Question

Let \(\mathfrak{A}=\langle\mathbb{N},<\rangle\) and \(\mathfrak{B}=\langle\mathbb{N}, \Delta\rangle\), where \(n \Delta m\) iff (i) \(n

Short answer

\(\sigma = \exists n\exists m (n < m \land \neg(n \Delta m))\).

Step-by-step solution

Understand the Structure \(\mathfrak{A}\)

The structure \(\mathfrak{A} = \langle \mathbb{N}, < \rangle\)\ represents the natural numbers with the usual ordering relation "less than". In this structure, the relation '<' functions as it typically does in mathematics, where for natural numbers \(n\) and \(m\), \(n < m\) if \(n\) comes before \(m\) on the number line.

Understand the Structure \(\mathfrak{B}\)

The structure \(\mathfrak{B} = \langle \mathbb{N}, \Delta \rangle\)\ is defined such that \(n \Delta m\) if either (i) both \(n\) and \(m\) are even and \(n < m\), or both are odd and \(n < m\), or (ii) \(n\) is even and \(m\) is odd. This relation separates natural numbers based on their parity and order.

Formulate a Sentence \(\sigma\)

We need to find a sentence \(\sigma\) such that it is true in \(\mathfrak{A}\) but false in \(\mathfrak{B}\). Consider the property that there exists at least one pair of numbers \(n\) and \(m\) such that \(n < m\). This statement is always true in \(\mathfrak{A}\).

Test \(\sigma\) in \(\mathfrak{B}\)

The sentence "There exist \(n\) and \(m\) such that \(n < m\)" should be false under \(\Delta\). In \(\mathfrak{B}\), given the definition of \(\Delta\), it is possible to construct such pairs. Thus, we revise \(\sigma\) to focus on parity differences, e.g., "There exist numbers with differing parities cumulatively ordered across \(<\)" which cannot hold ubiquitously in \(\Delta\) across all configurations.

Ensure Distinction

To ensure \(\sigma\) is true in \(\mathfrak{A}\) but false in \(\mathfrak{B}\), consider: \( \exists n \exists m (n < m \land eg(n \Delta m)) \). This captures that there exists an order pair in \(\mathfrak{A}\) which cannot be justified by \(\Delta\) due to parity conditions, potentially false in \(\mathfrak{B}\) due to inhibition across all pair types.

Key concepts

Order Relation

Order relations play a crucial role in mathematics as they define how numbers relate to each other in terms of size. In our exercise, we are dealing with two structures, \(\mathfrak{A}\) and \(\mathfrak{B}\). In \(\mathfrak{A} = \langle \mathbb{N}, < \rangle\), the natural numbers are organized with the usual 'less than' order. This essentially means that for any two natural numbers \(n\) and \(m\), \(n < m\) if \(n\) is less on the number line than \(m\). Understanding an order relation is foundational for grasping other operations and properties of numbers. For instance, order relations help determine sequences, establish bounds, and suggest convergence in more complex mathematical constructs. They simplify how we analyze numeric progressions by creating an intuitive hierarchy based on size.

Natural Numbers

Natural numbers are the simplest set of numbers used in mathematics, primarily for counting. They start from 1 and continue indefinitely: 1, 2, 3, and so on. Natural numbers form the backbone of our numerical understanding. In our models, both \(\mathfrak{A}\) and \(\mathfrak{B}\) use the set of natural numbers, \(\mathbb{N}\), as their base domain. This domain is crucial because it provides a straightforward framework for structuring various mathematical concepts.When we discuss natural numbers in terms of order relations, we're often talking about how these numbers relate to each other in terms of greater than, less than, or equal to. These numbers and their operations lay the groundwork for more complex numbers and operations encountered in higher mathematics.

Parity

Parity refers to whether a number is even or odd. In our exercise, this is a vital concept because \(\Delta\), the relation used in \(\mathfrak{B}\), heavily depends on the parity of natural numbers. An even number, such as 2, 4, or 6, can be divided by 2 without a remainder, whereas an odd number, like 1, 3, or 5, has a remainder of 1 when divided by 2.In \(\mathfrak{B} = \langle \mathbb{N}, \Delta \rangle\), a specific relation exists depending on whether numbers are even or odd. This emphasizes the significance of parity in creating logical distinctions in mathematical sets. When we define \(\Delta\) based on parity, it influences which number pairs satisfy this relation, providing a unique characteristic to this structural model.

Logical Structures

Logical structures in mathematics help us understand how different elements within a set relate based on defined rules or properties. In the context of this exercise, \(\mathfrak{A}\) and \(\mathfrak{B}\) serve as logical structures, each employing a different foundational relational approach: one with a simple order relation, and the other with parity-influenced relations. The idea is to find a logical statement that holds in some circumstances (such as in \(\mathfrak{A}\)) but not in others (like in \(\mathfrak{B}\)), highlighting distinctions between these structures. Logical structures are deeply tied to model theory by affecting truth values of statements depending on internal relations and partitions of the set. They provide insights into how assumptions and relations influence conclusions within mathematical systems, fostering a deeper understanding of theoretical principles.