Well-orderings are rigid proof

• wj2cho
In summary: This case is necessary because if f(a) was a fixed point, it would be an element of D that is less than a, contradicting the fact that a is the least element of D. Therefore, f(a) cannot be a fixed point.

wj2cho

Hi I've been trying to understand this proof, but there is one step that I don't get at all.

Proof: Suppose f is an automorphism of (E,<=). Consider a set D, a set of non-fixed points under f. If D is empty, f is an identity mapping. Suppose, toward a contradiction, that D is nonempty. Then D has a least element, say a. Since E is well-ordered, either f(a) < a or a < f(a). Since f(a) < a, f(a) is not an element of D. So f fixes f(a), hence f(f(a)) = f(a). But then f(a) = a since f is injective, contradicting that a is an element of D. The case a < f(a) follows similarly applying the inverse of f.

Why does f(a) < a imply that f(a) is not a fixed point?

wj2cho said:
Since f(a) < a, f(a) is not an element of D.
To be clear, the book should have said "Consider the case when f(a) < a.".

Is it possible that f(a) is not a fixed point? If it were not a fixed point, it would be an element of D that is less than a.. But a is defined as the least element of D, so this is impossible.

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Thank you very much!. In fact, the book did say "Consider the case when f(a) < a".

1. What is a well-ordering?

A well-ordering is a type of mathematical ordering in which every non-empty subset has a least element. This means that in a well-ordered set, every element has a unique position or rank, and there is no infinite descending chain of elements.

2. What does it mean for a well-ordering to be rigid?

A well-ordering is considered rigid if it cannot be changed or altered in any way while still preserving its properties. In other words, any attempt to change the ordering of elements in a well-ordered set will result in a different well-ordering.

3. How is the rigidity of well-orderings proven?

The rigidity of well-orderings is typically proven using a proof by contradiction. This involves assuming that a well-ordering can be changed or altered in some way, and then showing that this leads to a contradiction or violation of the definition of a well-ordering.

4. Why is the rigidity of well-orderings important?

The rigidity of well-orderings is important because it allows for a consistent and unchanging mathematical structure to be used in various applications and proofs. It also allows for the establishment of a unique and well-defined ordering of elements, which is necessary for many mathematical concepts and theories.

5. Can the rigidity of well-orderings be generalized to other mathematical structures?

Yes, the concept of rigidity can be applied to other mathematical structures, such as partial orders and total orders. However, the specific proof for rigidity may differ depending on the properties and definitions of these other structures.