When you see a green apple, can you conclude that all crows are black? If the Sun 4 billion years ago did not shine as brightly as it is now, why did not the earth's oceans of that era freeze? These and other paradoxes continue to excite lovers of logic and science.

Since ancient times, paradoxes have fascinated scientists and amateurs, stirring up the imagination and causing incessant controversy. Some of them only seem paradoxical, since the answers to them contradict common sense, others have not yet been resolved or cannot be resolved in principle.

## Maxwell's Demon

We are talking about a thought experiment with the help of which the great physicist James Maxwell showed the possibility of violating the second law of thermodynamics - one of the fundamental laws of modern science.

Imagine a vessel divided by an impenetrable septum into two parts - right and left. The partition has a hole with a door. The vessel is filled with gas with an undefined temperature.

Maxwell proposed a mental device (the so-called "demon") that opens a hole in order to let only molecules moving at an above average speed from the left side of the vessel to the right side. Thus, the demon divides the vessel into two zones: warm - with fast gas molecules, and cold - with slow ones.

This means that the entropy of a closed system has decreased, which contradicts the second law of thermodynamics. However, if you take a closer look at the model, it turns out that the proposed system is not closed. Indeed, for the implementation of such a demonic device, in reality, an additional supply of energy from the outside is required.

In 2010, Maxwell's thought experiment was even brought to life by the efforts of physicists from the University of Tokyo.

## Thompson lamp

The Thompson lamp paradox belongs to the class of supertasks, infinite sequences arising in a certain order of actions over a finite period of time. It was invented by the British philosopher of the 20th century James F. Thompson.

Imagine a desk lamp with a power off button. Let's say we turn on the lamp for a minute, then turn it off for 30 seconds, then turn it on again for 15 seconds, and so on, each time halving the time it takes to turn the lamp on and off. The question arises, will the lamp be turned on or off after 2 minutes?

It is impossible to give an answer to this paradox, because following the exact logic of the experiment, we must endlessly turn on and off the lamp without reaching the appointed time.

## The problem of two envelopes

This paradox has long been known to mathematicians, but in its current form it was only formulated in the 1980s. It consists of the following:

Two players are given one envelope each. Each of them contains a certain amount. It is only known that the amount of money in one envelope is twice the amount in the other. Then the players are given the opportunity to exchange envelopes.

Which is more profitable: keep the envelope you received or exchange it with your opponent? At first glance, both options are equally likely.

The paradox arises when the following reasoning arises: Let's say I have a sum of X in my hands. Another player may have an equally likely sum equal to 2X or X / 2. Therefore, in the case of an exchange, I will have the amount (2X + X / 2) / 2 = 5X / 4, that is, more than now. But in the case of an exchange, the same situation will arise - it will again become more profitable to take someone else's envelope, and from the point of view of both players.

## Boy or girl?

Suppose there are two children in a family, and one of them is a boy. Assuming that the probability of having a boy is 1/2, what are the chances that the second child will also be male?

The answer suggests itself intuitively: 50%. However, in reality, the odds are 1/3. There are three possibilities in total: an older brother and a younger sister, an older sister and a younger brother, and an older brother and younger brother. All three possibilities are equally probable, so the odds of each are 1/3.

However, this answer causes fierce controversy among mathematicians. Critics believe that, in fact, it is impossible to find an unambiguous solution to the problem if it is not known how exactly the information about this family was obtained.

## The crocodile's dilemma

The authorship of this ancient Greek sophism is attributed to Corax, and it consists in the following:

The crocodile snatched the baby from the mother and, in response to her pleas, asked her to guess whether he would return the baby to her or not. If the mother answers correctly, the child will be returned to her.

The paradox arises if the mother answers: "No, you will not return my child to me."

Now, in the case of the return of the baby, it turns out that the parent did not guess, therefore, the crocodile should have kept the child for himself. If the crocodile decides not to return the child, therefore, the mother told the truth, and he should have fulfilled his promise.

A stalemate arises in which the crocodile cannot return the child and cannot keep him. Of course, only if we are talking about a crystal honest speaking reptile.

## The weak young sun paradox

According to the generally accepted model of stellar evolution, 4 billion years ago, our Sun emitted 30% less energy than it does now. This means that the Earth in that epoch heated up much less, and the water on its surface should have frozen.

However, according to geological studies, our planet at that time was covered by oceans, and its climate was humid and warm. Some scientists refer to the possibility of a greenhouse effect, but in this case, the level of carbon dioxide and methane in the atmosphere should have exceeded the current one by hundreds and thousands of times. No evidence of this has ever been found.

## Hempel's paradox

The paradox, proposed by the German mathematician Karl Hempel in the 1940s, is also known as the "raven paradox."

It begins with the statement: "All crows are black." This sentence is logically equivalent to the theory: "All non-black objects are not crows."

Every time an observer sees a black crow, the first sentence gets empirical confirmation. When he sees a non-black object, for example, a green apple, he receives confirmation of the second statement.

The paradox arises from the equivalence of the two theories. Those. in fact, seeing a green apple gives us empirical evidence that all crows are black. However, this conclusion contradicts our feelings.

Observing non-black objects can increase our confidence that such objects are not ravens, but we do not get additional evidence of blackness of all ravens.