Question

Let \(\mathfrak{A}=\langle N,<\rangle, \mathfrak{B}=\langle N-\\{0\\},<\rangle .\) Show: \(\quad\) (i) \(\quad \mathbf{A} \cong \mathfrak{B}\); (ii) \(\mathfrak{a} \equiv \mathfrak{B}\); (iii) \(\mathfrak{B} \subseteq \mathfrak{A}\); (iv) \(\quad\) not\mathfrak \(\mathfrak{B} \prec \mathfrak{A}\).

Short answer

(i) \( \mathfrak{A} \cong \mathfrak{B} \); (ii) \( \mathfrak{A} \equiv \mathfrak{B} \); (iii) \( \mathfrak{B} \subseteq \mathfrak{A} \); (iv) \( \mathfrak{B} \not\prec \mathfrak{A} \).

Step-by-step solution

Define the Structures

Let's first understand the structures. \( \mathfrak{A} = \langle N, < \rangle \) is the set of natural numbers with the usual order. \( \mathfrak{B} = \langle N - \{0\}, < \rangle \) is the set of positive natural numbers with the same order.

Show \( \mathfrak{A} \cong \mathfrak{B} \)

Two structures are isomorphic if there exists a bijection between their sets that preserves the ordering. Define the function \( f: N \to N - \{0\} \) by \( f(n) = n+1 \). This function is a bijection because it maps each natural number \( n \) to a unique element in \( N - \{0\} \), and it preserves order since \( n < m \Rightarrow f(n) = n+1 < m+1 = f(m) \). Thus, \( \mathfrak{A} \cong \mathfrak{B} \).

Show \( \mathfrak{A} \equiv \mathfrak{B} \)

Two structures are elementarily equivalent if they satisfy the same first-order properties. Since they are isomorphic as shown above, they must satisfy the same first-order sentences, hence \( \mathfrak{A} \equiv \mathfrak{B} \).

Show \( \mathfrak{B} \subseteq \mathfrak{A} \)

The set \( N - \{0\} \) is a subset of \( N \) by definition, because each element in \( N - \{0\} \) is a natural number greater than 0, which are elements of \( N \). Thus, \( \mathfrak{B} \subseteq \mathfrak{A} \).

Show \( \mathfrak{B} \not\prec \mathfrak{A} \)

For \( \mathfrak{B} \prec \mathfrak{A} \) to hold, every property true in \( \mathfrak{B} \) must be true in \( \mathfrak{A} \), but not vice versa. However, \( \mathfrak{A} \) contains all the elements of \( \mathfrak{B} \) plus 0, making \( \mathfrak{A} \equiv \mathfrak{B} \). Therefore, \( \mathfrak{B} ot\prec \mathfrak{A} \).

Key concepts

Isomorphism

Isomorphism is a foundational concept in mathematical logic and model theory. Two structures, such as sets or groups, are considered isomorphic if they can be mapped onto each other by a bijection that respects their inherent structure and relations. In simpler words, if you can "relabel" one structure to get the other while preserving all relationships, they are isomorphic.

In our example, we have two structures:

• \( \mathfrak{A} = \langle N, < \rangle \) is the set of all natural numbers with the usual order\( < \).
• \( \mathfrak{B} = \langle N - \{0\}, < \rangle \) represents the set of positive natural numbers, also ordered by the usual \( < \).

To show they are isomorphic, we define a function \( f: N \to N - \{0\} \) that maps each natural number \( n \) to \( n + 1 \). This function is a bijection, meaning it's a one-to-one and onto mapping. Furthermore, if \( n < m \), then \( n+1 < m+1 \), maintaining the order. Therefore, this bijection proves \( \mathfrak{A} \) is isomorphic to \( \mathfrak{B} \). This concept is pivotal in understanding how different mathematical structures can have an identical form or "shape" while possibly labeling their elements differently.

Elementary Equivalence

Elementary equivalence is about two structures satisfying the same set of first-order logic sentences, which are statements describing properties and relations. If two structures are elementarily equivalent, any property expressed in first-order logic that holds in one must hold in the other.

In the given problem, since \( \mathfrak{A} \) and \( \mathfrak{B} \) are shown to be isomorphic, they automatically satisfy the same first-order statements, and hence \( \mathfrak{A} \equiv \mathfrak{B} \).

This occurs because isomorphisms ensure not just a structural similarity but also a logical equivalence in terms of first-order logic. All properties that can be described without needing a second or higher order description are preserved. This is a crucial aspect of model theory, as it extends our understanding of how structures relate beyond just physical appearance to their logical and property-based behavior.

Model Theory

Model theory intersects logic and abstract algebra to study the relationships between formal languages (like that of first-order logic) and their interpretations or "models." A model here refers to a set equipped with additional structure, such as an order or operation, that satisfies a given theory or set of formulas.

In our example:

• \( \mathfrak{A} \) and \( \mathfrak{B} \) serve as models of algebraic structures with the order relation.
• The concepts of isomorphism and elementary equivalence are special cases in model theory, exploring when different models of a theory are essentially the same or when they satisfy the same sentences.

Model theory aids in identifying when two seemingly different structures are indistinguishable in terms of the properties they model. It's powerful in deciphering complexities in theories like set theory, algebra, and beyond by formalizing the idea that appearance can be deceiving and that fundamentally different-looking structures can be closely related.

Ordered Structures

Ordered structures, as seen in the exercise, involve sets with a defined order relationship. Ordered structures help extend our understanding of mathematics by bringing order into otherwise unordered sets, providing a way to handle sequences, limits, and continuity.

In this exercise,:

• \( \langle N, < \rangle \) denotes the natural numbers with their conventional order, starting from 0.
• \( \langle N - \{0\}, < \rangle \) represents positive natural numbers starting from 1, keeping the same order.

Ordered structures enable complex mathematical discussions regarding convergence, functions, and more because they aren't just about the elements but how the elements are arranged or related.

Understanding the order within these structures reveals deeper insights into how systems evolve, interact, and can be manipulated. Whether dealing with number systems, geometries, or other forms of data, ordered structures contribute significantly to mathematical reasoning and problem solving.