proof of principle of transfinite induction
To prove the transfinite induction theorem
, we note that the class of ordinals
is well-ordered by ∈. So suppose for some Φ, there are ordinals α such that Φ(α) is not true. Suppose further that Φ satisfies the hypothesis
, i.e.
∀α(∀β<α(Φ(β))⇒Φ(α)). We will reach a contradiction
.
The class C={α:¬Φ(α)} is not empty. Note that it may be a proper class, but this is not important. Let γ=min(C) be the ∈-minimal element of C. Then by assumption
, for every λ<γ, Φ(λ) is true. Thus, by hypothesis, Φ(γ) is true, contradiction.
Title | proof of principle of transfinite induction |
---|---|
Canonical name | ProofOfPrincipleOfTransfiniteInduction |
Date of creation | 2013-03-22 12:29:06 |
Last modified on | 2013-03-22 12:29:06 |
Owner | jihemme (316) |
Last modified by | jihemme (316) |
Numerical id | 11 |
Author | jihemme (316) |
Entry type | Proof |
Classification | msc 03B10 |