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transfinite induction (Theorem)

Suppose $ \Phi(\alpha)$ is a property defined for every ordinal $ \alpha$, the principle of transfinite induction states that in the case where for every $ \alpha$, if the fact that $ \Phi(\beta)$ is true for every $ \beta<\alpha$ implies that $ \Phi(\alpha)$ is true, then $ \Phi(\alpha)$ is true for every ordinal $ \alpha$. Formally :

$\displaystyle \forall\alpha(\forall\beta(\beta<\alpha\Rightarrow \Phi(\beta))\Rightarrow \Phi(\alpha))\Rightarrow \forall\alpha(\Phi(\alpha)) $

The principle of transfinite induction is very similar to the principle of finite induction, except that it is stated in terms of the whole class of the ordinals.



"transfinite induction" is owned by jihemme. [ full author list (2) | owner history (1) ]
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See Also: principle of finite induction, induction, transfinite recursion

Other names:  principle of transfinite induction
Keywords:  well ordered set, well-ordering principle

Attachments:
proof of principle of transfinite induction (Proof) by jihemme
example of transfinite induction (Example) by CWoo
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Cross-references: class, terms, principle of finite induction, similar, implies, states, ordinal, property
There are 11 references to this entry.

This is version 7 of transfinite induction, born on 2002-02-25, modified 2002-06-01.
Object id is 2703, canonical name is TransfiniteInduction.
Accessed 7619 times total.

Classification:
AMS MSC: 03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic)
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Update: "Trans-inductive Posets" by Schneemann on 2006-02-18 14:28:58
Why restrict the principle of transfinite induction to well-ordered sets?

For an arbitrary poset, there remains only one necessary and sufficient condition to get the transfinite induction working. I suggest to call such posets trans-inductive.

I use the non-reflexive definition of posets, and the interval notations ("*" means, there's no lower bound.)
[*, y] = { x | x<y or x=y } and
[*, y) = { x | x<y }
__________________

Thm. Let (L, <) be a poset. The following are equivalent.

(i) Every totally ordered, nonempty subset S \subseteq L has a minimum.

(ii) If A \subseteq L, such that
for any x in L, if [*, x) \subseteq A, then x in A, Then A = L.

Def. A poset with the above property (i) is called trans-inductive.
__________________

Proof.

(i) => (ii): Assume A != L, but [*, x) in A always implies that x in A. Have a look at B = L-A, and a chain C \subseteq B that's maximal in B.
[*, min(C)) = \emptyset \subseteq A => min(C) in A.
=> contradiction :)

!(i) => !(ii): Assume there's a chain C \subseteq L with no minimum. Let f: (\NN, >) -> (C, <) be injective and order-preserving. Set A = {f(2k) | k in \NN}.
A breaks the rule of (ii).
__________________

See also the discussion on Tree (set theory).

Most topics in PlanetMath are taken from books or papers. Possibly the above is not in any book. I don't know if PlanetMath is the right place to introduce a new concept to mathematics, but I don't know another way. Write a paper??
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