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A very nice collection of questions.

Here are a few things one can say to get started.

• If $\kappa$ is a measurable cardinal and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, suppose that $M$ is a model of $T$ in which $\leq$ is a well-order and $M$ has size at least $\kappa$. Since $\kappa$ is measurable, we may by iterating the ultrapower maps find an elementary embedding $j:V\to N$ into a transitive class $N$ with critical point $\kappa$ and $j(\kappa)$ as large as desired. So $j(M)$ is a model of $T$, with the order $j(\leq)$ of size at least $j(\kappa)$ and a well-order in $N$ and hence also still a well-order in $V$.

• One can do it with less than a measurable. Specifically, if $\kappa$ is merely an unfoldable cardinal (this is consistent with $V=L$, so much smaller than measurable, but above indescribable), and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, note first that a downward Löwenheim-Skolem argument shows that $M$ has well-ordered models of size exactly $\kappa$. Now, we can place this model into a transitive model $N$ of size $\kappa$, and then apply unfoldability to get elementary embeddings $j:N\to \bar N$ with $j(\kappa)$ as large as desired. The same reasoning as before shows that $j(M)$ is a well-ordered model of large size.

You can get a kind of converse from the same idea, using the extension property characterization of unfoldability, if one allows the language to become larger.

Theorem. A cardinal $\kappa$ is unfoldable if and only if every theory $T$ of size at most $\kappa$ with a model of size at least $\kappa$ in which $\leq$ is a well-order, has well-ordered models of arbitrarily large size.

The idea is to write down the diagram of $\langle V_\kappa,\in,A,\leq\rangle$, where $\leq$ is a well-order of $V_\kappa$ and $A\subset V_\kappa$, and then get arbitrarily large extensions $\langle N,\in,A^*,\leq^*\rangle$, which will give you the extension version of the unfoldability property.

It seems to me that your questions are very closely related to how one often thinks about unfoldable cardinals.

A very nice collection of questions.

Here are a few things one can say to get started.

• If $\kappa$ is a measurable cardinal and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, suppose that $M$ is a model of $T$ in which $\leq$ is a well-order and $M$ has size at least $\kappa$. Since $\kappa$ is measurable, we may by iterating the ultrapower maps find an elementary embedding $j:V\to N$ into a transitive class $N$ with critical point $\kappa$ and $j(\kappa)$ as large as desired. So $j(M)$ is a model of $T$, with the order $j(\leq)$ of size at least $j(\kappa)$ and a well-order in $N$ and hence also still a well-order in $V$.

• One can do it with less than a measurable. Specifically, if $\kappa$ is merely an unfoldable cardinal (this is consistent with $V=L$, so much smaller than measurable, but above indescribable), and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, note first that a downward Löwenheim-Skolem argument shows that $M$ has well-ordered models of size exactly $\kappa$. Now, we can place this model into a transitive model $N$ of size $\kappa$, and then apply unfoldability to get elementary embeddings $j:N\to \bar N$ with $j(\kappa)$ as large as desired. The same reasoning as before shows that $j(M)$ is a well-ordered model of large size.

You can get a kind of converse from the same idea, using the extension property characterization of unfoldability, if one allows the language to become larger.

Theorem. A cardinal $\kappa$ is unfoldable if and only if every theory $T$ of size at most $\kappa$ with a model of size at least $\kappa$ in which $\leq$ is a well-order, has well-ordered models of arbitrarily large size.

The idea is to write down the diagram of $\langle V_\kappa,\in,A,\leq\rangle$, where $\leq$ is a well-order of $V_\kappa$ and $A\subset V_\kappa$, and then get arbitrarily large extensions $\langle N,\in,A^*,\leq^*\rangle$, which will give you the extension version of the unfoldability property.

It seems to me that your questions are very closely related to how one often thinks about unfoldable cardinals.

If you are only seeking end-extensions, rather than arbitrarily large extensions, then it is weak compactness, a weaker large cardinal, that you will want. An inaccessible cardinal $\kappa$ is weakly compact if and only if for every set $A\subset V_\kappa$, there is an elementary extension of $\langle V_\kappa,\in,A\rangle$ to a taller well-founded model $\langle M,\in,A^*\rangle$.

A very nice collection of questions.

Here are a few things one can say to get started.

• If $\kappa$ is a measurable cardinal and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, suppose that $M$ is a model of $T$ in which $\leq$ is a well-order and $M$ has size at least $\kappa$. Since $\kappa$ is measurable, we may by iterating the ultrapower maps find an elementary embedding $j:V\to N$ into a transitive class $N$ with critical point $\kappa$ and $j(\kappa)$ as large as desired. So $j(M)$ is a model of $T$, with the order $j(\leq)$ of size at least $j(\kappa)$ and a well-order in $N$ and hence also still a well-order in $V$.

• One can do it with less than a measurable. Specifically, if $\kappa$ is merely an unfoldable cardinal (this is consistent with $V=L$, so much smaller than measurable, but above indescribable), and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, note first that a downward Löwenheim-Skolem argument shows that $M$ has well-ordered models of size exactly $\kappa$. Now, we can place this model into a transitive model $N$ of size $\kappa$, and then apply unfoldability to get elementary embeddings $j:N\to \bar N$ with $j(\kappa)$ as large as desired. The same reasoning as before shows that $j(M)$ is a well-ordered model of large size.

