Or would it be a wrong thing to do? Rodrigo de Azevedo Show 1 more comment. Active Oldest Votes. The Moral : Man, number is weird. Noah Schweber Noah Schweber k 18 18 gold badges silver badges bronze badges. What it is, is a cardinal number , and these are very well-understood things - if extremely different from e. So, what do you mean by "categorize it in any set of numbers"?
It sounds like you have a specific notion of what "number" is in mind - can you clarify? Perhaps "something used to count things"? But then surely even the layperson can see the issues with that, provided they are given time to think a bit. But if you argue like that, there's not much preventing you from saying that everything is a number. Show 12 more comments. A number made by taking a natural number and adding 1 to it is a natural number.
There are no other natural numbers than the ones whose existence is implied by 1 and 2. Firstly there is no presumption that you can reverse the operation of adding 1. Or you didn't. For example, from the set of 1 and 2 you can make either the empty set, a set containing 1, a set containing 2 and a set containing 1 and 2.
Therefore the power of the set is four. Two to the power of the amount of members in the original set, is the power of that particular set. In addition, the power set of all the naturals is the power set of aleph-null. Moreover, making a list of all the subsets from natural numbers enables us to number all the subsets.
It works as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. What is important, is that it will yield an even bigger list with more subsets. This list, this new set of subsets, yields a bigger infinity than aleph-null.
Again, repeating this process, taking the power of the power sets of natural numbers will give even bigger infinities and so on. Besides the Hilbert Hotel Paradox, there are a few more examples of situations that go against our intuition about infinity in the first place. Infinity fits perfectly in our mathematical world. Nevertheless, in the physical world people still have some difficulties of understanding infinity when one tries to picture it.
Two other paradoxes help us to get a better idea of the concept of infinity. Picture a trumpet that is wide on the end and then, while moving to the other side of the object, the trumpet becomes thinner and thinner.
The surface of this trumpet is infinite. On the other hand the volume of the trumpet is finite. As someone would paint the trumpet he would need an enormous amount of paint and he would keep painting the object as it gets thinner and thinner.
Controversially, if he wanted to determine the volume he would just fill the trumpet with paint to see how much paint goes in the trumpet. If we think about physical paint one cannot paint the whole surface, due to the fact that the trumpet gets so thin at the end the size of the molecules of the paint are too large.
If we had mathematical paint this would be possible. At this point a lot of people thought there was something wrong with the idea of infinity, since they could not match the concept of infinity with the real world.
Another paradox is the St. Petersburg paradox. Assume a casino places one euro in a box. The question is how many euros one should bet for. Frequently, people would bet for around twenty euros. Even though, the chances of winning which is denoted by p in the formula decrease by a half each time a euro is added to the bet, whereas the payment for the casino doubles which is denoted by two to the power n in the formula.
This means the expected value of the amount of money a person can win is infinite. Cantor established the importance of one-to-one correspondence between the members of two sets.
Moreover, he defined these infinite and well-ordered sets and proved that the real numbers are more numerous than the natural numbers. In addition, he defined the cardinal and ordinal numbers and their arithmetic.
This article is written by Anne Dumoulin. Diophantine equations are polynomial equations over the integers. These are useful in many fields of mathematics such as group theory and operations research.
Wavelet analysis stems from the 19th century when Joseph Fourier studied the heat equation. From his work, a foundation was set for Fourier transformations FT.
However, FT requires conditions such as stationarity of the data set and it only gives frequency Different types of infinity. Mathematics Statistics. January 30, Some sets, even some sets containing an infinite number of elements, are countable such as the set of integers while other sets are not countable such as the set of real numbers.
All finite sets are countable and have a finite value for a cardinality. The set of natural numbers is an infinite set, and its cardinality is called aleph null or aleph naught.
Aleph null is a cardinal number, and the first cardinal infinity — it can be thought of informally as the "number of natural numbers. A set with cardinality less than or equal to is called a countable set. An example of another countable set is the set of even numbers,. The even numbers have a one-to-one correspondence with the natural numbers, namely. So the set of even numbers is countable. Note that if we add one element to the natural numbers to get say , the set still has cardinality by the mapping.
So we could say that. The same happens if we take away an element , leading us to the following nerdy joke:. Although is infinite, we can still go further.
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