In mathematics, there are results that may seem incredible or contradictory at first glance. These paradoxes are not mistakes – rather, they recall that intuition can fail in the field of exact sciences. Consider three of the most unusual mathematical paradoxes.
Hilbert Hotel Paradox
Imagine a hotel with an infinite number of rooms that is always completely busy. However, if you ask each guest to move to a room with a number, a unit is more than its current one, then the first number will be freed, and it will be possible to populate a new guest. Moreover, if an infinite number of new guests arrives, you can invite each guest to double the number of their current number. As a result, all odd numbers will be freed for new guests.
This situation described the German mathematician David Gilbert in 1925 to show that intuitive ideas about infinity are often misleading. In the real world, the statements of “all numbers are busy” and “no one else cannot be placed” means the same thing, but in the world of infinites this is not the case.
This Paradox is well known to many: the likelihood that in a group of 23 people, two of them coincide with a birthday exceeds 50%. This seems unlikely, given that the year is 365 days, but the problem arises due to an improper assessment of the number of pairs of possible coincidences. In a group of 23 people, 253 pairs can be formed, which makes the probability of a birthday coincidence quite high.
This paradox is of practical significance, for example, in cryptography. He demonstrates the risk of “conflicts