A Chain of Subsets of $\mathbb{R}$ Without any Good Countable Subchain

Question

Consider which $\bigl{(} A_i \bigr{)}_{i\in I}$ is a chain of subsets of $\mathbb{R}$. We say that a countable chain like $\bigl{(} B_n \bigr{)}_{n\in \mathbb{N}}$ is good if :

• for every $n\in \mathbb{N}$, the set $B_n$ be an element of the main chain $\bigl{(} A_i \bigr{)}_{i\in I}$
• for every $i\in I$ there exists an $n\in \mathbb{N}$ that $A_i \subset B_n $.

Give an example of $\bigl{(} A_i \bigr{)}_{i\in I}$ which for it, does not exist such a good chain $\bigl{(} B_n \bigr{)}_{n\in \mathbb{N}}$.

Such an example surely exists and surely exists of subsets with lebesgue zero measure, because if there does not exist then we can find a maximal zero measure set by using Zorn's lemma.

Answer 1

This has nothing to do with Zorn's lemma. Okay, almost nothing to do with Zorn.

Essentially we say that a chain $\mathcal A$ is good, if there is a countable subchain which is cofinal in $\mathcal A$.

So all you need is to show there is a chain which is not of countable cofinality.

For example $\aleph_1$ is regular, assuming "enough choice", and there is always a chain $(A_i)_{i<\omega_1}$, such a chain cannot be good if $\aleph_1$ is regular.

I don't see a natural approach to this using Zorn's lemma as a means for construction such a chain.

Answer 2

There are a number of reasonably simple such chains, all revolving around the same idea: that the first uncountable ordinal, $\omega_1$, is regular, that is, cannot be written as a countable union of countable ordinals.

Here's one natural class of examples: for $\eta<\omega_1$, let $A_\eta$ be the set of all reals which code well-orderings of $\omega$ of order type $<\eta$, or which do not code well-orderings of $\omega$ at all. Then clearly $A_\eta\subsetneq A_\theta$ whenever $\eta<\theta$, so we have a chain, but since $\omega_1$ is regular, there can be no good countable subchain. The specific example depends on the choice of "coding apparatus" for representing binary relations on $\omega$ as real numbers.

Another natural class: Via Zorn, let $\mathcal{C}=\{d_\eta: \eta<\omega_1\}$ be a a strictly increasing chain of Turing degrees. Then for each $\eta$ let $A_\eta=\{X: X\not\ge_T d_\eta\}.$ The specific example produced depends on the choice of the chain $\mathcal{C}$.

There is a serious difference between these two examples, even though they are similar in flavor: there are explicit coding methods, and hence nicely definable instances of the first type. However, it is consistent that there is no "nicely definable" chain $\mathcal{C}$ of $\omega_1$-many Turing degrees, so the latter class of examples is meaningfully less constructive than the former.

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Note that I needed to assume that $\omega_1$ is regular, in both parts. I believe this is necessary - that is, that it is consistent with ZF that every chain has a good countable subchain - but I can't prove it; I suspect Gitik's model provides an example, though.