Mapping the Infinite
Go beyond counting to discover how some infinities are larger than others. Master the art of pairing sets and exploring the boundaries of mathematical logic.
Like Duolingo, but for Mapping the Infinite. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
54 levels across 4 sections, about 108 minutes end to end, roughly 22 days at five minutes a day. It moves through Counting Without Numbers, Comparing Different Sizes of Forever, Building New Sets from Old, and The Logic of Mathematical Collections. It assumes you already know the basics.
Free forever · No credit card · iPhone & Android
Key ideas in Mapping the Infinite
- Pairing every item in one group with one in the other proves they are equal.
- If every item is matched and none are left over, the sizes are identical.
- A single leftover item proves the groups are different sizes.
- A 'perfect match' requires no leftovers on either side.
- The location of the leftover item tells you which group is bigger.
- You can never finish counting an infinite group to reach a final number.
- A complete match with no leftovers indicates perfect equality.
- A pairing rule can be applied to every item even if the list never ends.
- Understanding the consequence of leftover items in a pairing attempt
- Identifying the result of different pairing outcomes
- Why pairing is superior to counting for infinite groups
- Pairing is a step-by-step process of linking individual elements.
- The final proof of equality is the total absence of unmatched items.
- The process of proving equality through pairing
- Every item in the first group must connect to exactly one item in the second
- A perfect pairing means neither set has 'extra' elements
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Mapping the Infinite the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
If you have a bag of locks and a bag of keys, how can you prove the quantities are equal without counting them?
Get it right to open this lesson and 53 more in the app.
How to learn Mapping the Infinite, step by step
Go in order. 54 short levels across 4 sections, about 1 hour 48 minutes in all, five minutes a day.
- 1
Counting Without Numbers
- The Perfect Match: Pairing Elements
- When Sets Mirror Each Other
- Testing Equality with One-to-One Maps
- The Infinite Hotel Room Swap
- 2
Comparing Different Sizes of Forever
- Whole Numbers vs. Even Numbers
- Fitting Fractions into a Single Line
- Why Decimals Break the Pattern
- The Gap Between Countable and Uncountable
- Visualizing the Continuum
- Ordering the Magnitudes of Infinity
- 3
Building New Sets from Old
- The Collection of All Subsets
- Why a Power Set is Always Larger
- Creating an Endless Ladder of Infinities
- 4
The Logic of Mathematical Collections
- The Problem with the 'Set of All Sets'
- Defining Membership and Boundaries
- Choosing Elements from Infinite Piles
- The Unsolved Questions of Set Size
- How Infinity Shapes Modern Logic
4 sections · 18 units · 54 levels. Built to play, not to enroll.
Questions about Mapping the Infinite
How do I learn Mapping the Infinite?
Go in order: start with Counting Without Numbers, then Comparing Different Sizes of Forever, then Building New Sets from Old, then The Logic of Mathematical Collections. On Tomo that's 54 short levels, about 1 hour 48 minutes in all, played five minutes a day. Start free and play the first level in under a minute.
Is there a Duolingo-style app for Mapping the Infinite?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. Mapping the Infinite is one of its free courses, and you can type any other topic to get a course of your own.
Is the Mapping the Infinite course free?
Yes. Mapping the Infinite is playable on Tomo's free plan, which doesn't expire. No credit card, no waitlist. Download the app and start your first lesson.
How long does it take to learn Mapping the Infinite?
54 levels across 4 sections, about 1 hour 48 minutes of play in total. It's built for five minutes a day, so you finish it a little at a time instead of all at once.
Do I need any background to start Mapping the Infinite?
No. It's pitched at an intermediate level, so it starts where you are and builds up. If you can tap a phone, you can start.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Mapping the Infinite is taught in the Professor style: clear, structured, thorough. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start Mapping the Infinite today.
Download Tomo, search Mapping the Infinite, and play your first lesson in under a minute.