{"id":47,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/8-reductio-ad-absurdum\/"},"modified":"2025-10-13T18:07:53","modified_gmt":"2025-10-13T18:07:53","slug":"8-reductio-ad-absurdum","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/8-reductio-ad-absurdum\/","title":{"raw":"Reductio ad Absurdum","rendered":"Reductio ad Absurdum"},"content":{"raw":"

8.1 \u00a0A historical example<\/h2>\r\nIn his book, The Two New Sciences<\/span>,[10]<\/a><\/sup>\u00a0Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities or actual infinitesimals. One of his arguments can be reconstructed in the following way. \u00a0Galileo proposes that we take as a premise that there is an actual infinity of natural numbers (the natural numbers are the positive whole numbers from 1 on):\r\n

{1,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a02,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a03,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a04,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a05,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a06,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a07, \u00a0\u2026.}<\/p>\r\nHe also proposes that we take as a premise that there is an actual infinity of the squares of the natural numbers.\r\n

{1,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a04,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a09,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a016,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a025,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a036,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a049, \u00a0\u2026.}<\/p>\r\nNow, Galileo reasons, note that these two groups (today we would call them \u201csets\u201d) have the same size. \u00a0We can see this because we can see that there is a one-to-one correspondence between the two groups.\r\n\r\n\r\n\r\n\r\n\r\n
{1,<\/td>\r\n2,<\/td>\r\n3,<\/td>\r\n4,<\/td>\r\n5,<\/td>\r\n6,<\/td>\r\n7,\u00a0\u2026.}<\/td>\r\n<\/tr>\r\n
\u00a0\"arrow\"<\/td>\r\n\u00a0\"arrow\"<\/td>\r\n\u00a0\"arrow\"<\/td>\r\n\"arrow\"<\/td>\r\n\u00a0\"arrow\"<\/td>\r\n\u00a0\"arrow\"<\/td>\r\n\u00a0\"arrow\"<\/td>\r\n<\/tr>\r\n
\u00a0{1,<\/td>\r\n\u00a04,<\/td>\r\n9,<\/td>\r\n16,<\/td>\r\n25,<\/td>\r\n36,<\/td>\r\n49, ...}<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf we can associate every natural number with one and only one square number, and if we can associate every square number with one and only one natural number, then these sets must be the same size.\r\n\r\nBut wait a moment, Galileo says. \u00a0There are obviously very many more natural numbers than there are square numbers. \u00a0That is, every square number is in the list of natural numbers, but many of the natural numbers are not in the list of square numbers. \u00a0The following numbers are all in the list of natural numbers but not in the list of square numbers.\r\n

{2,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a03,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a05,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a06,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a07,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a08,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a010, \u00a0\u2026.}<\/p>\r\nSo, Galileo reasons, if there are many numbers in the group of natural numbers that are not in the group of the square numbers, and if there are no numbers in the group of the square numbers that are not in the naturals numbers, then the natural numbers is bigger than the square numbers. \u00a0And if the group of the natural numbers is bigger than the group of the square numbers, then the natural numbers and the square numbers are not the same size.\r\n\r\nWe have reached two conclusions: \u00a0the set of the natural numbers and the set of the square numbers are the same size; and, the set of the natural numbers and the set of the square numbers are not the same size. \u00a0That\u2019s contradictory.\r\n\r\nGalileo argues that the reason we reached a contradiction is because we assumed that there are actual infinities. \u00a0He concludes, therefore, that there are no actual infinities.\r\n

8.2 \u00a0Indirect proofs<\/h2>\r\n
\r\n

Key Terms<\/p>\r\n\r\n<\/header>\r\n

\r\n