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If I can pair them such that everything in list A has a pair in list B , but not vice-versa, then A is no larger than B , but it might still be the same size if the lists are infinite.
The lists [1, 2, 3] and [x, y, z] are the same size because I can pair them up [ 1, y , 2, x , 3, z ].
The positive integers and the negative integers are the same size because I can pair them up x , — x for any x. The list of all rational numbers fractions is the same size as the list of all integers, shown by interleaving digits pairs with Well, [1, 2, 3] is not the same size as [x, y]: no matter how I pair things up, there is always an extra number with no letter.
At first blush the list of integers appears to be larger than the list of positive integers since I can pair all the positives and leave all the negatives unpaired. But recall that just means that the list of positive integers is no larger than the list of integers. Is there a list that is larger? Georg Cantor presented several proofs that the real numbers are larger. The most famous of these proofs is his diagonalization argument.
Any real number can be represented as an integer followed by a decimal point and an infinite sequence of digits. Now we need to show that all pairings of infinite sequences of digits to integers of necessity leaves out some infinite sequences of digits. Thus, if one of our pairings was 17, 0. An example:. The number z above is a real number between 0 and 1 and is not paired with any positive integer. Since we can construct such a z for any pairing, we know that every pairing has at least one number not in it.
It is not clear that I care how many real numbers there are. However, Cantor diagonalization can be used to show all kinds of other things. For example, given the Church-Turing thesis there are the same number of things that can be done as there are integers. More simply, there are more problems than there are solutions. Diagonalization is so common there are special terms for it. A list that can be shown to be larger than the list of integers is called uncountably infinite , while lists that are the same size as the integers are countably infinite.
It feels like sleight of hand, some kind of trick. Let me try to outline some of the ways it could be a trick. In the end, whether you accept diagonalization or not is up to you. The majority of theoreticians in the world seem to accept it; indeed, not accepting it can earn a bit of ridicule.
When all is said and done, a proof is just a social construct, a particular kind of persuasive argument. Accept it only if it convinces you. Licensed under Creative Commons:. Diagonalizations Everywhere It is not clear that I care how many real numbers there are.
Is diagonalization wrong? Numbers from mathematics have symbolic definitions. Either way, every real number I can ever encounter can be expressed finitely, either by a finite description of defining equations or a finite precision real-world measurement. And if they have finite expressions, then they are countable.
Cantor's diagonal argument
A mong various philosophical categories, the finite and the infinite are of pivotal importance. The relationship between them is so close, sophisticated, and profound that neither can be perceived and studied without appealing to the other. Both notions are the subject of extensive mathematical research. Modern set theory is entirely devoted to the study of the most general properties of finite and infinite families of objects. A set of objects is finite if there exists a one-to-one correspondence, or bijection, between the set and some natural number n.
Cantor Diagonal Method
In set theory , Cantor's diagonal argument , also called the diagonalisation argument , the diagonal slash argument or the diagonal method , was published in by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. The diagonal argument was not Cantor's first proof of the uncountability of the real numbers , which appeared in Diagonalization arguments are often also the source of contradictions like Russell's paradox   and Richard's paradox. In his article, Cantor considered the set T of all infinite sequences of binary digits i. He begins with a constructive proof of the following theorem:. Next, a sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s 1 swapping 0 s for 1 s and vice versa , the 2nd digit as complementary to the 2nd digit of s 2 , the 3rd digit as complementary to the 3rd digit of s 3 , and generally for every n , the n th digit as complementary to the n th digit of s n. For the example above, this yields:.
Cantor’s Diagonalization Method
The Cantor diagonal method, also called the Cantor diagonal argument or Cantor's diagonal slash, is a clever technique used by Georg Cantor to show that the integers and reals cannot be put into a one-to-one correspondence i. However, Cantor's diagonal method is completely general and applies to any set as described below. Given any set , consider the power set consisting of all subsets of. Cantor's diagonal method can be used to show that is larger than , i.
Cantor's Diagonal Argument