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What is the equation for infinity?
Infinity is NOT a number, there is no FORMULA for it. Infinity cannot exist in any sets where it is defined. Infinities are different even for different sets! The infinity in natural numbers set {1, 2, 3, ...}, integers {0, 1, -1, 2, -1,....}, rational numbers {1/2, -1, 4/5,....}, real {pi, sqrt(2), 1, 4/5, ....}, the complex numbers {1, 5 +3i, ...} and many other infinite fields, rings, groups have their infinity defined as their "cardinality", or the abstract sense, the "size" of it. It quite obvious the infinity of natural numbers is much smaller than that of the real. It is said that the "cardinality" of the set of all "infinities" is the largest of all infinities. (Equal to the number of all possible sets). Why can't infinity be a number? Well, to be a "number" we must be able to add it together, and ta special element "0" such that "0" + a for any a is still a (or multiply, with element "1" instead, needed concept for concept of number, or number axioms). It's easy to prove that this element is UNIQUE, (or the equivalent class of it is unique, a and b are in the same equivalent class means a = b in the set) Now with our understanding and definition of infinity, "infinity" != 0, or that means there is no element in this set, we can't define operations within the empty set. But also, "infinity" + 1 = "infinity". We just said 0 is unique. Contradiction. So, no infinity is NOT a number. Please note, there is a difference between infinity as an ELEMENT then infinity as a NUMBER. An element of a set does not require to be part of the operation, this set will not be a well defined number set.
Which sets does the number zero belong?
There are an infinity of possible answers: the integers, rationals, reals, complex numbers, the set {0,1,-3}, the set containing only the element 0;
Does a set with no element belong to any kind of set?
I believe you are talking about subsets. The empty set (set with no elements) is a subset of any set, including of the empty set. ("If an object is an element of set A, then it is also an element of set B." Since no element is an element of set A, the statement is vacuously true.)
What is the supremum of empty set?
it's negative infinity!
What is the set of all even natural numbers?
Infinity.
