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I was wondering about a way to make really large countable ordinals. It turns out that in certain models of ZFC (for example pointwise definable ones) every ordinal is definable with NO parameters, which means you can't expect your ordinal to exist in every model of ZFC without having some FOST equivalent characterization.

Of course, one can simply make an axiom (a scheme with constant a symbol) to create such an ordinal which is not definable within ZFC (unless something with $\omega$-consistency IDK what really). One can then make a scheme with more constant symbols that are all ordinals which cannot be defined from previous ordinals.

A sort of "limit" of this is making an axiom stating the existence of an elementary embedding from $\omega_1$ to itself with a critical point. I believe this could quite easily be proven inconsistent, but I'm not sure how to start.

I call this "The Axiom of Countable Ineffability" or $\text{CI}$ for short. Here's the formal definition of $\text{CI}(Q)$ for a set of formulae $Q$:

1. There is some ordinal $\alpha\in\omega_1$ such that $j(\alpha)\neq\alpha$
2. For every $\alpha\in\omega_1$, $j(\alpha)\in\omega_1$.
3. For every $\varphi\in Q$, an axiom stating that for any $\alpha_0,\alpha_1...\in\omega_1$ with $\varphi^{\omega_1}(\alpha_0,\alpha_1...)$, $\varphi^{\omega_1}(j(\alpha_0),j(\alpha_1)...)$

Of course, because it only quantifies over $\varphi^{\omega_1}$ for FOST $\varphi$, for which there exists a set-sized truth predicate in $V$ (defined on the set of all finite sequences of countable ordinals), this can be stated as a single FOST axiom (as long as the class $\{\varphi^{\omega_1}:\varphi\in Q\}$). In fact, this is true for any uncountable ordinal, not just $\omega_1$.

The Questions

1. What is the consistency of $\text{ZFC}+\text{CI}(\Sigma_n)$?
2. What is the consistency of $\text{ZFC}+\text{CI}$?

I was wondering about a way to make really large countable ordinals. It turns out that in certain models of ZFC (for example pointwise definable ones) every ordinal is definable with no parameters, which means you can't expect your ordinal to exist in every model of ZFC without having some FOST (language of first-order set theory) equivalent characterization.

Of course, one can simply make an axiom (a scheme with constant a symbol) to create such an ordinal which is not definable within ZFC (unless something with $\omega$-consistency, I don’t know what really). One can then make a scheme with more constant symbols that are all ordinals which cannot be defined from previous ordinals.

A sort of "limit" of this is making an axiom stating the existence of an elementary embedding from $\omega_1$ to itself with a critical point. I believe this could quite easily be proven inconsistent, but I'm not sure how to start.

I call this "The Axiom of Countable Ineffability" or $\text{CI}$ for short. Here's the formal definition of $\text{CI}(Q)$ for a set of formulae $Q$:

1. There is some ordinal $\alpha\in\omega_1$ such that $j(\alpha)\neq\alpha$
2. For every $\alpha\in\omega_1$, $j(\alpha)\in\omega_1$.
3. For every $\varphi\in Q$, an axiom stating that for any $\alpha_0,\alpha_1...\in\omega_1$ with $\varphi^{\omega_1}(\alpha_0,\alpha_1...)$, $\varphi^{\omega_1}(j(\alpha_0),j(\alpha_1)...)$

Of course, because it only quantifies over $\varphi^{\omega_1}$ for FOST $\varphi$, for which there exists a set-sized truth predicate in $V$ (defined on the set of all finite sequences of countable ordinals), this can be stated as a single FOST axiom (as long as the class $\{\varphi^{\omega_1}:\varphi\in Q\}$). In fact, this is true for any uncountable ordinal, not just $\omega_1$.

The Questions

1. What is the consistency of $\text{ZFC}+\text{CI}(\Sigma_n)$?
2. What is the consistency of $\text{ZFC}+\text{CI}$?

I was wondering about a way to make really large countable ordinals. It turns out that in certain models of ZFC (for example pointwise definable ones) every ordinal is definable with NO parameters, which means you can't expect your ordinal to exist in every model of ZFC without having some FOST equivalent characterization.

Of course, one can simply make an axiom (a scheme with constant a symbol) to create such an ordinal which is not definable within ZFC (unless something with $\omega$-consistency IDK what really). One can then make a scheme with more constant symbols that are all ordinals which cannot be defined from previous ordinals.

A sort of "limit" of this is making an axiom schema with a function symbol $j$, where $j$ is an elementary embedding from $\omega_1$ to itself with critical point $\theta$. I believe this could quite easily be proven inconsistent, but I'm not sure how to start.

I call this "The Axiom Schema of Countable Ineffability" or $\text{CI}$ for short. Here's the formal definition of $\text{CI}(Q)$ for a set of formulae $Q$:

1. There is some ordinal $\alpha\in\omega_1$ such that $j(\alpha)\neq\alpha$
2. For every $\alpha\in\omega_1$, $j(\alpha)\in\omega_1$.
3. For every $\varphi\in Q$, an axiom stating that for any $\alpha_0,\alpha_1...\in\omega_1$ with $\varphi^{\omega_1}(\alpha_0,\alpha_1...)$, $\varphi^{\omega_1}(j(\alpha_0),j(\alpha_1)...)$

Alternatively, MK could be used to state this in V, for which it shares the name $\text{CI}(Q)$. This axiom is not a schema, just a singular FOST axiom.

The Questions

1. What is the consistency of $\text{ZFC}+\text{CI}(\Sigma_n)$?
2. What is the consistency of $\text{ZFC}+\text{CI}$?
3. Is $\text{MK}+\text{CI}(\Sigma_n)$ equivalent to $\text{MK}+V\models(\text{ZFC}+\text{CI}(\Sigma_n))$? Is it equiconsistent?

