Quiz Genciencia: Infinitos – On infinite subsets (some clues).
In the comments that were left in the post in which it is posed the question of whether there are infinities greater than others, I found a recurring theme which is that the part always has to have fewer elements than the whole; This conception was radically modified in the 19th century, but it had remained in people’s minds since the Greeks.
It is interesting to study some properties of numbers, especially when their quantity is infinite: The whole idea is quite simple: it is about objects that can be placed together, like inside a bag, and that are distinguishable from each other, even if it is intellectually. So one can have a set of people, for example, or a set of numbers. In particular, the sets of numbers are of interest and within these there are some more relevant ones:
- Natural Numbers, represented by the letter , made up of 1, 2, 3, 4, …
- Whole numbers, represented by the letter , which include the natural ones and add the negatives and the 0:…, -3, -2, -1, 0, 1, 2,…
- Rational Numbers, represented by the letter , which include the previous ones and add all those that can be written as a division: -4/5, 3/8, 1, 2, 8/3, 9/5, … It is important to note that any number with a comma, which present a certain periodicity, for example 0.333333… 0.142857142857… can be written in division form (1/3 and 1/7, in the previous examples.)
- Real Numbers, represented by the letter , which include the above and add all the others, for example , , etc. etc.
As you can see, each of the sets above is infinite, that is, I cannot tell how many elements it has. If someone tells me that such a set has n elements, and shows me a list, I will always be able to find an element “n + 1”. It is interesting then to see how one works with sets that are infinite; there is a story known as “Hilbert’s Hotel” that tells the following:
“There was a hotel that had infinite rooms. One day a new guest arrives to stay there, but the concierge tells him that he was unlucky, that they were all full. The indignant guest calls the manager, and asks him how it was possible in a hotel with infinite rooms. The manager agrees with him, but says there is nothing he can do, so the guest responds quickly: ‘I know what can be done; whoever is in room 1 he sends to room 2, to the from room 2 to room 3 and so on, then room 1 will be free for me. ‘ The manager found this solution wonderful and did so. “Some days later another guest arrives and asks to stay, to which they reply that the hotel was full, but don’t worry, they knew how to fix it. So this guest says that there was a problem, that he was not alone, but with a group of friends … and that it was an infinite group. The manager, again dismayed, did not know what to do, but the guest, also very clever, tells him not to worry, to send the one from room 1 to 2, from room 2 to 4, from room 3 to the 6 and so on. That way all the odd-numbered rooms would be free for his infinite friends. ”
This beautiful story, which although it seems to be a stretch, is showing 2 very important properties of infinite sets. First of when adding an element to the infinite set (first part of the story) infinity is unchanged. The hotel remains the same and with the same number of rooms. The same would have happened if 10, 20, 30, … elements were added (it was enough to send the guest from room 1 to 11, 21, 31, etc.) The second part shows that adding infinite elements to the set also does not change the total amount. At the same time it shows something very peculiar, and that is that the number of even (or odd) rooms in the hotel is the same as the number of total rooms. In general one would be tempted to think that the part is always less than the whole, but for an infinite set this would be failing. This is where the suspicions of mathematicians arose, which led to a theory of infinity and a truly surprising development of mathematics.
Something that we are very used to doing is counting. When you go to the supermarket, for example, and buy 5 apples, how did you know that there were 5? He simply placed each of the apples next to a natural number (in technical language he bijected between the elements of two sets) and determined which was the greater. This same method of counting can be transferred to infinite sets. In other words, what we will try to do is build a relationship between the natural numbers and the others, thus demonstrating that there are as many natural numbers as there are integers, etc. etc.
It is easy to see (in the hotel example) that there are as many natives as there are even natives, or odd natives. It is enough to assign each natural n the number 2n + 1 or 2n-1. That way there will be neither natural nor even (or odd) free, each one will be linked to another. With the integers it is the same case, since it is enough to order them in an intelligent way so that they can be “counted”, for example: 0, 1, -1, 2, -2, 3, -3,…. That way we can also assign each natural a whole number and vice versa.
For this reason, it can be said that generating a bijection is the way to control whether both sets have the same amount of elements, the same infinity. The next step would be to see if from the list of sets of numbers above (or any that may occur to them), one has more elements than another, thus it will have been proven that indeed not all infinities are equal.
Next week the full resolution will come, which by the way some people already mentioned in the comments, only without explaining too much.
Updated 06/11/2008: The solution is available.