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 (
)
is greater than all other extended real
numbers. Minus 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
Undefined operations
Notice that
is not equivalent to 
.
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
If 
then
If 
then
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Paradoxes
based on arithmetic with infinity
Let us look more attentively
at the following arithmetic
property of infinity:
This property explains
a famous Paradox
of Hilbert’s Hotel.
Hilbert’s 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
.
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.
