Problem 3 - A History of Mathematics: An Introduction

Question

Show that Zermelo's axiom of separation resolves Russell's barber paradox as well as the Richard paradox (Exercise 1) in the sense that certain "sets" are now excluded from discussion.

Step-by-Step Solution

Answer

Answer: Zermelo's axiom of separation resolves Russell's barber paradox and the Richard paradox by excluding problematic sets that lead to self-referential contradictions. In the case of Russell's barber paradox, the axiom prevents the formation of a set of all men who do not shave themselves, which would lead to a contradiction. Similarly, in the Richard paradox, the axiom disallows the formation of the set of all real numbers defined by finite English phrases, which would also result in a contradictory situation. By eliminating such sets from set theory, Zermelo's axiom of separation helps prevent paradoxical consequences and maintain consistency in set theory.

Step 1: Understanding Zermelo's Axiom of Separation

Zermelo's axiom of separation states that for any proposition P and any set S, there exists a set S' containing all elements of S that satisfy the proposition P. In other words, given a set S, we can form a subset S' by selecting the elements of S that meet a specific condition. Mathematically, this is represented as: S' = {x ∈ S | P(x) is true}.

Step 2: Understanding Russell's Barber Paradox

The Russell's barber paradox is a logical paradox formulated by the philosopher Bertrand Russell. It considers a barber who shaves all men in a village who do not shave themselves and only these men. The paradox arises when we try to determine whether the barber shaves himself or not. If he does, then he must not, because he only shaves men who do not shave themselves. But if he does not, then he must, because he shaves all men who do not shave themselves.

Step 3: Applying Zermelo's Axiom of Separation to Russell's Barber Paradox

The paradox arises due to our attempt to define a set of all men who do not shave themselves, which leads to a contradiction. With Zermelo's axiom of separation, we can avoid this paradoxical consequence by recognizing that the set cannot be consistently defined. The proposition "x does not belong to x" (i.e., a man who does not shave himself) leads to a contradiction when we try to define a set R for which: R = {x | x ∉ x}. Since this set cannot be consistently defined, Zermelo's axiom of separation excludes it from being a valid set in set theory, thus resolving the paradox.

Step 4: Understanding the Richard Paradox

The Richard paradox is another self-referential paradox, proposed by the French mathematician Jules Richard. It involves defining a real number R by considering a list of all real numbers that can be defined by finite English phrases, ordered alphabetically. The paradox arises when considering the real number R obtained by adding 1 to the nth digit of the nth real number in the list and dividing by 10. This real number R, although defined by a finite phrase, must not be in the list because its nth digit is different from the nth digit of the nth real number in the list. This leads to a contradiction since it would mean R should be in the list and not in the list at the same time.

Step 5: Applying Zermelo's Axiom of Separation to the Richard Paradox

In this case, Zermelo's axiom of separation helps resolve the Richard paradox by recognizing that the set of all real numbers that can be defined by finite English phrases cannot be consistently defined as a set. This is because attempting to create a set based on this definition leads to a self-referential contradiction, similar to the barber paradox. By excluding such problematic sets from discussion in set theory, Zermelo's axiom of separation allows us to avoid contradictions, thus resolving both Russell's barber paradox and the Richard paradox.

Key Concepts

Zermelo's Axiom of Separation

Zermelo's Axiom of Separation is a fundamental principle in set theory. It allows for the creation of subsets from any given set, based on a specified condition. Given any set \( S \) and a proposition \( P(x) \), the axiom states that there exists a subset \( S' \) comprised of all elements \( x \) in \( S \) that satisfy \( P(x) \). For example:

• If \( S \) represents all animals in a zoo, and \( P(x) \) is the property of having stripes, then \( S' \) would be the subset of all striped animals.
• In mathematical terms, \( S' = \{ x \in S \mid P(x) \text { is true} \} \).

This axiom is crucial for resolving some of the paradoxes that arise in set theory by ensuring that only well-defined sets are discussed.

Russell's Barber Paradox

Russell's Barber Paradox is a self-referential puzzle that was introduced by the philosopher Bertrand Russell. It centers around a barber in a village who shaves all those who do not shave themselves. The paradox comes into play when determining if the barber shaves himself.

• If the barber shaves himself, he should not, because he only shaves those who do not shave themselves.
• If he does not shave himself, then he should, because he shaves those who do not shave themselves.

This riddle creates a logical loop, making it impossible to solve under conventional assumptions. However, Zermelo's Axiom of Separation can help avoid such contradictions by preventing the formation of a set defined in this manner, thus keeping logical consistency intact.

Richard Paradox

Introduced by French mathematician Jules Richard, the Richard Paradox explores the complexities of self-reference in the realm of mathematics. It involves constructing a list of all real numbers definable by finite English descriptions and creates a new number, \( R \), with specific properties.

• For each real number, change the \( n \)th digit and redefine it.
• Although \( R \) is described through a phrase of finite length, it paradoxically cannot belong to the original list.

Zermelo's Axiom of Separation helps navigate this conundrum by excluding the creation of such inconsistent sets from mathematical discourse. This action precludes contradictions from arising through paradoxical set definitions.

Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. It forms the foundation of modern mathematical analysis and provides a unified framework for numbers and functions. Set theory is used in:

• Defining numbers, sequences, and functional spaces.
• Explaining concepts in topology, algebra, and mathematical analysis.

A central component of set theory is the axiomatization of sets, which aims to prevent paradoxes like the ones associated with self-reference by rigorously specifying the properties and rules for forming sets. Zermelo's Axiom of Separation plays a pivotal role by helping to guide the construction of acceptable and logically sound sets.

Logical Paradoxes

Logical paradoxes are statements or problems that defy conventional logic by resulting in contradictions or seemingly unresolvable dilemmas. Common characteristics include self-reference, infinite regress, or circular reasoning. Two famous examples are Russell's Barber Paradox and the Richard Paradox.

• These paradoxes often highlight limitations within certain logical systems or set definitions.
• Resolving them typically involves redefining the criteria or systems with principles like Zermelo's Axiom of Separation.

By addressing and resolving logical paradoxes, mathematicians refine concepts and advance the boundaries of theoretical understanding, ensuring consistency and coherence in mathematical systems.