Question

Let \(\mathfrak{A}_{1}=(\mathbb{N}, \leq)\) and \(\mathfrak{A}_{2}=\langle\mathbb{Z}, \leq\rangle\) be the ordered sets of natural, respectively integer, numbers. Give a sentence \(\sigma\) such that \(\mathfrak{A l}_{1} \models \sigma\) and \(\mathfrak{A}_{2} \models \neg \sigma\). Do the same for \(\mathfrak{A}_{2}\) and \(\mathfrak{B}=\langle\mathbb{Q}, \leq\rangle\) (the ordered set of rationals). N.B. \(\sigma\) is in the language of posets; in particular, you may not add extra constants, function symbols, etc., defined abbreviations are of course harmless.

Short answer

Use least element property for naturals vs. integers, and density property for integers vs. rationals.

Step-by-step solution

Understand the Problem

We need to find a property (sentence \(\sigma\)) that holds true in the natural numbers (\(\mathfrak{A}_{1}\)), but not true in integers (\(\mathfrak{A}_{2}\)). Similarly, find another property such that it holds in integers but not in rationals (\(\mathfrak{B}\)).

Identify a Property for Naturals vs. Integers

In \(\mathfrak{A}_{1} = (\mathbb{N}, \leq)\), the least element exists, which is 0. In \(\mathfrak{A}_{2} = (\mathbb{Z}, \leq)\), there is no least element. We can use the property of having a least element.

Formulate the Sentence for Naturals vs. Integers

Construct the sentence \(\sigma\) expressing "there exists a least element", say \( \forall x (0 \leq x) \). This holds in naturals but not in integers, as there's no least element in integers.

Identify a Property for Integers vs. Rationals

In integers \( \mathfrak{A}_{2} = (\mathbb{Z}, \leq) \), every two elements are either equal or distinct. In contrast, between any two different rationals, there's always another rational, showcasing density.

Formulate the Sentence for Integers vs. Rationals

Construct \(\sigma\) depicting "there do not exist two distinct elements with nothing in between them", such as \( \forall x \forall y (x < y \to \exists z (x < z < y)) \). This holds in rationals but not in integers.

Key concepts

Natural Numbers

Natural numbers, denoted as \(\mathbb{N}\), represent a sequence of positive integers starting from zero and extending infinitely, meaning they are the most basic set of numbers we deal with. Natural numbers are crucial in everyday counting and ordering activities. Here are some important traits:

• They include numbers like 0, 1, 2, 3, and so on.
• Every natural number has a unique successor, which makes them countable.
• The set begins with a least element, which is 0, especially relevant in ordered sets.

When we consider natural numbers as an ordered set, the presence of a definite starting point makes it unique. This characteristic differentiates them from other sets like integers or rationals, which do not have an obvious least element.

Integer Numbers

Integer numbers, represented as \(\mathbb{Z}\), expand the concept of natural numbers by including negative numbers. Integers combine both positive and negative counting numbers along with zero. This inclusion gives the set of integers a broader range and definition:

• Include numbers like -2, -1, 0, 1, 2, and so forth.
• They have no smallest or greatest element, as they go infinitely in both directions.
• Integers remain discrete; between two consecutive integers, there are no integers.

In the context of ordered sets, the absence of a least element is a distinctive trait for integers. Unlike natural numbers, integers can extend indefinitely without a starting boundary, making sets like \(\mathbb{Z}\) display different ordered properties, such as the lack of a unique smallest element.

Rational Numbers

Rational numbers, denoted as \(\mathbb{Q}\), form a set of numbers that can be represented as fractions of integers, such as \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\). This definition allows rational numbers to form a dense set. Some key points about rational numbers include:

• Can be positive, negative, or zero, much like integers.
• They fill the number line completely between any two numbers, no matter how small the interval.
• Between any two rational numbers, there exists another rational number, which embodies their densely packed nature.

In ordered sets, the density of rational numbers sets them apart from integers. Every gap you perceive in integers is filled by rationals, highlighting their continuity and helping rational numbers build robust comparisons in mathematical scenarios.