I was wondering about a way to make really large countable ordinals. It turns out that in certain models of ZFC (for example pointwise definable ones) every ordinal is definable with NO parameters, which means you can't expect your ordinal to exist in every model of ZFC without having some FOST equivalent characterization.

Of course, one can simply make an axiom (a scheme with constant a symbol) to create such an ordinal which is not definable within ZFC (unless something with $\omega$-consistency IDK what really). One can then make a scheme with more constant symbols that are all ordinals which cannot be defined from previous ordinals.

A sort of "limit" of this is making an axiom stating the existence of an elementary embedding from $\omega_1$ to itself with a critical point. I believe this could quite easily be proven inconsistent, but I'm not sure how to start.

I call this "The Axiom of Countable Ineffability" or $\text{CI}$ for short. Here's the formal definition of $\text{CI}(Q)$ for a set of formulae $Q$:

1. There is some ordinal $\alpha\in\omega_1$ such that $j(\alpha)\neq\alpha$
2. For every $\alpha\in\omega_1$, $j(\alpha)\in\omega_1$.
3. For every $\varphi\in Q$, an axiom stating that for any $\alpha_0,\alpha_1...\in\omega_1$ with $\varphi^{\omega_1}(\alpha_0,\alpha_1...)$, $\varphi^{\omega_1}(j(\alpha_0),j(\alpha_1)...)$

Of course, because it only quantifies over $\varphi^{\omega_1}$ for FOST $\varphi$, for which there exists a set-sized truth predicate in $V$ (defined on the set of all finite sequences of countable ordinals), this can be stated as a single FOST axiom (as long as the class $\{\varphi^{\omega_1}:\varphi\in Q\}$). In fact, this is true for any uncountable ordinal, not just $\omega_1$.

The Questions

1. What is the consistency of $\text{ZFC}+\text{CI}(\Sigma_n)$?
2. What is the consistency of $\text{ZFC}+\text{CI}$?

I was wondering about a way to make really large countable ordinals. It turns out that in certain models of ZFC (for example pointwise definable ones) every ordinal is definable with NO parameters, which means you can't expect your ordinal to exist in every model of ZFC without having some FOST equivalent characterization.

Of course, one can simply make an axiom (a scheme with constant a symbol) to create such an ordinal which is not definable within ZFC (unless something with $\omega$-consistency IDK what really). One can then make a scheme with more constant symbols that are all ordinals which cannot be defined from previous ordinals.

A sort of "limit" of this is making an axiom schema with a function symbol $j$, where $j$ is an elementary embedding from $\omega_1$ to itself with critical point $\theta$. I believe this could quite easily be proven inconsistent, but I'm not sure how to start.

I call this "The Axiom Schema of Countable Ineffability" or $\text{CI}$ for short. Here's the formal definition of $\text{CI}(Q)$ for a set of formulae $Q$:

1. There is some ordinal $\alpha\in\omega_1$ such that $j(\alpha)\neq\alpha$
2. For every $\alpha\in\omega_1$, $j(\alpha)\in\omega_1$.
3. For every $\varphi\in Q$, an axiom stating that for any $\alpha_0,\alpha_1...\in\omega_1$ with $\varphi(\alpha_0,\alpha_1...)$, $\varphi(j(\alpha_0),j(\alpha_1)...)$

Alternatively, MK could be used to state this in V, for which it shares the name $\text{CI}(Q)$. This axiom is not a schema, just a singular FOST axiom.

The Questions

1. What is the consistency of $\text{ZFC}+\text{CI}(\Sigma_n)$?
2. What is the consistency of $\text{ZFC}+\text{CI}$?
3. Is $\text{MK}+\text{CI}(\Sigma_n)$ equivalent to $\text{MK}+V\models(\text{ZFC}+\text{CI}(\Sigma_n))$? Is it equiconsistent?

I was wondering about a way to make really large countable ordinals. It turns out that in certain models of ZFC (for example pointwise definable ones) every ordinal is definable with NO parameters, which means you can't expect your ordinal to exist in every model of ZFC without having some FOST equivalent characterization.

Of course, one can simply make an axiom (a scheme with constant a symbol) to create such an ordinal which is not definable within ZFC (unless something with $\omega$-consistency IDK what really). One can then make a scheme with more constant symbols that are all ordinals which cannot be defined from previous ordinals.

A sort of "limit" of this is making an axiom schema with a function symbol $j$, where $j$ is an elementary embedding from $\omega_1$ to itself with critical point $\theta$. I believe this could quite easily be proven inconsistent, but I'm not sure how to start.

I call this "The Axiom Schema of Countable Ineffability" or $\text{CI}$ for short. Here's the formal definition of $\text{CI}(Q)$ for a set of formulae $Q$:

1. There is some ordinal $\alpha\in\omega_1$ such that $j(\alpha)\neq\alpha$
2. For every $\alpha\in\omega_1$, $j(\alpha)\in\omega_1$.
3. For every $\varphi\in Q$, an axiom stating that for any $\alpha_0,\alpha_1...\in\omega_1$ with $\varphi^{\omega_1}(\alpha_0,\alpha_1...)$, $\varphi^{\omega_1}(j(\alpha_0),j(\alpha_1)...)$

Alternatively, MK could be used to state this in V, for which it shares the name $\text{CI}(Q)$. This axiom is not a schema, just a singular FOST axiom.

The Questions

1. What is the consistency of $\text{ZFC}+\text{CI}(\Sigma_n)$?
2. What is the consistency of $\text{ZFC}+\text{CI}$?
3. Is $\text{MK}+\text{CI}(\Sigma_n)$ equivalent to $\text{MK}+V\models(\text{ZFC}+\text{CI}(\Sigma_n))$? Is it equiconsistent?