Robert J. Marks: That’s a weird, counterintuitive — but quite real — consequence of the concept of infinity in math. The Library can be encoded any number of times, including all slightly misspelled variants , in any irrational number:
All documents in English can be reduced to a sequence of ones and zeros. ASCII (American Standard Code for Information Interchange) uses eight bits for each letter, character and punctuation mark in the English language. For example, the letter B is assigned the eight bit number 01000010. The name “Bob” in ASCII is 010000100110111101100010. In generating bits for the irrational number by coin flipping, will we eventually reproduce this ASCII code for “Bob”? It might take some time, but most assuredly the answer is yes. What about the ASCII version of this entire article? Eventually yes. The entire book Moby Dick? The King James version of the Bible? Yes and yes. The entire contents of the Library of Congress?
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. A math nerd would say this is assured with probability one.
What, though, about numbers like ½ =0.5? This doesn’t contain the Library of Congress. But randomly choosing 0.5, equal to 0.100000000000000000…in binary, has a zero chance of occurring. It requires an infinite number of zeros in a row after the first one. That has a zero chance of happening. This is also true of all the rational numbers between zero and one that don’t contain the entire Library of Congress.
The cause of our strange observation is infinity. When infinity is assumed to exist, weird things can happen. Most real numbers require infinite precision to define. And, as discussed, infinity does not exist.Robert J. Marks, “4. How Almost All Numbers Can Encode the Library of Congress” at Mind Matters News
Takehome: Re 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. That’s why infinity is a concept in math but not in the real world. Note: You should ask, how do we come to have concepts that are not part of the real world?
Here are the earlier instalments in this series:
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.
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)