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Werg22
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Since the positive reals aren't denumerable (or so do I read) and the positive rationals are, then that must mean the positive irrationals aren't. How do we go on about proving this?
Werg22 said:Huh... The whole point is show that P is not denumerable in order to prove that R isn't either... The starting point is P not R.
HallsofIvy said:That strikes me as an awfully clumsy way to prove that R is not denumerable! It's easy to use Cantor's diagonal method to prove that R is not denumerable. That same method wouldn't work with P. (The number you produce by changing the "diagonal" digit might be rational.)
Positive irrationals are numbers that cannot be expressed as a ratio of two integers and are greater than zero. Examples include pi, the square root of 2, and the golden ratio.
A set is denumerable if it can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, ...). This means that every element in the set can be counted and listed in a specific order.
Proving that the positive irrationals are not denumerable is an important concept in mathematics because it helps us understand the size and complexity of different sets. It also has implications in other areas of math, such as calculus and geometry.
One way to prove this is by using a proof by contradiction. Assume that the positive irrationals are denumerable and create a list of all possible positive irrationals. Then, show that there is at least one positive irrational number missing from the list, which contradicts the initial assumption.
Yes, another method is using a diagonalization argument. This involves creating a new number by changing the digits along the diagonal of a list of positive irrational numbers. This new number will be different from every number on the list, proving that the list is incomplete and therefore the positive irrationals are not denumerable.