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...

" P equals NP? "  is a million dollars unsolved problem . Introduction " P" is the class of problems where we have algorithms that solves the problem in polynomial time. " NP"  is the class of problems having solutions that can be proven in polynomial time once you "guess" the solution. This is analogous with the sorting problem: it's easier to check the correct sorting of a list than to actually sort a list. Of course, if you can fully solve the problem in polynomial time, you can also prove it in polynomial time, so P is included in NP. However, NP seems to contain some very hard problems that cannot be currently solved in polynomial time. It is believed that NP contains some problems that can never be possibly solved in P. I will not do a formal proof, more like an argument or a sketch. You might have more patience and skills to develop some of these ideas even if they might have flaws in the current form. You can take it a...