Riddles and infinities.
Riddles and other things that many people don't know are among the things that I love. Here are some of them.
1. Cantor showed us that there are 3 inifinities of diffeent sizes. He called them aleph 0, aleph 1, and aleph 2. Since we can't count in order to see the differences he used a 1-1 match up. As an example If there is a bus where all the people were sitting and all the seats were full we would know that the number of people and seats are equal.
The aleph 0 set contains the natural numbers (1, 2, 3.....). If we compare them to only the even numbers (2, 4, 6....) we see that every natural number has an even number that matches up (simply multiply by 2) and vice versa. these 2 infinities are equal in size. Any other set that we can list that will contain them all (fractions (1/1, 1/2, 2/2, 2/1 ......) for example) are also aleph 0, because we can match them up with 1, 2, 3.... simply by numbering the list.
Aleph 1 - You can't make a list of the decimal numbers (.00001, .00002 ....) that will contain all of them. One way to show this is to write down a specail number for checking. You change the first digit of the first decimal number, the second digit of the second one, and so on. If someone claims that this special number is exactly like #128 of the list you can point out that the 128th digit is different. So aleph 1 is larger. It has more. All the points of the universe is also a set of aleph 1. It can be shown that all of them can be matched up with the points in a line of one meter !
Aleph 2 - is the set of all curved lines. There are an infinite number of curved lines that can connect 2 points while in the decimal system there is only one straight line. This is why you can make an infinite number of keys to a lock where no other key will fit.
2. Achillis and the turtle, - Achilles can run 10 times faster than the turtle. In a race between them the turtle is given a fore of 10 k"m. By the time that Achillis runs 10 k"m the turtle has run 1 k"m. By the time that Achillis runs 1 k"m the turtle has run 100 meters. Every distance that Achilles runs the turtle runs 1/10th.
Achilles can't reach the turtle or overtake him. How come ?
I will give the answer to this and 2 other riddles to anyone that asks me.
3. The barber. There is a town in which all the men have no beards. Either they shaved themelves or the one barber shaved them. The question is who shaved the male barber. He also shaves himself and is shaved by the barber. But I wrote above "or". Can't be both.
I would have liked to leave the answer to this question also to tell only those who contact me, but I think that the answer is very important.
Simply - there is NO town with this description. In my chapter on "Types of problems and solutions" I mentioned that the solution of unsolvable problems is their recogniztion as such. They have to be ignored.
4. 1=2 - Here are the calculations
x=y
xy = y squared
xy - x squared = y squared - x sqared
x(y-x) = (x+y)(x-y)
x = x+y = 2x
1 = 2
In every step above the same was done on both sides of the equation. That's acceptable - right ? So where is the mistake ? Can 1 sometimes be equal to 2 ?
The answer will be given to those that ask.
5. The hats - There are 3 black hats and 2 red ones. There are 3 men who are sitting a row. .The first one can see the other 2 and is asked what is the color of his hat. He says "I don't know".
The middle man is asked the same and he also says "I don't know". The third man (who can't see the others) is also asked, and he says "I know". What is the color of his hat and how come he knows ?
The answer will be told to those that ask.
I could continue but these are the things that I like most to tell people in addition to my other chapters of this page.
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