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Why is the Set of Natural Numbers Undecidable?

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A system in mathematics defining numbers and arranging numbers on different bases is the number system. A number system can be defined as the proper way of representing numbers on the number line. Number system has different bases and there are majorly four types of bases known, they are, binary number system, decimal number system, octal number system, and hexadecimal number system. The most commonly used number system in mathematics is the decimal number system, the digits belonging from 0-9. The decimal system has base 10. There are different types of numbers defined based on the properties of numbers. Let’s learn about natural numbers,

Natural numbers

Natural numbers are defined as the numbers that generate from 1 and go up to infinity. It can be said that no negative numbers or 0 are involved in the set of natural numbers. Natural numbers are also known as positive numbers or countable numbers. They are called countable numbers since while counting in real life, only positive numbers are used, for instance, there are 5 apples in the basket, here, 5 is a natural number. The set of natural numbers is shown as,

N = {1, 2, 3, … ∞}

Why is the set of natural numbers undecidable?

Answer:

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.

Conceptual Problems

Question 1: Are odd number set and even number set are the subsets of natural number set? Explain how.

Answer:

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.

Question 2: What is the difference between natural numbers and whole numbers?

Answer:

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.

Question 3: Why natural numbers are called countable numbers?

Answer:

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.

Question 4: Which of the given below numbers are natural numbers:

12, 77, -9, 55/3, 20.

Answer:

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.

Question 5: Which of the following is not a natural number:

0, 11, 13/3, 14/7, 23.

Answer:

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

Question 6: Let X be the set of Natural numbers, and Y be the set of the whole numbers. Explain what will be X – Y?

Solution:

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}

Question 7: Let X be the set of Natural numbers, and Y be the set of the whole numbers. Explain what will be X ∩ Y?

Solution:

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


Last Updated : 30 Dec, 2023
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