Answer:
Theorem 2.27 states that if E is a well-founded relation on P, then there exists a unique function p from P into the ordinals such that for all x E P, p(x) = sup{p(y) +1: y Ex}.
To understand this theorem, let's break it down step by step:
1. Well-founded relation: A relation E on set P is said to be well-founded if every non-empty subset of P has a minimal element with respect to E. In simpler terms, it means that there is no infinite descending chain in P.
2. Function p: The theorem states that there exists a function p that maps elements from set P to the ordinals. An ordinal is a generalization of natural numbers and represents a well-ordered set.
3. p(x) = sup{p(y) + 1: y Ex}: For every element x in set P, the value of p(x) is equal to the supremum (the least upper bound) of the set {p(y) + 1: y Ex}.
To understand this equation, let's consider an example:
Suppose P is the set of integers, and E is the "less than" relation. In this case, the theorem states that there exists a function p from the integers to the ordinals such that for every integer x, p(x) is equal to the supremum of {p(y) + 1: y < x}.
Let's say we want to find the value of p(5). We consider all integers y that are less than 5 (in this case, 0, 1, 2, 3, and 4). We calculate p(y) + 1 for each y and take the supremum of these values.
For example, if p(0) = 1, p(1) = 2, p(2) = 3, p(3) = 4, and p(4) = 5, then the set {p(y) + 1: y < 5} would be {2, 3, 4, 5, 6}. The supremum of this set is 6, so p(5) = 6.
It's important to note that the uniqueness of function p is also guaranteed by the theorem. This means that there is only one function that satisfies the given conditions.
In summary, Theorem 2.27 states that if a relation E on set P is well-founded, then there exists a unique function p that maps elements from P to the ordinals, and for every x in P, p(x) is equal to the supremum of {p(y) + 1: y Ex}. This theorem provides a mathematical framework to study well-founded relations and their associated functions.
Explanation: