0
Infinties are actually different sizes-which may sound stupid but the decimal infinity is longer then the whole number infinity-because you can have a much wider variety of combinations of decimal numbers. However, infinties are generally known to be bigger then both a Google-plex to the power of a google-plex, or even bigger then a graham's number, which is the biggest number anyone has been able to think of.
Still curious? Ask our experts.
Chat with our AI personalities
Rafa
Ezra
FranAdd your answer:
Earn +20 pts
What is the cardinality of the set of rational numbers?
Infinity but (are you ready for this?) the smallest of infinities, which is Aleph-zero.
What kind of number is infinity?
"Infinity" means different things in different context. As the size of a set, there is actually not one number infinity, but several: some infinities can be larger than others.
What is the difference between countable infinity and uncountable infinity?
First we need to recognize that any form of infinity is just a mathematical construct. We are not ever going to have to deal with actual infinities, but what is very useful is understanding limiting behavior as something or other goes towards infinity. It is here that the difference between countable and uncountable infinities matters immensely. ; countable : can be put into a one to one correspondence with the natural numbers ; uncountable : cannot be put into a one to one correspondence with the natural numbers
What is infinity plus 1?
Infinity is not a number. It is another way of saying that something gets larger and larger without an end. So, you cannot add 1 to some thing which is not a number. However, in mathematics, there are infinities that could be added, multiplied, or subtracted. There are cardinal, ordinal, surreal and hyperreal numbers to all of which 1 can be meaningfully added. For infinite cardinal numbers, adding one does not change the number. For ordinal numbers, the number can be changed depending on whether you add 1 to the number, or the number to 1 - there is no commutativity. With hyperreal numbers, one can do pretty much anything one would expect of a real number. As to the surreal ones, the arithmetic operations produce magic results increasing the number of various infinities to fantastic proportions. In general, to add 1 to a number means to take the successor of that number. The successor of a set A is the set which consists of all the elements of A along with A itself. In other words the successor of A is A U {A}, and it has "one" more element than A. So, infinity + 1 = infinity U {infinity}.
Can you make a generalization about the relationship between the number of elements in a set and the number of subsets?
For a set with a finite number, n, of elements, the number of subsets in 2^n. This includes the null set and the set itself. Things get a bit complicated if the original set has infinitely many elements. It is still 2^k but the complications arise because of infinities and transfinite numbers.
