**The world is full of paradoxical. **

I know that I know nothing. So once said Socrates.

This statement in itself is ironic, because it demonstrates the complexity of the meaning of a single word.

It also explains the understanding of the vision of the world, one of the founders of Western philosophy: you should question everything you think you know.

Indeed, the deeper you dig, the more paradoxes around you will begin to see.

**1. To get somewhere you must first travel halfway, then half of the remaining half, then half the remaining distance and so on to infinity: so that movement is impossible.**

The dichotomy paradox is considered the brainchild of the ancient Greek philosopher Zeno, which was allegedly created for evidence that the universe is unique and that any change, including motion, is impossible (the same was the opinion of his teacher Parmenides).

People intuitively reject this paradox for the past many years.

From a mathematical point of view, the solution to which came in the 19th century, is to make that half plus one quarter plus one eighth plus one-sixteenth, and so on down to one. This is similar to the number 0,999…., that’ll be a 1.

But this theoretical solution actually does not explain how an object reaches its destination. The solution to this question is more complex and is still not clear, given the theory of the 20th century about the matter, time and space are indivisible.

**2. At any time a moving object is indistinguishable from a stationary, so movement is impossible.**

This paradox is called the paradox of the arrow, and this is another argument of Zeno against motion. The problem here is that at one point in time is 0 seconds, and therefore, the motion in this case is zero.

Zeno argued that if time was made of moments, the fact that the movement is not happening in any particular moment, I would say that it is not happening at all.

As with the dichotomy paradox, the arrow paradox actually hints at modern ideas of quantum mechanics. In the book “Reflections on relativity” (“Reflections of Relativity”) Kevin brown notes that in the context of the special theory of relativity, an object in motion is different from a fixed object.

Relativity requires that objects moving at different speeds, in different ways seemed to an outside observer, and that they themselves had different views about the world.

**3. If you restored the vehicle by replacing all its wooden parts, is it still the same ship?**

Another classic paradox from Ancient Greece, “Ship of Theseus” is a paradox about the contradictions of identity. It is well described by Plutarch.

The ship on which Theseus and the youth of Athens returned from Crete had 30 oars, which were stored until the time of Demetrius Valeria. And all because when the old wooden boards begin to decay, they were replaced by new, more sturdy.

They stayed so long that this ship became a constant topic of discussion among philosophers who talked about the logic of things that change. One group of philosophers said that the ship remained the same, while like other philosophers insisted that after replacing logs, the ship was different.

**4. Can an omnipotent create a rock too heavy that he himself can lift?**

How can there be evil if God is omnipotent? How can we be free if God is omniscient?

These are just a few of the many existing paradoxes concerning the application of logic to questions of the divine subject.

Some people may refer to these paradoxes, thus explaining why they do not believe in a Supreme being. However, others say that they are unimportant and for various reasons do not work.

**5. There is an infinitely long horn which has finite volume but infinite surface area.**

Moving toward the problem, which appeared in the 17th century, we get one of the many paradoxes associated with the geometry and infinity.

“Horn of Gabriel” is formed by taking the curve y = 1/x and rotate around a horizontal axis, as shown in the figure.

Using calculus methods that allow you to calculate areas and volumes thus constructed figures, we can see that the infinitely long horn actually has a finite volume equal to the number PI, but an infinite surface area.

In other words, the horn will fit a certain amount of paint, but to paint the entire surface, you will need an infinite amount.

**6. Heterological word is a word which describes itself. And describes themselves whether the word “heterological”?**

This is one of the many paradoxes that had long haunted the minds of modern mathematicians and logicians.

An example of a heterological word may be the word “verb”, which is not a verb, in fact (in contrast to the “noun” which is a noun). Another example could be the word “long” which is not a long word (as opposed to the word “short”, which is a short word).

So “heterological” is heterologiczne word or not? If it would be a word that does not describe itself, then it would describe itself. But if it had been the word that describes itself, it would not have described himself.

This is due to Russell’s paradox, which asks whether a certain set itself as an element.

Creating such self-destructing variety, Bertrand Russell (Bertrand Russell) and other scholars have demonstrated the importance of establishing careful rules when creating sets, which laid the Foundation for 20th century mathematics.

**7. Pilots can “go” out of combat mode if they are psychologically unfit, but anyone who wants “out” of combat duty, proves him to be normal.**

“Catch -22” – is a satirical novel about the Second world war Joseph Heller (Joseph Heller), which describes the situation when someone needs something that you can only get when he doesn’t need it.

This so-called paradox of samoregulacji. The protagonist of the novel Yossarian is faced with this paradox in the evaluation of pilot activities, but in the end, wherever he went, he always saw a paradoxical and oppressive rules.

**8. In each figure there is something interesting.**

1 is the first nonzero natural number, 2 is the smallest Prime number, 3 is the first odd Prime number, 4 is the smallest composite number, etc. When you finally get to the number that will seem uninteresting, then it will be interesting due to the fact that it seemed to you boring.

The paradox of interesting numbers were based on an inaccurate definition of the word “interesting”, what makes it more stupid variant of the heterological paradox and Russell’s paradox, which rely on controversial somereference.

A quantum computing researcher Nathaniel Johnston (Nathaniel Johnston) has found a clever solution to the paradox. Instead of relying on the intuitive concept of the word “interesting” as in the original paradox, he defined an interesting whole number as such appears in the online encyclopedia of integer sequences.

