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that the number of elements of [itex] \mathbb{R} [/itex] (seen as a set, obviously) is bigger than the number of elements of [itex] \mathbb{N} [/itex] ...? 
Daniel.
Daniel.
In mathematics, "R is Bigger Than N Set Elements" is a statement that means the cardinality (number of elements) of the real numbers set (R) is greater than the cardinality of any given set containing a finite number (N) of elements. This means that there are infinitely more real numbers than elements in the given set.
Yes, it is possible to prove that R is bigger than N set elements using a mathematical proof. This proof involves showing that there is no one-to-one correspondence (bijection) between the real numbers set and the set containing N elements. This means that there are always more real numbers than elements in the given set, no matter how large N may be.
The concept of infinity is essential in proving that R is bigger than N set elements. This is because the real numbers set is considered to be an uncountable infinite set, while any set containing a finite number of elements is considered to be a countable infinite set. By definition, an uncountable infinite set is always bigger than a countable infinite set.
One example that can help understand this concept is comparing the real numbers set to the set of whole numbers (1, 2, 3, ...). While both sets are infinite, the real numbers set contains all the numbers between any two whole numbers, making it infinitely larger than the set of whole numbers.
Yes, there are practical applications of proving R is bigger than N set elements. This concept is used in various fields of mathematics, such as calculus, analysis, and topology. It also has applications in computer science, specifically in the field of cryptography, where the uncountable infinite set of real numbers is used to create encryption keys that are difficult to crack.