数理思维数理思维

論ParadoxA paradox is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition. Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together.Zeno’s Paradox( 諾 論):Imagine the Greek hero Achilles in a footrace with a plodding reptile. Because he is such a fast runner, Achilles graciously allows the reptile a head start of a hundred meters. Suppose each racer runs at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred meters, bringing him to the reptile's starting point; during this time, the reptile has "run" a (much shorter) distance, say one meter. It will then take Achilles some further period of time to run that distance, in which said period the reptile will advance farther; and then another period of time to reach this third point, while the reptile moves ahead. Thus, whenever Achilles reaches somewhere the reptile has been, he still has farther to go. Therefore, swift Achilles can never overtake the reptile. Explanation:As the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. It does not necessarily require an infinite amount of time to traverse an infinite sequence of distances.Hilbert’s Paradox( 論):In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. Now, imagine a hotel with an infinite number of rooms. Suppose an infinite number of guests came along and all the rooms are occupied. When there comes a new guest, you may move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., then you can fit the newcomer into room 1. Therefore the hotel is not full.Explanation:In an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent(the pigeonhole principle does not hold for infinite sets). If the cardinal numbers of two infinite sets are the same, then their cardinal numbers remain the same if a finite set is added into one of them.Cantor’s Paradox( 論):We have learned that from any set S we can construct another set2S–the set of all subsets of S -with larger cardinal number. Now consider the set S of all sets. Its cardinal number should be the largest that can possibly exist. However, the set of all subsets of S mush have a larger cardinal number than the set S itself. Hence, there must be a larger cardinal number than the largest one.Gödel’s Paradox( 論):On May 4, 1934, A makes the single statement “Every statement that A makes on May 4, 1934 is false.”This statement cannot be true because it says of itself that it is false. But it also cannot be false because, to be false, A would have had to make a true statement on May 4, 1934. But he made only the one statement.Russell’s Paradox(羅 論) :Each concept determines a distinction between those objects to which it applies and those to which it does not. For instance, the concept of a fruit applies to an apple but not to a potato. Sometimes a concept happens to apply to itself. For example, the concept of abstract notion is itself an abstract notion. This is clearly a rather exceptional situation. Therefore, a concept will be called normal if it does not apply to itself.Consider the concept of normal concept and we ask if it does or does not apply to itself.Suppose it does. Then it is a normal concept and so, because of the definition of normal concept, it does not apply to itself.Therefore we are forced to conclude that our concept does not apply to itself. But then, by the very same definition, it is normal and so it clearly does apply to itself. This contradicts our first conclusion.Another version of Russell’s ParadoxBarber Paradox (理 論)裡 理 不 不理 不不不Berry’s Paradox (or The Word Paradox 數 論)數 例 12 年 數 數數 數 不 數 句 數 不句 數 句 了 六 了數The Liar’s Paradox說句句 句There is no rule without an exception.( 例 )The only thing to fear is our own fear.( )The only lesson we learn from history is that men do not learn from the lessons of history.(歷 歷 )不不(heterological word) 不(heterological) 不。