Numbers that are bigger than human imagination
But what about infinity? To the average person, infinity seems like an obvious concept. In fact, from this point of view, infinity is not a number but something that continues forever; But whether the human mind can understand it is another question.
The author and philosopher Edmund Burke wrote in the 1700s that “infinity” tends to fill the brain with a kind of pleasant dread, which is the purest work and the truest test of the sublime. For Burke, the concept of infinity was a combination of fear and wonder, pleasure and pain simultaneously; And except in the world of fantasy, one can rarely face such a feeling in the world of reality.
However, a century later the logician Georg Cantor turned the concept of infinity into something confusing. He showed that some infinities are larger than others; but how? To understand why, think of numbers as sets. If you want to compare all natural numbers (1, 2, 3, 4, etc.) in one set and all even numbers are in another set, then every natural number can be paired with a similar even number. This pair shows that two sets which are both infinite, have the same size or so called “countable infinity”.
However, Cantor shows that this cannot be done for natural numbers and real numbers; Because real numbers have an infinity of intermediate decimal numbers (for example, 0.123, 0.1234, 0.12345, and so on).
If you try to pair numbers with any set, you will always find a real number that is not paired with a natural number. Real numbers are “infinitely uncountable”; Therefore, the infinities are different from each other. In general, it is difficult to accept the above concepts. Just imagine what happens to the mind when it tries to imagine such large numbers.