Paradoxes and Infinities
- Grades 7-11
This course explores conundrums and analyzes a range of mathematical and philosophical paradoxes. You’ll consider Zeno’s paradoxes of space and time, such as The Racecourse, in which Achilles continually travels half of the remaining distance and seemingly can never reach the finish line. To address these types of paradoxes, you’ll be introduced to the concepts of infinite series and limits, and then explore paradoxes of set theory, self-reference, and truth, and paradoxes of probability and inductive reasoning. Finally, you’ll examine the concept of infinity and its paradoxes, and learn how to demonstrate that some infinities are bigger than others. Through these investigations, you’ll acquire skills and concepts that are foundational for higher-level mathematics, and leave the course with more nuanced problem-solving skills, an enriched mathematical vocabulary, and insight into some of the most perplexing questions ever posed.
Typical Class Size: 16-18
- Formulate critical and quantitatively reasonable thoughts orally and in writing with clear and concise arguments
- Create and justify proofs of theorems like that of Cantor, Banack-Tarski, and Gödel
- Construct and rigorously examine paradoxes and open-ended problems
- Elaborate and create self-referential paradoxes
- Explain and justify the difficulties of classical philosophical problems
- Determine whether a set is finite, countably infinite or uncountably infinite
- Identify, examine, and test computer problems that are solvable or unsolvable
- Explore the hierarchy of larger and larger infinities
Summer Dates & Locations
Testing and Prerequisites
|Required Level||CTY-Level||Not required|
Students must achieve qualifying scores on an advanced assessment to be eligible for CTY programs. If you don’t have qualifying scores, you have several different testing options. We’ll help you find the right option for your situation.Sign up for Testing Learn More
Cost and Financial Aid
- Nonrefundable Application Fee - $50 (Waived for financial aid applicants)
- Nonrefundable International Fee - $250 (outside US only)
Financial Aid is available
We are committed to serving all talented youth regardless of financial circumstances. Financial assistance is available based on need.
About Mathematics at CTY
Explore the study of shapes
Many of our courses allow students to describe the world around them in basic and profound ways. Our younger students learn about shape, scale, and proportion in Geometry and Spatial Sense. Middle School students explore beautiful real-world applications of lines; analyze data based on curves that fit a uniform, symmetric and bell-shaped, or skewed pattern in Data and Chance. And advanced students explore the underlying mathematics and fundamental characteristics of shapes, distance, and continuous deformations in our proof-based Topology course.
Dive deep into logic and reasoning
Our courses in formal logic give you the tools to question the world around you. Inductive and Deductive Reasoning introduces younger students to different types of reasoning, as well as the strengths and weaknesses inherent in various forms of critical analysis. Older students explore how logical reasoning can explain (or fail to explain) counter-intuitive results in Paradoxes and Infinities, or take a more rigorous approach to formal logic in Mathematical Logic.