.
- Divide by 9 and x = 0.
- Now add 0.9999... so 0.9999... = 1 - 0 = 1.
That makes sense from the real-number perspective. However, to an infinitesimal beast, it would be as amusingly wrong as 2+2=5 was to us. The mistake started from step 5. That 0.9999... is a different one to them, as the discrepancy between that and 1 is ten times as great. For them, it’s true that x is less than all real numbers, but it is not 0. To imagine this, it is obvious to us that 1 ≠ 0, even though it is less than all infinite numbers.