A paradox is a true result that is surprising to our human sensibilities. These are the kinds of things on the list you provide. There is nothing wrong with paradoxes of this sort.
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Indeed, having our intuition turned on its head is in my opinion one of the great things about mathematics. Logical contradictions are not permissible in mathematics, since one can derive any statement true or false from a contradiction. Contradiction which is naturally derived from assumptions, e. Russell's paradox, or Cantor's paradox. These are sentences which exhibit the inconsistency of a definition. For example Burali-Forti shows that there is no set which contains "all the ordinals".
Therefore any system of definitions in which every collection is a set will be inconsistent granted we can define an ordinal in this system, which is something we expect to be able to do in a reasonable set theory.
Paradoxes in Mathematics
Counterintuitive results which show how strange mathematics can be. In such model we can partition the real numbers into more parts than sets.
Yes, we can cut a set into a strictly greater number of parts than we have elements to partition. Sounds strange? Well, this is just one of the things that can go wrong when not assuming the axiom of choice! In classical logic logical paradoxes are not allowed but in paraconsistent logic they are allowed.
The Role of Paradoxes in the Evolution of Mathematics | Mathematical Association of America
There are several types of paradox. There is one that we may call a phenomenological paradox, one where the mathematical results contradict basic truths about what the mathematics is supposed to model. For example, The Banach-Tarski paradox can be considered such a paradox. Then there are logical paradoxes, that is a statement that is provably both true and false. In classical logic where every statement is either true or false, but not both it can easily be shown that a logical contradiction entails every other statement and thus if a logical contradiction exists in a logical system then that system is quite useless.
Therefore, paradoxes must be banished. Phenomenological paradoxes are banished by either fine-tuning the axioms so that the paradoxical result no longer follows or by accepting the result as true and announce that our intuition has been refined. Logical paradoxes are usually resolved by carefully fine-tuning the axioms or by somehow disallowing the paradoxical creatures away from the discussion.
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No , there isn't. Within sound mathematics, one cannot obtain inconsiatnt results. Then again, we simply cannot prove consistency of mathematics and it has been proven that we cannot, at least not if it is true. Thus - paradoxically - yes , there is room in mathematics for logical paradoxes, though it is just some tiny little corner and we know there is none, but we cannot prove that. You can quibble about what precisely is meant by the term "paradox," but if I understand your meaning, the so-called liar paradoxes can be seen as logical paradoxes.
Bertrand Russell classified known logical paradoxes into seven categories. Ball, W. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp.
Base rate fallacy
Bunch, B. Mathematical Fallacies and Paradoxes. New York: Dover, Carnap, R. Introduction to Symbolic Logic and Its Applications. Church, A. New York: Rowman and Littlefield, p. Curry, H. Foundations of Mathematical Logic. Assume now that the school only admits students who are at band 9 or 10 in at least one of the skills. If we look at the whole population, the average academic rank of the weakest sportsperson and the best sportsperson are both equal 5. However, within the set of admitted students, the average academic rank of the elite sportsperson is still that of the whole population 5.
This is where unexpected trends can occur through random chance alone in a data set with a large number of variables. Recommended Celebrating pi - our favourite non-whole number Mathematics. When looking at many variables and mining for trends, it is easy to overlook how many possible trends you are testing.
While each pair is extremely unlikely to look dependent, the chances are that from the half million pairs, quite a few will look dependent. In a group of 23 people assuming each of their birthdays is an independently chosen day of the year with all days equally likely , it is more likely than not that at least two of the group have the same birthday. People often disbelieve this, recalling that it is rare that they meet someone who shares their own birthday.
If you just pick two people, the chance they share a birthday is, of course, low roughly 1 in , which is less than 0. So by looking across the whole group you are testing to see if any one of these pairings, each of which independently has a 0. These many possibilities of a pair actually make it statistically very likely for coincidental matches to arise.
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