Measure content performance. Develop and improve products. List of Partners vendors. Share Flipboard Email. Anne Marie Helmenstine, Ph. Chemistry Expert. Helmenstine holds a Ph. She has taught science courses at the high school, college, and graduate levels. Facebook Facebook Twitter Twitter. Updated February 22, Featured Video. Cite this Article Format.
Helmenstine, Anne Marie, Ph. Math Glossary: Mathematics Terms and Definitions. Bayes Theorem Definition and Examples. Biography of Giordano Bruno, Scientist and Philosopher.
Timeline of Greek and Roman Philosophers. The Beautiful, the Sublime, and the Picturesque. Your Privacy Rights. To change or withdraw your consent choices for ThoughtCo. At any time, you can update your settings through the "EU Privacy" link at the bottom of any page. The power set of a set is the set of all the different subsets you can make from it. For example, from the set of 1 and 2, I can make a set of nothing, or 1, or 2, or 1 and 2.
The power set of 1,2,3 is: the empty set, 1, and 2, and 3, and 1 and 2, and 1 and 3, and 2 and 3, and 1,2,3. As you can see, a power set contains many more members than the original set. Two to the power of however many members the original set had, to be exact. Imagine a list of every natural number. Now the subset of all, say, even numbers would look like this: yes, no , yes, no, yes, no, and so on.
The subset of all odd numbers would look like this. And how about every number—except 5. Or, no number—except 5. Obviously this list of subsets is going to be, well, infinite. But imagine matching them all one-to-one with a natural. The way to do this is to start up here in the first subset and just do the opposite of what we see. As you can see, we are describing a subset that will be, by definition, different in at least one way from every single other subset on this aleph-null-sized list.
Even if we put this new subset back in, diagonalization can still be done. The power set of the naturals will always resist a one-to-one correspondence with the naturals. The point is, there are more cardinals after aleph-null. Wait … what are we doing? Of course we can. This is math, not science! Its consequences just become what we observe. We are not fitting our theories to some physical universe, whose behaviour and underlying laws would be the same whether we were here or not; we are creating this universe ourselves.
If the axioms we declare to be true lead us to contradictions or paradoxes, we can go back and tweak them, or just abandon them altogether, or we can just refuse to allow ourselves to do the things that cause the paradoxes. The axiom of infinity is simply the declaration that one infinite set exists—the set of all natural numbers.
Are we going to have to add a new axiom every time we describe aleph-null-more numbers? The Axiom of Replacement can help us here.
Infinity is firmly rooted in mathematics. But according to Justin Moore , a math researcher at Cornell University in Ithaca, New York, even within the field there are slightly different uses of the word. There isn't just one type of infinity, either. Counting, for example, represents a type of infinity that is unbounded—what's known as a potential infinity.
In theory, you can go on counting forever without ever reaching a largest number. However, infinity can be bounded, too, like the infinity symbol, for example. You can loop around it an unlimited number of times, but you must follow its contour—or boundary. All infinities may not be equal, either. At the end of the 19th century, Cantor controversially proved that some collections of counting numbers are bigger than the counting numbers themselves.
Since the counting numbers are already infinite, it means that some infinities are larger than others. He also showed that some types of infinities may be uncountable, as opposed to collections like the counting numbers. Without infinity, many mathematical concepts would fall apart. Because they are finite, and infinity is We can sometimes use infinity like it is a number, but infinity does not behave like a real number. Which is mathematical shorthand for " negative infinity is less than any real number, and infinity is greater than any real number".
No, because we really don't know how big infinity is, so we can't say that two infinities are the same. If you continue to study this subject you will find discussions about infinite sets, and the idea of different sizes of infinity.
That subject has special names like Aleph-null how many Natural Numbers , Aleph-one and so on, which are used to measure the sizes of sets. But there are more real numbers such as Infinity is a simple idea: "endless".
Most things we know have an end, but infinity does not.
0コメント