![]() In other words, you could assume the continuum hypothesis to be true and never encounter a contradiction. ( Kurt Gödel proved that there is such incompleteness in every meaningful formulation of mathematics in 1931.) They are provably unprovable: you can neither prove nor disprove the conjecture by the usual mathematical means. ![]() As it turns out, the continuum hypothesis belongs to the statements that elude our basic mathematical framework. It states that there is no set whose cardinality lies between that of the natural numbers and that of the real numbers.īut Cantor could not prove his assumption-nor could anyone else. Finding no such set, the mathematician formulated his famous “continuum hypothesis” in 1878. Just how large is the set of real numbers? Cantor asked himself this same question when he was investigating whether a set of numbers existed that would be larger than the natural numbers but smaller than the real numbers. A Provably Unprovable Assumptionīesides the cardinality ℵ 0 of the natural numbers, there is at least one more (uncountable) infinity-and it might be a better choice than ℵ 0 in the competition to find the largest number. Cantor could therefore conclude that there are “uncountably many” real numbers. Thus, the list cannot be complete, which contradicts the original assumption. It does not appear in the list because it always differs in at least one digit from each number listed. In this way, an irrational number with an infinite number of decimal places will be obtained. (If you hit a 9 you can change the value to a 0.) To set up Cantor's Diagonal argument, you can begin by creating a list of all rational numbers by following the arrows and ignoring fractions in which the numerator is greater than the denominator. Credit: Buckyball Design Source Wikipedia For the third, you increase the third decimal place of the third number by 1, and so on. The second decimal place is obtained by calculating the second decimal place of the second number plus one-that is, 5. He did it in the following way: the first decimal place of the new number corresponds to the first decimal place of the first number in the list plus one-that is, 4 in the above example. But Cantor demonstrated that he could construct another number between 0 and 1 that does not appear in the list. If we can truly count all of the values, our list must contain every real number between 0 and 1. The only important thing is that it is complete. It is sufficient to assume that all real values between 0 and 1 are countable (which, we will soon see, is wrong).Īccordingly, you could write all these values in an infinitely long list, with one below the other. To proceed, you do not have to consider all real numbers. The Infinity of Real Numbers Exceeds Natural NumbersĬantor proved this fact with his second “diagonal argument.” This is a proof by contradiction: you start with the assumption that there are countably infinite real numbers and derive a contradictory statement from this idea (that is, “There cannot be countably infinite real numbers”). If, in addition to the rational numbers, you also allow irrational values, such as the square root of 2, pi or Chaitin’s constant, then the set suddenly becomes so large that you can no longer enumerate its elements-even if the list is infinitely long. The sets presented so far all have the same cardinality. So instead of submitting the answer “infinity” to the largest numbers competition, you could offer ℵ 0. The cardinality of the natural numbers is called “countably infinite” and is represented by ℵ 0 (spoken as “aleph zero”). The smallest infinity owes its name to this fact. That means you could-at least theoretically-number the elements of these sets (if you had infinite time and leisure). At the end of the 19th century, mathematician Georg Cantor laid a foundation for the mathematical concept of infinity by thinking about quantities and their size. It took humankind several millennia to realize this idea and cast it into a neat theory. Thus, infinity would not be a guaranteed winner in a largest number contest. Furthermore, there are differences even with infinite values: infinity does not always equal infinity. ![]() ![]() Therefore, a statement such as ∞ + 1 makes no sense. For example, the number line is infinite, regardless of whether you start it at –∞, 0 or 1. The answer is no one because infinity is not an ordinary number that follows the usual rules of calculation. What about ∞ + 1, ∞ 2 or ∞ ∞? If people put forward these replies to the largest natural number question, who would be right? But even if we allowed infinitely large values, this response would cause problems. What is the largest natural number possible? By using the word natural, I have ruled out the possibility that you simply answer infinity (∞) to torpedo the guessing game.
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