You can get a kind of converse from the same idea, using the extension property characterization of unfoldability, if one allows the language to become larger. The idea will be to write down the diagram of $\langle V_\kappa,\in,A,\leq\rangle$, where $\leq$ is a well-order of $V_\kappa$ and $A\subset V_\kappa$, and then get arbitrarily large extensions $\langle N,\in,A^*,\leq^*\rangle$, which will give you the unfoldability property.

In fact, it seems to me that your questions are very closely related to how one often thinks about unfoldable cardinals.

A very nice collection of questions.

Here are a few things one can say to get started.

• If $\kappa$ is a measurable cardinal and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, suppose that $M$ is a model of $T$ in which $\leq$ is a well-order and $M$ has size at least $\kappa$. Since $\kappa$ is measurable, we may by iterating the ultrapower maps find an elementary embedding $j:V\to N$ into a transitive class $N$ with critical point $\kappa$ and $j(\kappa)$ as large as desired. So $j(M)$ is a model of $T$, with the order $j(\leq)$ of size at least $j(\kappa)$ and a well-order in $N$ and hence also still a well-order in $V$.

• One can do it with less than a measurable. Specifically, if $\kappa$ is merely an unfoldable cardinal (this is consistent with $V=L$, so much smaller than measurable, but above indescribable), and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, note first that a downward Löwenheim-Skolem argument shows that $M$ has well-ordered models of size exactly $\kappa$. Now, we can place this model into a transitive model $N$ of size $\kappa$, and then apply unfoldability to get elementary embeddings $j:N\to \bar N$ with $j(\kappa)$ as large as desired. The same reasoning as before shows that $j(M)$ is a well-ordered model of large size.

You can get a kind of converse from the same idea, using the extension property characterization of unfoldability, if one allows the language to become larger.

Theorem. A cardinal $\kappa$ is unfoldable if and only if every theory $T$ of size at most $\kappa$ with a model of size at least $\kappa$ in which $\leq$ is a well-order, has well-ordered models of arbitrarily large size.

The idea is to write down the diagram of $\langle V_\kappa,\in,A,\leq\rangle$, where $\leq$ is a well-order of $V_\kappa$ and $A\subset V_\kappa$, and then get arbitrarily large extensions $\langle N,\in,A^*,\leq^*\rangle$, which will give you the extension version of the unfoldability property.

It seems to me that your questions are very closely related to how one often thinks about unfoldable cardinals.

A very nice collection of questions.

Here are a few things one can say to get started.

• If $\kappa$ is a measurable cardinal and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, suppose that $M$ is a model of $T$ in which $\leq$ is a well-order and $M$ has size at least $\kappa$. Since $\kappa$ is measurable, we may by iterating the ultrapower maps find an elementary embedding $j:V\to N$ into a transitive class $N$ with critical point $\kappa$ and $j(\kappa)$ as large as desired. So $j(M)$ is a model of $T$, with the order $j(\leq)$ a well-order in $N$ and hence still a well-order in $V$.

• One can do it with less than a measurable. Specifically, if $\kappa$ is merely an unfoldable cardinal (this is consistent with $V=L$, so much smaller than measurable, but above indescribable), and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, note first that a downward Löwenheim-Skolem argument shows that $M$ has well-ordered models of size exactly $\kappa$. Now, we can place this model into a transitive model $N$ of size $\kappa$, and then apply unfoldability to get elementary embeddings $j:N\to \bar N$ with $j(\kappa)$ as large as desired. The same reasoning as before shows that $j(M)$ is a well-ordered model of large size.

You can get a kind of converse from the same idea, using the extension property characterization of unfoldability, if one allows the language to become larger. The idea will be to write down the diagram of $\langle V_\kappa,\in,A,\leq\rangle$, where $\leq$ is a well-order of $V_\kappa$ and $A\subset V_\kappa$, and then get arbitrarily large extensions $\langle N,\in,A^*,\leq^*\rangle$, which will give you the unfoldability property.

A very nice collection of questions.

Here are a few things one can say to get started.

• If $\kappa$ is a measurable cardinal and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, suppose that $M$ is a model of $T$ in which $\leq$ is a well-order and $M$ has size at least $\kappa$. Since $\kappa$ is measurable, we may by iterating the ultrapower maps find an elementary embedding $j:V\to N$ into a transitive class $N$ with critical point $\kappa$ and $j(\kappa)$ as large as desired. So $j(M)$ is a model of $T$, with the order $j(\leq)$ of size at least $j(\kappa)$ and a well-order in $N$ and hence also still a well-order in $V$.

• One can do it with less than a measurable. Specifically, if $\kappa$ is merely an unfoldable cardinal (this is consistent with $V=L$, so much smaller than measurable, but above indescribable), and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, note first that a downward Löwenheim-Skolem argument shows that $M$ has well-ordered models of size exactly $\kappa$. Now, we can place this model into a transitive model $N$ of size $\kappa$, and then apply unfoldability to get elementary embeddings $j:N\to \bar N$ with $j(\kappa)$ as large as desired. The same reasoning as before shows that $j(M)$ is a well-ordered model of large size.

You can get a kind of converse from the same idea, using the extension property characterization of unfoldability, if one allows the language to become larger. The idea will be to write down the diagram of $\langle V_\kappa,\in,A,\leq\rangle$, where $\leq$ is a well-order of $V_\kappa$ and $A\subset V_\kappa$, and then get arbitrarily large extensions $\langle N,\in,A^*,\leq^*\rangle$, which will give you the unfoldability property.

In fact, it seems to me that your questions are very closely related to how one often thinks about unfoldable cardinals.