Philosophy of Everything

This article contains my personal reflections on irrational numbers, especially the informational property of their infinite strings of decimal - strings that often look close to random at a shallow look. It starts with an introduction of known facts, it continues with some less discussed properties and finishes with some philosophical considerations. Introduction Example of irrational numbers are numbers like √2 and ℼ (PI). Irrational numbers cannot be represented as a fraction like, 0.25 is represented by 25/100 or 1/4. Such numbers have an infinite number of decimals that are not repeating any groups we would take. Numbers like 0.(321) are not irrational, even if they have an infinite number of decimals, because they are repeating: 0.321 321 321 321 ... Such numbers with repeating decimal groups can be represented as a simple fraction: 0.321 = 321/999 an therefore are considered "rational". Rational numbers can be represented as fractions (like 234/789), and the...

Please check my informal introduction to "Kolmogorov complexity" if you are not familiar with the subject. In this article I will approach some less discussed ramifications of it and hopefully some new ideas: an example of a Kolmogorov compression that would require an arbitrary high amount of working memory before shrinking to the compressed string (conjecture). 1. Root cause why "Kolmogorov complexity" is uncomputable Kolmogororov complexity is uncomputable because Turing machines have a halting problem :) Let's start with the observation the the Halting problem of a Turing machine having a limited band is actually decidable, unlike the infinite band one. A "finite band Turing machine" is actually a finite state machine that will either stop without going through the same state, or will cycle infinitely. Having a limited number of possible states, it cannot run too much without repeating a previous state. As soon as the machine returns to...

I want to give you an intuition about  Kolmogorov complexity , without being very formal. Basically, Kolmogorov complexity defines a measure of how small can you compress a certain string of characters. You can think about it like measuring how small can you make a file (string of characters) by compressing it with ZIP or similar. A file storing a very redundant information (like "aaaaaaaaaaaaaaaa....") can be compressed in a much smaller size compared with a random one. For calculating the "Kolmogorov complexity", the "compressed" form is considered to be something like a self-extract archive: the shortest program that can output the initial string of characters (the "file/string to be compressed"). The Kolmogorov complexity is more general and includes classical compressing algorithms. For example a Huffman code is not very powerful in compressing a string like "123456789101112131415161718192021222324..." (concatenated consecut...