Transition Year Maths 17. Infinity and Paradoxes


Mathematics cannot be applied to everything, and cannot explain everything. A paradox is a set of statements which contain contradictions. The existence of paradoxes causes problems for mathematical logicians which they cannot solve.


Russell’s Paradox


An English man called Bertrand Russell (1872 – 1970) tried to show that all mathematics could be reduced to pure logic, but he came across a contradiction which is now known as Russell’s Paradox. An example of Russell’s Paradox:


20 people live on an island. Some people cut their own hair, the rest use the barber who only cuts the hair of people who do not cut their own hair. 10 people never cut their own hair. Draw a Venn diagram of the people on the island:


Let A = people who cut their own hair Let B = people who do not cut their own hair A


B


The circles are separate because there is no overlap (or intersection) between the two groups of people.


In which circle is the barber?


He cannot be in circle B because he only cuts the hair of people who do not cut their own hair. Is it possible to work out how many people cut their own hair on this island? If the statements at the start of the example are true, it is impossible to work out how many people cut their own hair. This is a paradox.


PROJECT 17.1 Hair Cutting / Venn Diagram


20 people live on an island; 10 cut their own hair, 3 people sometimes cut their own hair, but otherwise use the barber. The rest always use the barber, who only cuts the hair of those who do not cut their own hair. Discuss this problem in small groups of 2 – 4 and try to agree on a Venn diagram with 2 overlapping circles showing in which circle each of the 20 people should be placed. Is it possible? In which area is the barber?


Paradoxes like these have led to philosophies of numbers, such as intuitionism. An intuitionist will claim that there are statements which are neither true nor false.


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