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If we have a family of classes $(\mathfrak{M}_\alpha)_{\alpha\in D}$ of $\in$-structures with $D$ being a limit ordinal or the class of ordinals, and a family $(\chi_{\alpha,\beta})_{\alpha<\beta(<D)}$ being a direct system (see [here][1] for a definition of a direct system) of end extensions (an end extension $\chi:\langle A,E\rangle\rightarrow\langle B,F\rangle$ being a 1-1 $\in$-homomorphism such that $bF\chi(a)\Leftrightarrow b=\chi(a')$ for some $a'Ea$), how can one define the direct limit of the $(\mathfrak{M}_\alpha,\chi_{\alpha,\beta})$ so that the result is still a class ?

I suppose that the $\chi_{\alpha,\beta}$ are not necessarily compatible, in the sense that given $\alpha$, $\chi_{\alpha,\beta}(x)$ is not necessarily equal to $\chi_{\alpha,\beta'}(x)$ for $\beta\neq\beta'$.

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    $\begingroup$ Don't cross-post. math.stackexchange.com/questions/797938/… $\endgroup$
    – Asaf Karagila ♦
    Commented May 16, 2014 at 17:29
  • $\begingroup$ Ok, I deleted the other post. $\endgroup$
    – fyusuf-a
    Commented May 18, 2014 at 19:15
  • $\begingroup$ Thanks. Generally speaking, you can flag the question and ask for it to be migrated, there's no need to cross-post. $\endgroup$
    – Asaf Karagila ♦
    Commented May 18, 2014 at 19:18

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You can define it the same way as you define a direct limit of sets, using Scott's trick to form equivalence classes. Whenever you have an equivalence relation defined on a class $X$, you can form equivalence classes that are sets by sending $x\in X$ to the set of all $y\in X$ that are equivalent to $x$ and have minimal rank (in the sense of the cumulative hierarchy).

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  • $\begingroup$ In fact in the precise case I am interested in, the $(\mathfrak{M}_\alpha)_{\alpha\in D}$ are defined by transfinite recursion. Except if we consider that it is possible to define classes by transifinite recursion (I would like a bibliography on that if it exists), the ordinals in $D$ are "naive" ordinals in the meta-language. One has to recall that the equivalence relation is on $\sqcup_\alpha |\mathfrak{M}_\alpha|$ whose elements are $(\alpha,m)$ with $m\in|\mathfrak{M}_\alpha|$. Hence $\sqcup_\alpha |\mathfrak{M}_\alpha|$ is not a class. $\endgroup$
    – fyusuf-a
    Commented May 18, 2014 at 19:23
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    $\begingroup$ It sounds to me like your real question has nothing to do with taking the direct limit and is instead about simply defining the sequence $(\mathfrak{M}_\alpha)$ at all. In general I believe you are correct that recursive definitions cannot be made for proper classes, but it may be possible to do so in the specific case you are interested in (for instance, if $\mathfrak{M}_{\alpha}\cap V_\gamma$ can be defined in terms of only $\mathfrak{M}_{\beta}\cap V_\gamma$ for $\beta<\alpha$ instead of the whole of $\mathfrak{M}_{\beta}$). $\endgroup$
    – Eric Wofsey
    Commented May 18, 2014 at 19:48
  • $\begingroup$ Ok, I asked a new question, could you give it a look, please ? $\endgroup$
    – fyusuf-a
    Commented May 20, 2014 at 11:51

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