![]() At the same time, however, there must also be infinitely many real numbers between any two points on the number line, no matter how close together those points may be. It's fairly easy to see that there must be infinitely many real numbers on the number line. In the meantime, there are some other rather interesting things to consider. ![]() What is the first positive real number after zero? You could spend an awful lot of time thinking about that one! But we'll come back to the question of why rational numbers are countably infinite, whereas real numbers are not, later. Imagine, for example, that we want to start counting positive real numbers, starting with zero. Once you really start to think about this question, you begin to realise intuitively why we can't count the real numbers. So why can't we count the reals? Well, the key word here is continuum. But they can nevertheless be counted, and we'll see why in due course. how on Earth do you count rational numbers? I mean, they're fractions. Whole numbers, integers and rational numbers are all countably infinite! In fact, it's not only the natural numbers. For that reason, mathematicians tend to describe the set of natural numbers as being countably infinite. Nevertheless, we could continue to count. ![]() It is of course true that we would never actually finish counting, no matter how long we kept going. When we talk about the set of natural numbers, for example, we know that we can start at one (or zero, depending on how you choose to define the natural numbers) and count to. The word set implies that we are dealing with something that can be counted. “There really is room for creativity.Although we sometimes talk about the set of real numbers, mathematicians more often refer to it in terms of a field or continuum. That would mean that the behavior of the universe - and everything in it - can’t be predetermined, Gisin says. Given the appropriate equations and the conditions of the world, classical physics says, everything can, in principle, be calculated, and therefore, predetermined.īut if the world is described by numbers that have randomness baked into them, as Gisin suggests, that would knock classical physics from its deterministic perch. Standard classical physics, the branch of physics that governs the everyday, human-sized world, leaves no room for free will. Most physicists don’t give much thought to philosophical puzzles like this one, but Gisin’s argument has big implications for the seemingly unrelated concept of free will. After some number of digits, for example, the thousandth digit, or maybe even the billionth digit, their values are essentially random. Instead, Gisin argues March 19 on, only a certain number of digits of real numbers have physical meaning. Such numbers (for example, 1.9801545341073… and so on) contain an infinite amount of information: You could imagine encoding in those digits the answers to every fathomable question in the English language - and more.īut to represent the world, real numbers shouldn’t contain unlimited information, Gisin says, because, “in a finite volume of space you will never have an infinite amount of information.” Gisin - known for his work on the foundations and applications of quantum mechanics - takes issue with real numbers that consist of a never-ending string of digits with no discernable pattern and that can’t be calculated by a computer.
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