And that sets of tens of thousands of mathematical sequences like the Prime numbers, Fibonacci numbers, Pythagorean triples, etc.

Based on this definition, the first uninteresting number, the smallest integer that does not appear in any of the sequences – 11 630. As the encyclopedia on a continuous basis, adding new sequences, some of them include the formerly uninteresting numbers.

**9. The bar always has at least one customer for which it is true that if he drinks, then drink all.**

Conditional statements in formal logic are sometimes contradictory interpretations, and the paradox of alcoholism is a good example. At first glance, the paradox assumes that one person makes to drink the rest of the bar.

In fact, all of this suggests that it would be impossible for all at the bar drinking, if every single customer did not drink. Therefore, there is at least one customer (that is, the last who doesn’t drink) that drinking could make it so you could say that I drink everything.

**10. From the ball which can be cut into a finite number of parts, really make the other two ball of the same size.**

The paradox of Banach-Tarski is based on the many strange and contradictory properties of infinite sets and geometric rotations.

Part that you can cut the ball in will look very weird, so the paradox only works in the abstract mathematical realm. It would be great if it was possible to take, for example, an Apple, cut it into pieces and assemble two identical but smaller to share with a friend.

But the physical “balls” of the material world can’t be parsed as a mathematical sphere.

**11. Potato weighing 100 grams is 99 percent water. If it dries up by 1 percent, then its new weight is 50 grams.**

Even when working on outdated methods with finite quantities, math can lead to strange results.

To understand the potato paradox, you need to carefully look at the number contained in the potato water.

Because potatoes is 99 percent water, the dry components is equal to 1 percent. Weight of potatoes – 100 grams, therefore the weight of dry material – 1 gram.

When 100 grams of potatoes dried to 98 percent water, 1 gram of the dry component is converted into 2 grams. A gram is two percent of the 50 grams, and this should be the new weight of the potato.

**12. If a room of 23 people, the chances are very high that at least two of them were born in the same day.**

Another amazing mathematical result: the paradox of a birthday comes from a careful analysis of the associated probabilities.

If the room are two people, then the likelihood is that they have birthday in the same day equals 1/365 (ignoring leap years), because in addition to the birthday of one person in a year there are still 364 other days, any of which may be the birthday of the second person.

If the room is three people, the probability that they all have different birthdays is equal to 364/365 x 363/365. That is when we know the birthday of the first person to choose the dates of birth of the second remains 364 days and for the third – 363 days.

Continuing thus, we reach the number 23 in person, and find that the probability that all people will have different birthdays falls below 50 percent, so the probability of two identical birthdays is significantly increased.

**13. From friends most people have more friends than them.**

It seems impossible, but when you look at the question from a mathematical point of view, everything becomes clear. A good example of this paradox is the social networking in which most people have few friends. But some of them are very sociable people, so friends in them very much.

These people often “seem” as “friends of my friends”, so they raise the average number.

**14. A physicist involved in the invention of a time machine, visiting the “old” version of themselves. This version gives him the idea to create a time machine, and the “young” version uses these ideas to create a direct device, since reverting to an old version of myself.**

Time travel, if possible, can lead to very weird situations.

The Bootstrap paradox is the opposite of the classic grandfather paradox. In order to back and not allow yourself to travel back in time, some of the information and the objects are returned in time, and give you the opportunity later to get back the young version of himself.

And here comes the question: how does the first time appeared the information and object. This paradox was discussed in 1941. Robert Heinlein (Robert Heinlein) was one of the first who raised this issue.

The use of this paradox is not uncommon in science fiction, and the name of the paradox took just from a story by Robert Heinlein.

**15. If on Earth there is nothing unique, then in our galaxy there must be many alien civilizations. However, people have not yet found evidence of other intelligent life in the Universe.**

Some people believe silence is our Universe a paradox. One of the fundamental assumptions of astronomy: planet Earth – this is quite common planet with a common solar system in the shared galaxy, which is not something cosmically unique.

A NASA satellite found that in our galaxy there is probably about 11 billion Earth-like planets. Given this, life, like us, had to develop somewhere not too far from us (at least on a cosmic scale).

But despite the existence of powerful telescopes, people are unable to detect the existence of any technological civilization anywhere in the Universe. Noisy civilization: humanity broadcasts TV and radio signals that are clearly artificial.

Such a civilization as ours, should give signs of its existence, that people would have found if they existed.

Moreover, the civilization that emerged millions of years ago (pretty recent from a space point of view), had enough time to at least begin colonizing the galaxy, meaning that evidence of its existence must be even more.

Indeed, possessing this amount of time, a colonizing civilization would be able to colonize the entire Galaxy. Physicist Enrico Fermi (Enrico Fermi), in whose honor was named this paradox, once during lunch break, colleagues asked: “Where are they?”

One of the solutions to the paradox challenges the above idea and suggests that complex life is extremely rare in the Universe. Another theory claims that technological civilization is inevitably destroyed by nuclear war or ecological destruction.

A more optimistic solution is the idea that the aliens are intentionally hiding from us, until we become more socially and technologically Mature. Another theory is that alien technology is so advanced that we don’t even recognize them.

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