Class Notes from 10/22/2013

I apologize if I paraphrased or got something else incorrect. I compiled the notes to the best of my knowledge :)

October 22, 2013

An easy guide to infinite chocolate (YouTube Video)

http://youtu.be/7Yo6mwSB1As

Manifestations:

1. Mirrors
2. Yin Yang
3. Cycle of Life
4. Can't Stop Time
5. Tech/Science
6. Germs/Bacteria
7. Time in the Present Tense
8. Ascending/Descending
9. Thoughts
10. Shapes
11. Galaxy “Adam's Quote”
12. Creativity
13. Light Rays
14. Evolution
15. Energy
16. Roman Columns
17. Droste
18. Sculpture in DC
19. Klein Bottle
20. Recycling Symbol
21. Music

How difficult was the categorization of the manifestations of infinity?

• Abstract
• Concrete
• Formal
• Measurement
• Life

It was interesting to note that many people felt that it was more easily categorized at home. However, there were differences in opinion on how they are categorized. For example, the naming of each category seemed subjective. John suggested, that he could have simplified further with less categories (e.g.: getting rid of the “formal” category), and some others agreed.

Two previous readings

Comparing the Infinite: Pairing up collections via a One-to-One Correspondence. (pp. 146-157)

To Infinity & Beyond: To Infinity and Beyond (pp. 61-67)

Try to summarize the concepts presented in the two articles mentioned above. What are the implications of a one-to-one relationship?

One big take-away was the way of solving the problem. In the hotel with an infinite number of buses for example, it provided a different way of looking at things. Instead of calculating the number of remaining rooms, or the number of floors in the “infinite hotel”, you could just figure out a system of organizing people from the buses linearly to file into the hotel.

Scenario: the hotel is infinitely large, and it's already full. With a one-to-one correspondence, we were always able to solve the problem. This is because of the one-to-one relationship between the number of buses, or people- and the number of hotel rooms.

In the second reading, Cantor proved essentially two different “sizes” of infinity. For example...

For the incremental value between 0 and 1 you could write:

1 → 0.713234092358092834

2 → 0.281043205984534098

3 → 0.500000000000000000

4 → 0.111535493485394859

This was monumental because it disproves the ability to calculate the value in between any two real numbers (right...?)

… So what do we know about infinity?

||N|| < ||R||

Continuum hypothesis – the idea that there is no infinite set with a cadinal number in between the smaller infinity, and the larger infitinty.

Professor Carter

Zeno – A Greek philosopher from mid-late 400 BC known for his paradoxes.

Dichotomy A – Kitty and the food.

At some point the kitty has to traverse past the halfway point of the room. Then another half, etc. until it becomes aparent that the kitty will never be able to reach the food.

Achilles and the Tortoise- Tortoise is given a head start, and even though Achilles is moving faster, the tortoise moves during the time that Achilles catches up to him. This continues through an incremental period until it shows that Achilles will never quite catch up to the tortoise.

The Stadium-

The Arrow-