Mathematical Induction, Transfinite Induction, and Induction Over The Continuum

Abstract


Introduction
In conducting mathematical activities, reasoning in mathematics plays a fundamental role as the foundation or method for performing mathematics. The commonly known methods are deductive reasoning and inductive reasoning. Deductive reasoning is a process where specific facts are derived from a generally known truth, while inductive reasoning in mathematics is based on the principle of induction. The principle of induction asserts that a universal truth is demonstrated by showing that if a specific case is true and subsequently a sufficient number of cases are also true (Singh, Yadav:2017). An example of inductive reasoning in mathematics is prominently seen in mathematical induction.
Mathematical induction states that a statement holds true for every natural number ∈ ℕ if it satisfies the following properties: The statement is true for = 1; usually denoted as (1) is true.
(ii). If the statement is true for = , then the statement is also true for = + 1; typically written as "if ( ) is true, then ( + 1) is also true." Mathematical induction turns out to be inadequate as an inductive method used in mathematics for certain cases related to the concept of true infinity (actual infinity), such as the properties of transfinite arithmetic in set theory and the Heine-Borel property in real line topology. To overcome this issue, an inductive method in mathematics that goes beyond mathematical induction needs to be developed in order to obtain valid truths about many mathematical problems related to true infinity. The inductive methods to be discussed in this article are transfinite induction (Goldrei: 1996) and induction over the continuum (Kalantari: 2007).

Method
The method used in this research is a literature review.

Discussion
This section will begin with mathematical induction. According to (Hungerford: 1974), the principle of mathematical induction is as follows: The principle of Mathematical Induction. If is a subset of the set of natural numbers denoted by ℕ and satisfies the property 0 ∈ , and one of the following statements holds true: The principle of mathematical induction operates within set theory. This means that when proving that = ℕ, the set must satisfy certain properties. Various literature describes the principle of mathematical induction, with Hungerford providing a more specific formulation, as follows: Mathematical Induction. If is a subset of the set of natural numbers denoted by ℕ, which satisfies the following properties: (ii). ∈ ⟹ + 1 ∈ for all ∈ ℕ. then = ℕ.
In Hungerford's formulation of mathematical induction, the set of all natural numbers denoted by ℕ is not initiated from the number 1, but rather from the number 0. However, there is a difference in mathematical induction in logic, where the set of all natural numbers denoted by ℕ starts from the number 1. The mathematical induction in the logical version can be expressed as follows (Herstein: 1986): Mathematical Induction. Statement holds true for every natural number ∈ ℕ if it satisfies the following properties: (i). Statement is true for = 1; usually denoted as (1) being true.

(ii).
If statement is true for = , then statement is also true for = + 1; typically written as "if ( ) is true, then ( + 1) is also true." Then, an example of the usage of logical version mathematical induction is as follows: Example 1.1. Prove that for every natural number , the following equality holds: 1 + 2 + 3 + ⋯ + = ( + 1) 2
Assume that ( ) holds true for = .
Next, we will examine transfinite induction as one of the accepted inductive methods in mathematics, applied to a statement that holds true for all ordinal numbers (both finite or infinite ordinal numbers, as well as transfinite ordinal numbers). Before introducing what transfinite induction is, we first introduce the concept of ordinal numbers and some of their properties, referring to (Jech: 2003).

Definition 1.2. Let be a set and < be a relation on P. The relation < is said to be a partial order if
it satisfies the following properties: (i). ≮ for every ∈ (the relation < is anti-reflexive).
And the pair ( , <) is called a partially ordered set.
Next, the concept of special elements in a partially ordered set ( , <) will be introduced. Definition 1.3. Given a partially ordered set ( , <) and X a non-empty subset of , and ∈ , then: is called a maximal element of if ∈ and ∀ ∈ , ≮ .
(ii). is called a minimal element of if ∈ and ∀ ∈ , ≮ .
(iii). is called the greatest element of if ∈ and ∀ ∈ , ≤ .
(iv). is called the least element of if ∈ and ∀ ∈ , ≤ .
is called the supremum of if a is the smallest element among all upper bounds of .

(viii). is called the infimum of if a is the largest element among all lower bounds of .
In a partially ordered set ( , <), the ordering relation < is said to be linearly ordered or totally ordered if ∀ , ∈ one of the following holds < or = or > .
Definition 1.4. If the ordering relation < on a partially ordered set ( , <) is linearly ordered and every non-empty subset of has a least element, then the set is said to be well-ordered.
Next, the idea to construct ordinal numbers is by introducing a relation on them; meaning that given two ordinal numbers and , the partially ordered relation < is defined as follows: < ≝ ∈ from this concept, an understanding of transitive sets is obtained.

Definition 1.6. A set is called an ordinal number if it is both transitive and well-ordered.
Usually, ordinal numbers are denoted by , , , … and the class of all ordinal numbers is denoted by .
From the concept of ordinal numbers, further constructions of ordinal numbers are made, such as finite ordinals and transfinite ordinals.
Constructing finite ordinal numbers.
Next, the concept of the successor of a set X will be introduced as follows: The set + will be called the successor of the set .
Through the concept of the successor of a set, finite ordinal numbers will be defined as follows:  Ordinal numbers are classified into two categories as follows: (i).

Successor ordinal numbers. An ordinal number is called a successor ordinal if there
exists an ordinal number such that = + = + 1.
Examples of successor ordinal numbers include all finite ordinal numbers, as well as + 1, + 2. + 3, … and so on.
Just like in mathematical induction, which provides a condition for a statement to hold true for all natural numbers, a similar concept is introduced to examine the validity of a statement that applies to all ordinals. This concept is known as transfinite induction. According to (Andre: 2018), transfinite induction is defined as follows: Transfinite Induction. Let { | ∈ } be a class indexed by ordinals, and let be a property of elements. Suppose ( ):"element satisfies property " and for every ∈ the following holds: The definition of transfinite induction above can be simplified as follows:

Transfinite Induction. A statement is true for all ordinal numbers ∈ if it satisfies the following properties:
(i). The statement P is true for = 0; usually denoted as (0) being true.
(iii). If is a limit ordinal ( = sup{ | < } and is true for all with < , then is also true for

; It is usually denoted as if ( ) is true for all with < , then ( ) is also true.
Some examples of statements that can be proven true using transfinite induction are the properties of transfinite arithmetic. Transfinite arithmetic refers to arithmetic operations involving both finite and transfinite ordinals.

Transfinite Arithmetic.
(i). Addition of ordinal numbers. Given two ordinal numbers, and , the addition of these two ordinal numbers is defined as follows: Thus, ( ) is true for = 0.
From (i), (ii) and (iii), it is evident that the statement ( ) is true for all ordinal numbers .
Some properties of transfinite arithmetic, specifically those related to the exponentiation of ordinal numbers, can be proven using transfinite induction. One of these properties states that for any three ordinal numbers , and , we have + = . .
Define the following set: We will prove that = [ , ) using induction over the continuum.
It is evident that is continuous at .
Consider and given above.
By induction over the continuum, we conclude that = [ , ).
A similar argument applies to the continuity of at as well.

Conclusion
The conclusion of this article is that mathematical induction can be applied to a statement that holds true for all natural numbers. Furthermore, transfinite induction can be applied to a statement that holds true for all ordinal numbers. However, induction over the continuum cannot yet be applied to a specific statement, but it can be said that something is proven true if it holds for every point in the interval [ , ).
The suggestion that can be drawn from this article is to develop a new method of induction in mathematics in addition to the three types of induction discussed earlier.