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I've been told once or twice that the following holds:

There is a model of $ZFC+MA+\neg CH$ in which there is a $\mathfrak{c}$-universal linear order embedded in $(\omega^\omega, \le^\ast)$

Moreover, several people have attributed the construction of such a model to Hugh Woodin. The context that seems the most natural for this construction to appear is in his work on automatic-continuity, however for the life of me, I can't seem to find it.

It might also be the case that the embedding is into $\mathcal{P}(\omega)/fin$.

Does anybody know of a reference for this? Or if Woodin did indeed construct it (I only ask because my searches have yielded little fruit)?

Edit:

In this context a $\mathfrak{c}$-universal linear order $(L, \prec)$ has the following property: $\vert L \vert \le \mathfrak{c}$ and for every linear order $(\ell, <)$, with $\vert \ell \vert \le \mathfrak{c}$ there is an embedding $\varphi_\ell:\ell \rightarrow L$ respecting the linear order of $\ell$.

Thought I should share what I can find/know:

So far I've been able to find models for the following

$ZFC+ \neg CH$ and there is such a $\mathfrak{c}$-universal linear order in $(\omega^\omega,\le^\ast)$.

This is due to Laver, http://www.sciencedirect.com/science/article/pii/S0049237X08716306

In addition, there is also

$ZFC+MA+\neg CH$ and no such order is embedded in $(\omega^\omega,\le^\ast)$

There seem to be several models for this one, Laver cites two, one from Solovay and one from Kunen. I can only assume the Solovay model is the same constructed in Theorem 5.7 (p. 201, "Hausdroff Gaps and Limits", by Frankiewicz and Zbierski)

In addition there is

$CH\implies$ $(\omega^\omega,\le^\ast)$ is contains a $\mathfrak{c}$-universal linear order

Which should be contained in (or atleast hinted at in) "Model Theory", by Chang and Keisler. If not there, the lack of $(\omega,\omega)$-gaps in $(\omega^\omega,\le^\ast)$ should produce it rather quickly.

So yeah, it would seem that the only link missing in this puzzle is the one I can't find the reference for.

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I've been told once or twice that the following holds:

There is a model of $ZFC+MA+\neg CH$ in which there is a $\mathfrak{c}$-universal linear order embedded in $(\omega^\omega, \le^\ast)$

Moreover, several people have attributed the construction of such a model to Hugh Woodin. The context that seems the most natural for this construction to appear is in his work on automatic-continuity, however for the life of me, I can't seem to find it.

It might also be the case that the embedding is into $\mathcal{P}(\omega)/fin$.

Does anybody know of a reference for this? Or if Woodin did indeed construct it (I only ask because my searches have yielded little fruit)?

Edit:

In this context a $\mathfrak{c}$-universal linear order $(L, \prec)$ has the following property: $\vert L \vert \le \mathfrak{c}$ and for every linear order $(\ell, <)$, with $\vert \ell \vert \le \mathfrak{c}$ there is an embedding $\varphi_\ell:\ell \rightarrow L$ respecting the linear order of $\ell$.

Thought I should share what I can find/know:

So far I've been able to find models for the following

$ZFC+ \neg CH$ and there is such a $\mathfrak{c}$-universal linear order in $(\omega^\omega,\le^\ast)$.

This is due to Laver, http://www.sciencedirect.com/science/article/pii/S0049237X08716306

In addition, there is also

$ZFC+MA+\neg CH$ and no such order is embedded in $(\omega^\omega,\le^\ast)$

There seem to be several models for this one, Laver cites two, one from Solovay and one from Kunen. I can only assume the Solovay model is the same constructed in Theorem 5.7 (p. 201, "Hausdroff Gaps and Limits", by Frankiewicz and Zbierski)

In addition there is

$CH\implies$ there $(\omega^\omega,\le^\ast)$ is a $\mathfrak{c}$-universal linear orderin $(\omega^\omega,\le^\ast)$

Which should be contained in (or atleast hinted at in) "Model Theory", by Chang and Keisler. If not there, the lack of $(\omega,\omega)$-gaps in $(\omega^\omega,\le^\ast)$ should produce it rather quickly.

So yeah, it would seem that the only link missing in this puzzle is the one I can't find the reference for.

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Thought I should share what I can find/know:

So far I've been able to find models for the following

$ZFC+ \neg CH$ and there is such a $\mathfrak{c}$-universal linear order in $(\omega^\omega,\le^\ast)$.

This is due to Laver, http://www.sciencedirect.com/science/article/pii/S0049237X08716306

In addition, there is also

$ZFC+MA+\neg CH$ and no such order is embedded in $(\omega^\omega,\le^\ast)$

There seem to be several models for this one, Laver cites two, one from Solovay and one from Kunen. I can only assume the Solovay model is the same constructed in Theorem 5.7 (p. 201, "Hausdroff Gaps and Limits", by Frankiewicz and Zbierski)

In addition there is

$CH\implies$ there is a $\mathfrak{c}$-universal linear order in $(\omega^\omega,\le^\ast)$

Which should be contained in (or atleast hinted at in) "Model Theory", by Chang and Keisler. If not there, the lack of $(\omega,\omega)$-gaps in $(\omega^\omega,\le^\ast)$ should produce it rather quickly.

So yeah, it would seem that the only link missing in this puzzle is the one I can't find the reference for.

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