Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The following were 2 problems given to us on transfinite induction, The transfinite induction I saw on books and books etc was using ordinals but the definition we were given is different and I still couldn't find a proper example on how to use it to a question

1) Let $\preceq$ be a well ordering on $[0,1]$ and for $t \in [0,1]$ let $x_t =\sup(B_t \cup \{t\})$ (sup is taken with respect to the usual ordering on real numbers)

where $B_t=\{x\in W : x \prec t\}$

Show that there exists $a \in [0,1]$ such that $x_t =1$ for each $t \succeq a$

2) Let $(W,\preceq)$ be a well ordered set and $t \in W$. if $f \colon W \to W$ is an order preserving bijection show that $f$ is the identity function on $W$.

Definition of the transfinite induction given: Let $(W,\preceq)$ be a well ordered set and $A \subseteq W$ and $B_t \subseteq A \implies t \in A$ Then $A=W$ (where $B_t=\{x\in W \mid x \prec t\}$)

Any help would be appreciated

share|cite|improve this question
You'll have to do better than that, I'm afraid! First get rid of all the typos, then format it properly using TeX. If you don't make an effort, why should we? – TonyK May 31 '13 at 14:21
What's Bt? in the first example. You should probably learn a bit about LaTeX and MathJax, and while at it please denote the well-order and the usual order of $[0,1]$ differently. Otherwise it's impossible to read. – Asaf Karagila May 31 '13 at 14:22
In order to help you out adhering to @TonyK and Asaf's suggestions, let me point you to some basic information about writing maths at this site. See e.g. here, here, here and here. – Lord_Farin May 31 '13 at 14:24
Thanks for the comments I would try to make it better formatted – user68099 May 31 '13 at 15:10

Ordinals are well-ordered sets, and every well-ordered set is isomorphic to a unique ordinal. So the idea of using general well-orders and ordinals is the same.

Let me give you some hint about the second question (because I can understand it without further clarifications).

We define $A=\{t\in W\mid f(t)=t\}$, and we want to show that $A=W$.

Suppose $t\in W$ such that $\{t'\in W\mid t'<t\}\subseteq A$. Then $f(t')=t'$ for all $t'<t$. Let $x=f^{-1}(t)$, if $x<t$ then by the induction assumption $x\in A$, and therefore $f(x)=x<t$. So we have to have $t\leq x$. Because $f$ is order-preserving we have that $t\leq x\rightarrow f(t)\leq f(x)=t$.

Because $f$ is a bijection it is impossible to have $f(t)<t$ (as those already have preimages) and therefore $f(t)=t$.

share|cite|improve this answer
Thanks a lot for the answer found it very clear – user68099 May 31 '13 at 15:22

Your Answer


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.