Problem 10 - Analysis with an introduction to proof

Question

For any set \(A\), define the successor of \(A\), denoted \(S(A)\), by \(S(A)=A \cup\\{A\\}\). One way to "construct" the natural numbers and zero from set theory is by the following correspondence: $$ 0 \leftrightarrow \varnothing, 1 \leftrightarrow S(\varnothing), 2 \leftrightarrow S(S(\varnothing)), 3 \leftrightarrow S(S(S(\varnothing))), \ldots $$ Note that the axiom of infinity guarantees that all of these successor sets exist. Show that \(5 \leftrightarrow\\{0,1,2,3,4\\}\).

Step-by-Step Solution

Answer

In this exercise, we showed that \(5 \leftrightarrow \\{0,1,2,3,4\\}\) by repeatedly applying the successor function to the empty set representing 0. We found that the set representation of 5 corresponds to the set \(\\{0,1,2,3,4\\}\).

Step 1: Represent 0 and 1 using the successor function

According to the correspondence provided, 0 is represented by the empty set, which is denoted by \(\varnothing\). We can directly write it as: \[0 \leftrightarrow \varnothing\] To represent the number 1, we simply apply the successor function to the representation of 0: \[1 \leftrightarrow S(\varnothing)\]

Step 2: Determine the representation of 2, 3, and 4

Now we need to apply the successor function repeatedly to find the set representations of the numbers 2, 3, and 4: 2 \(\leftrightarrow S(S(\varnothing))\) 3 \(\leftrightarrow S(S(S(\varnothing)))\) 4 \(\leftrightarrow S(S(S(S(\varnothing))))\)

Step 3: Define the set representation of 5

Following the pattern, the set representation of the number 5 is: \[5 \leftrightarrow S(S(S(S(S(\varnothing)))))\]

Step 4: Unwrap the set representation of 5

Now, let's unwrap the set representation of 5 by applying the successor function to each element: \(S(S(S(S(S(\varnothing))))) = S(S(S(S(\varnothing )))\cup \\{S(S(S(\varnothing)))\\})\) \(= S(S(S(\varnothing ))\cup \\{S(S(S(\varnothing)))\\})\cup\\{S(S(S(S(\varnothing ))))\\}\) \(= S(S(\varnothing )\cup\\{S(S(S(\varnothing)))\\})\cup\\{S(S(S(S(\varnothing)))),S(S(S(S(S(\varnothing)))))\\}\) \(= S(\varnothing)\cup\\{S(S(\varnothing)),S(S(S(\varnothing)))\\}\cup \\{S(S(S(S(\varnothing)))),S(S(S(S(S(\varnothing)))))\\}\) \(= \varnothing\cup\\{S(\varnothing),S(S(\varnothing)),S(S(S(\varnothing))),S(S(S(S(\varnothing)))),S(S(S(S(S(\varnothing)))))\\}\) \(= \\{0,1,2,3,4\\}\)

Step 5: Show that 5 corresponds to the set {0, 1, 2, 3, 4}

We have now unwrapped the set representation of the number 5 using the successor function and found that it is equal to the set {0, 1, 2, 3, 4}. Therefore, we can conclude that: \[5 \leftrightarrow\\{0,1,2,3,4\\}\]

Key Concepts

Axiom of Infinity

In set theory, the Axiom of Infinity is a fundamental principle that asserts the existence of an infinite set. Simply put, this axiom states that there is at least one set that has no end, allowing us to build an endlessly expanding collection of elements. This concept is crucial when we talk about the natural numbers, as the entire idea of counting is predicated on the infinity of the counting process.

For instance, without the Axiom of Infinity, we wouldn't be able to guarantee that for every natural number, there is always another one that comes after it. This is where the Axiom of Infinity shows its power - it forms the basis that supports the existence of an unbounded sequence, like the natural numbers 0, 1, 2, and so on. It’s like having an infinite ladder, with the Axiom of Infinity assuring us that there will always be another rung to climb.

Set Representation of Natural Numbers

The set representation of natural numbers is a fascinating method of describing numbers using the language of sets. In this system, we start with the most basic set, the empty set denoted by \(\varnothing\), and define each subsequent natural number in terms of the previous one by using the successor function.

For example, following this method, the number 0 is represented by the empty set. The number 1 is then the successor of the empty set, or \(S(\varnothing)\). For 2, we take the successor of the successor of the empty set, expressed as \(S(S(\varnothing))\), and so on. Each natural number is thus associated with a unique set built upon the concept of 'the next' or 'succession'. By using this set-based framework, we give a robust structural backing to the otherwise abstract notion of 'counting'.

Construction of Natural Numbers from Set Theory

The construction of natural numbers from set theory is like building blocks, where each block is placed atop the previous one to reach new heights. In this context, 'heights' are the natural numbers, and our 'blocks' are sets. Starting with the empty set representing zero, we apply the successor function to construct the natural numbers one at a time.

Successor Function and Number Construction

It's important to note that the successor function, denoted as \(S\), takes a set and adds to it the set itself as a new element, like so: \(S(A) = A \cup \{A\}\). By applying \(S\) repeatedly starting from \(\varnothing\), we can construct an entire sequence of natural numbers.

• 0 corresponds to \(\varnothing\)
• 1 corresponds to \(S(\varnothing)\)
• 2 corresponds to \(S(S(\varnothing))\)
• ... and so on

With each iteration, we are effectively 'naming' the next number in the sequence, ensuring a straightforward and logical progression that mirrors the inherent structure of the natural numbers.