Question

Show that the ordening \(<\), defined by \(x

Short answer

Ordering is transitive: if \(x < y < z\), then \(x < z\) holds in PA.

Step-by-step solution

Understand the Definition of Transitivity

To show that the ordering is transitive, we need to prove \( \forall x y z(x < y \wedge y < z \rightarrow x < z)\)under Peano Arithmetic (PA). The ordering "\(<\)" is given as \(x < y := \exists u(y = x + Su)\)meaning \(y\) is the successor of some number added to \(x\).

Identify What Needs to be Proved

Using the definition of ordering, we need to show that for any natural numbers \(x, y, z\), if \(y = x + S u\) for some \(u\) and \(z = y + S v\) for some \(v\), then there exists a \(w\) such that \(z = x + Sw\).

Express the Hypotheses in Terms of Peano Addition

Given:1. \(y = x + Su\) for some \(u\).2. \(z = y + Sv\) for some \(v\).Our goal becomes proving \(z = x + Sw\) for some \(w\).

Use Associative and Commutative Properties

Substitute the given equations:\(z = (x + Su) + Sv\). Now we simplify using the properties of addition:\[(x + Su) + Sv = x + (Su + Sv) = x + S(u + v)\] due to the properties of associative and commutative laws of addition in Peano Arithmetic.

Conclude With Existence of \(w\)

From \(z = x + S(u + v)\), we can see that there is some \(w = u + v\) such that \(z = x + Sw\), proving the transitivity of the ordering under PA.

Key concepts

Transitive Property

In mathematics, the transitive property is a simple yet powerful concept. It states that if a relation holds between a first and a second element, and also between the second and a third element, then it must hold between the first and third element as well. This is often written as: if \( a < b \) and \( b < c \), then \( a < c \). In Peano Arithmetic, this property is essential because it allows the comparison of natural numbers to be logical and consistent. By proving transitivity in the context of Peano Arithmetic, we establish a reliable framework for reasoning about the order of natural numbers. This is crucial when defining other mathematical properties and performing proofs.

Order Relation

An order relation is a way of arranging elements in a set in a specific sequence. In Peano Arithmetic, the order relation \(<\) is defined by the formula \( x < y := \exists u (y = x + Su) \). This means \( y \) is greater than \( x \) if there exists a natural number \( u \) such that \( y \) equals \( x \) plus the successor of \( u \). This formulation allows us to understand the relation between two numbers based on their differences. By using the successor function, Peano Arithmetic establishes a clear and logical ordering of natural numbers. This order relation is foundational in proving other mathematical concepts, ensuring systematic development of number theory.

Natural Numbers

Natural numbers are the building blocks of arithmetic and number theory. They are the non-negative integers starting from 0, such as 0, 1, 2, 3, and so on. In Peano Arithmetic, natural numbers are defined using a set of axioms, laying the groundwork for all arithmetic operations. These axioms include the existence of a first natural number (0) and the successor function, which helps define the next number in the sequence. Natural numbers are used extensively in various mathematical disciplines. They are essential for counting, order, and constructing more complex number systems. Understanding their properties, such as transitivity and order relations, allows for a deeper appreciation and manipulation of mathematical ideas.

Successor Function

The successor function is a fundamental concept in Peano Arithmetic. It is a function defined on the set of natural numbers that assigns to every natural number \( n \) its immediate follower, denoted as \( S(n) \). This essentially means if \( n \) is a natural number, then the successor \( S(n) \) is \( n + 1 \). This function is crucial because it underpins the process of generating all natural numbers starting from 0. Each number is constructed by successively applying the successor function. The definition of the successor is integral in defining the order relation \(<\), where a number \( y \) is greater than \( x \) if \( y = x + Su \). The successor function, therefore, plays a key role in establishing the basic arithmetic properties like transitivity that are foundational to Peano's Arithmetic.