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I have a question about the following:

enter image description here

I think this should say "... if $I$ finite or if there exists a well-order ..." because if $I$ is a set like this enter image description here

also lets one disjointify $A_i$ with $i \in \{a,b,c,d\}$. Or am I missing something? Thanks!

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    $I$ is finite or admits a well-order if and only if $I$ admits a well-order.2012-10-22

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If $I$ is finite, there is a well-ordering of $I$ in ZF. Thus, the finite case is automatically covered by ‘if there exists a wellorder relation on $I$’.

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    I'm sorry for the (temporary) unaccept, I was too quick on the trigger: isn't a well-order a linear order that is also well-founded? My $I$ is certainly not a linear order. How can I make it into a well-order nonetheless? Thank you for your help!2012-10-22
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    @MattN.: Any order on a finite $I$ that you may already have in hand can be ignored: it’s a theorem of ZF that if $I$ is finite, there is a binary relation $\preceq$ such that $\langle I,\preceq\rangle$ is a well-order. Of course if you happen already to have a well-ordering of your finite $I$, or something that can easily be turned into one, as the one in your picture can be by making $a, you can certainly use it.2012-10-22
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    Oh, right. The order it comes with doesn't matter -- I can pick an index set of same cardinality to index and pick it so that it has a well-order (if possible). In particular, if it's finite, it's in bijection with a subset of $\omega$ and therefore well-ordered.2012-10-22