Recently, I found myself in a frustrating yet amusing situation with an internet service provider (ISP) that will remain unnamed. They promised me a certain price for their services, but ended up charging me more. This discrepancy reminded me of the Zermelo-Fraenkel set theory, a foundational system in modern mathematics known for its precision and logical rigor.
For those unfamiliar, Zermelo-Fraenkel set theory is a collection of axioms that underpin much of modern mathematics. To grasp its essence, you only need to know two things: it exists, and it uses a unique way to define numbers. For instance, the number 2 is represented as “the set that contains the set that contains only the set containing nothing as well as the set containing nothing.” It’s a bit of a mathematical joke, much like this entire story.
The first axiom states that two sets are equal if they have the same elements. However, my ISP offered the same services at different prices, violating this axiom. Despite this, I persisted in my quest to get the original price.
The second axiom implies that a set cannot be a member of itself. Yet, the ISP claimed that their best offer was the only option available, contradicting this principle. When I asked to speak to a manager, it was like saying, “Your logical system is flawed.”
My goal was simple: get the internet service for the promised price of $40 instead of $50. The manager offered a new deal: internet plus a router for $45. When I asked if I could take the deal without the router, they said no, violating the third axiom, which allows for the creation of subsets.
The ISP seemed to have mastered the fifth axiom, which involves combining sets, as they called it “bundling.” However, they stumbled over the seventh axiom, the axiom of infinity, which is more a critique of mathematics than ISPs. In our finite universe, nothing can be infinite, not even their lack of customer service.
The manager’s breakdown of the $45 offer into $40 for internet and $5 for the router revealed another violation. The possible service combinations should include all possible subsets, known as the power set, but they didn’t.
Despite these logical inconsistencies, I remembered that mathematical truths depend on the axioms we accept. So, I asked if I could take the $45 option and return the router to avoid the extra charge. The ISP representative’s response was music to a mathematician’s ears: “I can’t tell you you can’t do that.”
This story, partly based on real events, was first shared at the festival of Bad Ad-Hoc Hypotheses (BAHFest), where the aim is to entertain and enlighten about the workings of science. For more entertaining stories, you can explore Audible, the sponsor of this video. Audible offers a vast selection of audiobooks, including “How Not To Be Wrong” by Jordan Ellenberg, a witty exploration of using math to avoid mistakes.
Engage in a role-play exercise where you and your classmates simulate a customer service interaction between an ISP and a customer. Use the axioms of Zermelo-Fraenkel set theory to guide your dialogue. Try to identify and discuss any logical inconsistencies in the ISP’s responses.
Design a comic strip that humorously illustrates the parallels between the ISP scenario and Zermelo-Fraenkel set theory. Focus on the axioms and how they are violated in the story. Share your comic with the class and discuss the mathematical concepts depicted.
Participate in a debate where one side defends the ISP’s pricing strategies while the other side uses set theory to argue against them. Prepare your arguments by identifying specific axioms that are violated and propose logical solutions.
Write a short story or skit that incorporates elements of set theory into a real-world scenario. Use humor to highlight the absurdity of certain situations, similar to the ISP story. Present your story to the class and discuss the mathematical principles involved.
Create an interactive quiz that tests your understanding of Zermelo-Fraenkel set theory and its application to the ISP scenario. Include questions that require identifying violations of axioms and propose alternative solutions. Share the quiz with your peers for feedback.
Set – A collection of distinct objects, considered as an object in its own right. – In mathematics, a set is often used to group numbers or objects together, such as the set of all even numbers.
Theory – A coherent group of propositions formulated to explain a group of facts or phenomena in the natural world and repeatedly confirmed through experiment or observation. – The theory of relativity revolutionized our understanding of space, time, and gravity.
Axioms – Basic assumptions or self-evident truths that serve as the foundation for a system of reasoning or a mathematical theory. – Euclidean geometry is based on five fundamental axioms that define the properties of space.
Numbers – Mathematical objects used to count, measure, and label. – Complex numbers are an extension of the real numbers and include the imaginary unit i, where i² = -1.
Elements – Individual objects or members of a set. – In the set of natural numbers, each number is considered an element of that set.
Subsets – A set that contains some or all elements of another set. – The set of all even numbers is a subset of the set of all integers.
Infinity – A concept in mathematics that describes something without any bound or larger than any natural number. – In calculus, limits are used to describe the behavior of functions as they approach infinity.
Combinations – Selections of items from a larger pool, where the order of selection does not matter. – In probability theory, combinations are used to calculate the number of ways to choose a subset of items from a larger set.
Power – The result of raising a number to an exponent, indicating how many times to multiply the number by itself. – In physics, power is also defined as the rate at which work is done or energy is transferred.
Mathematics – The abstract science of number, quantity, and space, either as abstract concepts or as applied to other disciplines such as physics and engineering. – Mathematics is essential for developing models that predict the behavior of physical systems.
