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Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
Infinity paradoxes
Set theory paradoxes
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Infinity and paradoxes

Galileo's type of paradoxes

Arithmetic with infinite quantities

Paradoxes based on arithmetic with infinity

Paradoxes based on undefined operations with infinity

The Infinite Circle

Galileo's type of paradoxes

In Galileo's scientific work, the "Two New Sciences", the famous Italian scientist wrote about the contradiction with perfect squares.
There are positive integer numbers 1, 2, 3, 4, 5 ... .
Numbers 1, 4, 9, 16, 25 ... are their perfect squares.

Some positive integer numbers are perfect squares, while others are not; therefore, quantity of all positive integer numbers, including both squares and non-squares, must be bigger than just the squares. But for every square there is exactly one number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other.
(This was an early proof by one-to-one correspondence of infinite sets).

Galileo did not resolve this contradiction, and concluded that the ideas of less, equal, and greater did not make sense when applied to infinite sets.
In the nineteenth century, Cantor, using the same methods, showed that ideas of less, equal, and greater applied to infinite sets.

With finite sets, a part is always smaller than the whole. But with infinite sets one part of the set can be just as large as the whole. Often it looks as a paradox, but from the mathematical point of view there is no paradox.
Galileo's paradox
Paradox of even and natural numbers
Paradox of odd and natural numbers

are demonstration of this surprising property of infinite sets.

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Arithmetic with infinite quantities

Infinity is not a real number but the extended real number line. Infinity (infinity) is greater than all other extended real numbers. Minus infinity (--infinity) is less than all other extended real numbers.

Infinity obeys a different arithmetic than finite numbers. In 19-th century Georg Cantor described a way to do arithmetic with infinite quantities. His definition was: a collection is infinite, if some of its parts are as big as the whole.

Infinity and minus infinity have the following arithmetic properties:

Infinity with itself

infinity + infinity = infinity

infinity *infinity = infinity

-infinity * (-infinity) = infinity

-infinity + (- infinity) = infinity

infinity *(- infinity) = - infinity

Undefined operations

0 * infinity = undefined operation

0 * (- infinity) = undefined operation

infinity + (- infinity) = undefined operation

infinity - infinity = undefined operation

(+-infinity) / (+- infinity) = undefined operation

(+- infinity) ** 0 = undefined operation

1 ** (+- infinity) = undefined operation

Notice that

x/infinity =0 is not equivalent to 0*infinity=x.
If the second were true, it would have to be true for every x, and all numbers would be equal. This is what is meant by being undefined, or indeterminate.

Operations involving infinity and real numbers

- infinity < x < infinity

x +  infinity =  infinity

x+(- infinity) = -  infinity

x -  infinity =  infinity

x - (-  infinity) =  infinity

x /  infinity = 0

x / (- infinity) = 0

If then

x *  infinity =  infinity

 x*(- infinity) =  infinity

If then

x *  infinity =  infinity

x * (-  infinity) =  infinity

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Paradoxes based on arithmetic with infinity

Let us look more attentively at the following arithmetic property of infinity:

- infinity < x < infinity

x +  infinity =  infinity

This property explains a famous Paradox of Hilbertís Hotel. Hilbertís Hotel is a hotel with an infinite number of rooms and infinite number of guests. Every room is occupied. A new visitor arrives.

Can he be accommodated?

Yes, arithmetic with infinite quantities allows to do it.

Let x = 1. Then we have 1 + = .

What will be if infinite number of guests arrive? Can they be accommodated?

Yes, they will be accommodated thanks to property

infinity + infinity = infinity.

Hotel is full, and yet it has an infinite number of vacancies.

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Paradoxes based on undefined operations with infinity

The outstanding mathematician J.E. Littlewood described the Ross-Littlewood paradox in 1953. It was the first example of paradoxes based on undefined operation with infinity like . Littlewood himself offered one solution. But playing around undefined operation it is possible to get any wanted result. Paradox of wizard and mermaid and Paradox of enchantress and witch can be seen as other examples with undefined operation.

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The Infinite Circle

The curvature of a circle's circumference decreases as the size of the circle increases. For example, the curvature of the earth's surface is so negligible that it appears flat. The limit of decrease in curvature is a straight line.
An infinite circle is therefore a straight line.

by Tetyana Butler

Complex functions Tutorial
Complex analysis is studying the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers.
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