Positive integer set {1, 2,3, ...} and positive even set {2,4,6, ...} are evenly matched!
That is, the number of elements in the set is the same!
We can count positive even numbers in an infinite set by one-to-one correspondence with natural numbers:
For every positive integer, there is an even number corresponding to it. For any even number, there is a positive integer. From this point of view, the two sets now look the same size, that is, evenly matched.
What's going on here? (In fact, this unique feature of infinite set was put forward by Galileo in 1683, so it is sometimes called Galileo paradox. )
Georg Cantor (1845- 19 18), a great mathematician, was born in St. Petersburg and is famous for establishing set theory.
If the elements in a set can correspond to natural numbers one by one, then we say that the set is countable. If we can sort or list the elements in a set in some way, then the set is countable, because any list can be marked, that is, every item is related to the natural number 1, 2, 3, ... All finite sets are of course countable. The real problem comes from infinite sets.
Cantor pointed out in a paper "On the Properties of Real Algebraic Number Sets" published in 1874 that integers, rational numbers and even algebraic numbers are countable.
As we know, algebraic number is the solution of algebraic equation, and the general formula of algebraic equation is where n is a positive integer and ai is an integer. For any algebraic equation, add all the absolute values of the coefficients (ai) and n, and we call the obtained value the height of the equation. For a certain height (such as 5), there are finite equations, and each equation has at most n solutions. So all algebraic numbers can be arranged according to their high solutions. Therefore, algebraic numbers are countable.
What should I do if I exceed the quantity? Can transcendental numbers be listed in a table in some way? This seems highly improbable! We didn't even check whether a particular number is a transcendental number!
What about real numbers with algebraic numbers and transcendental numbers? Are real numbers countable?
1874, Cantor proved that algebraic numbers are countable, and he also proved that real numbers are uncountable.
Cantor finally realized that there are at least two kinds of infinity: countable infinity and uncountable infinity, that is, the infinity of natural numbers and the infinity of continuum.
There are two different infinite potentials in front of us: one is suitable for natural numbers, rational numbers and algebraic numbers; Another potential applies to real numbers and continuum.
The difference between countable infinity and uncountable infinity has proved to be extremely useful, and even imagining a simple infinity is shocking enough.
Cantor made other amazing discoveries when he explored infinite sets. He found that we can establish a one-to-one correspondence between the continuum (real numbers on a straight line) and points on a plane or even points in N-dimensional space.
189 1 year, Cantor published another proof about uncountable real numbers, and since then, this proof has been jaw-dropping. Cantor's proof involves sets rather than numbers. This idea is called diagonal proof, diagonal process, diagonal argument or diagonalization.
In 1895, Cantor chose to add the subscript 0 to the first letter of the Hebrew alphabet to represent the radix of a set of countable natural numbers (and therefore any countable infinite set). Cantor called it the first transition.
Cantor proved that the cardinal number of the continuum is:
Cantor proved that the elements of any nonempty set cannot correspond to the elements of its power set. This fact is obvious for finite sets, but not for infinite sets. This conclusion is now called Cantor theorem, and it is also the main result of the paper introducing diagonalization skills in 189 1. Just as a set has a power set, a power set can have its own power set, and so on. All these sets have different cardinality.
Cantor's continuum hypothesis is expressed in mathematical language as follows:
The profound significance of all this lies in that the cardinal number of countable sets is not only smaller than that of Lian's sequel,
It is very, very, very small:
In fact, the only difference between a continuum and a countable set is whether it contains transcendental numbers. This shows that there are many transcendental numbers, accounting for the vast majority of real numbers.
Including some practical mathematical proofs in Turing's thesis, the core problem lies in the difference between countable sets and uncountable sets, as shown in the following figure.