Decreasing sequence Aleph's

**Dinkydoe** · December 24th 2009, 12:12 PM

I need to show: there does not exist a sequence:

$\left\langle X_n:n\in \omega \right\rangle$ such that $(\forall n)(\mathcal{P}(X_{n+1})\preceq X_n)$ .

Supposing there exists such a sequence:
We can observe that for any $n\in\omega: X_n \prec \mathcal{P}(X_n)\preceq X_{n-1}\prec \cdots \prec \mathcal{P}(X_1)\preceq X_0$ . Thus we have $X_n \prec X_k$ for all $0\leq k < n$ . In particular, for any $n$ we have $\mathcal{P}^n(X_n)\preceq X_0$ .

In a previous excercise I had to show that (without the axiom of choice): $\aleph(X) \preceq \mathcal{P}(\mathcal{P}(\mathcal{P}(X)))$ . And since $\aleph(X)$ is the smallest ordinal $\alpha$ such that $\alpha\not\preceq X$ we obtain: $\aleph(X)\prec \aleph(\mathcal{P}(\mathcal{P}(\mathcal{P}(X))))$ .

From this I conclude we get a strictly decreasing sequence Aleph's: Since $\aleph(X_n) \preceq \mathcal{P}(\mathcal{P}(\mathcal{P}(X_n)))\preceq X_{n-3}\prec \aleph(X_{n-3})$ . Thus the subsequence $\left\langle X_{3n}:n\in \omega\right\rangle$ is a sequence such that $\aleph(X_n) \prec \aleph(X_{n-1})$ and $\aleph(X_0) = \bigcup_{n\in\omega} \aleph(X_n)$ .

Those were some of my conclusions: However I don't see where the contradiction kicks in. Can anyone offer me some suggestions? Probably there can not exist a infinite decreasing sequence of ordinals, intuitively that might sound logical but I can't proof it.

Since the set of ordinals are wellordered, our sequence: $\left\langle \aleph(X_n):n\in \omega \right\rangle$ should contain a smallest ordinal. But can't we simply define that one as $\bigcap_{n\in \omega} \aleph(X_n)$ ?

**clic-clac** · December 25th 2009, 12:19 PM

Hi

Yep you can find a contradiction when considering a strictly decreasing sequence of ordinals:

Assume $(\alpha_n)_{n\in\omega}$ is a sequence of ordinals such that $\alpha_n>\alpha_{n+1}$ for any $n\in\omega.$

What about $X=\bigcup\{\alpha_n\ ;\ n\in\omega\}$ ?
Take $x\in X,$ there is a $k\in\omega$ such that $x=\alpha_k$ and $\alpha_{k+1}$ is a strictly lower element of $X:\ X,$ a set of ordinals, has no minimum. Contradiction.

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