Mapping the Infinite
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.
For the part of you with thirty open tabs that never became anything.
Free during early access · 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.
Where Mapping the Infinite takes you
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.
- 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.
You pick the voice
Mapping the Infinite is taught in the 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.
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