为什么自然数集不可判定？

数学中定义数字和在不同基础上排列数字的系统是数字系统。数字系统可以定义为在数轴上表示数字的正确方式。数字系统有不同的基数，已知的基数主要有四种，它们是二进制数字系统、十进制数字系统、八进制数字系统和十六进制数字系统。数学中最常用的数字系统是十进制数字系统，数字属于0-9。十进制系统以 10 为底。根据数字的属性定义了不同类型的数字。让我们学习自然数，

自然数

自然数被定义为从 1 到无穷大的数字。可以说自然数集合中不包含负数或0。自然数也称为正数或可数数。它们被称为可数数，因为在现实生活中计数时只使用正数，例如篮子里有 5 个苹果，这里 5 是自然数。自然数集如下所示，

N = {1, 2, 3, ... ∞}

为什么自然数集是不可判定的？

回答：

Before explaining this problem, it is essential to learn what is undecidable. The term “decidable” means that a particular theory or set of rules has logical meaning or consequences. Undecidable or unsolvable is used for the nature of the infinite and how mathematics can be sustained definitely.

The infinities have various sizes and the theories are based on axioms which are the principles considered to be true and they are proved. These principles are considered to have sets that are complete and are solvable. Natural numbers have a standard form of sets, that is, {1, 2, 3, …N}given by Ernst Zermelo and Abraham Fraenkel. Georg cantor who is the founder of set theory came up with the hypothesis of infinite. He claimed that the set of even infinite numbers could vary from each other.

Therefore, a set of natural numbers which is actually an infinite set can be compared to another infinite set. For example, the set of the real number contained all negative, real, natural numbers, which is far bigger than the set of natural numbers. Are there any other infinities between these two infinities? The scientist was unable to prove this conjecture that there are no other infinities. Hence, it can be said that the set of natural numbers is undecidable or unsolvable.

概念问题

问题1：奇数集和偶数集是自然数集的子集吗？解释如何。

回答：

The set of natural numbers is, N = {1, 2, 3, … ∞}

The set of even numbers can be written as, E = {2, 4, 6, …n}

Where, n is the even number.

The set of odd numbers can be written as, O = {1, 3, 5, … n + 1}

Since, n is even, n + 1 is the odd number.

Therefore, it can be clearly seen that the even and odd sets both are subsets of natural numbers.

问题2：自然数和整数有什么区别？

回答：

Natural numbers generate from 1 and end at infinity while whole numbers generate from 0 and go up to infinity. It can be said that natural numbers and whole numbers are the same if 0 is removed from the whole numbers.

问题3：为什么自然数称为可数数？

回答：

Natural numbers are called countable numbers since these numbers can be used to count objects. For example, there are 50 students in the class. Here, 50 is the natural number and is countable in nature.

问题4：下列数字中哪些是自然数：

12、77、-9、55/3、20。

回答：

Since -9 is negative in nature and 55/3 is not a whole number. These two are not natural numbers. The natural numbers are 12, 77, 20.

问题5：下列哪个不是自然数：

0、11、13/3、14/7、23。

回答：

11, 14/7 = 2, and 23 are clearly seen as the natural numbers. Whereas, 0, 13/3 are not natural numbers.

问题 6：设 X 是自然数的集合，Y 是整数的集合。解释 X – Y 是什么？

解决方案：

X = {1, 2, 3, 4, …}

Y = {0, 1, 2, 3, 4, …}

It is obvious that since whole numbers are a 0 added to the natural numbers, the subtraction of the set of natural numbers from the whole number will provide 0.

Therefore, X – Y = {0}

问题 7：设 X 是自然数的集合，Y 是整数的集合。解释 X ∩ Y 是什么？

解决方案：

X = {1, 2, 3, 4, …}

Y = {0, 1, 2, 3, 4, …}

It is obvious that since whole numbers are a 0 added to the natural numbers, the intersection of the set of natural numbers with the whole number will give the set of natural numbers.

Therefore, X ∩ Y = Y