Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Lemma I.10.10 in Kunen's set thoery (2011) claims that a set $A$ is well-orderable iff $A$ is bijective with an ordinal.

He does not prove this, and one direction is clear to me.

However, I can't see why $A$ bijective with an ordinal implies $A$ is well-orderable. By the way he develops this within ZFC except foundation, power set, and choice.

Any help is appreciated -Thanks!

share|improve this question
    
Oh, I'd have thiought that this is the "one direction is clear to me". The well-order of the cardinal induces a well-order on $A$ via the bijection. –  Hagen von Eitzen Jan 16 '13 at 8:40

1 Answer 1

up vote 4 down vote accepted

If $f\colon A\to \alpha$ is a bijection then $a\prec b\iff f(a)\in f(b)$ is a well ordering of $A$.

share|improve this answer
    
Of course, how silly of me! thanks –  user52534 Jan 16 '13 at 8:45
    
I would think the other direction, requiring replacement in an essential way, and building bijections via approximations, is the one that is not immediate... Maybe "clear" simply means "proved in the book". –  Andres Caicedo Jan 16 '13 at 19:16
    
@Andres: Yeah, that's what Hagen's comment said and that was what I thought when I started reading the question as well... –  Asaf Karagila Jan 16 '13 at 19:22

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.