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• A cardinal $\kappa$ is $\Sigma_n$-correct if all $\Sigma_n$ assertions are absolute between $V$ and $V_\kappa$. These are the cardinals to which assertions of that complexity are reflected. There is a closed unbounded class $C_n$ of $\Sigma_n$ correct cardinals, also known as the $C^{(n)}$-cardinals. But beware, one cannot necessarily intersect these class clubs to form $\bigcap_n C^{(n)}$, the class of fully correct cardinals, since the definition of being $\Sigma_n$ is not uniform in $n$, but increasingly complex as $n$ increases.

• Reflection is equivalent to the axiom of replacement over the other axioms of set theory. This is because if $A$ is a set and you have witness objects $b$ for every element $a\in A$, then by reflection there is some $V_\alpha$ in which those witnesses reside and for which the witnessing property is absolute. So now the set of witnesses can be picked out via separation as a subset of $V_\alpha$.

• Reflection is also equivalent to the principle of transfinite recursion. I explain on my blog.

• Philosophical ideas based on reflection are often mentioned, by some of the authors you mention in your post, as providing philosophical justification for the existence of large cardinals. For example, one early idea is that if we accept reflection as a fundamental principle, then ZFC is saying essentially that the class of ordinals is a regular strong limit, and so we should expect there to be some rank initial segment $V_\kappa$ in which this is true, making $\kappa$ an inaccessible cardinal. Further arguments continue with this to much larger large cardinals. See work of Reinhardt, and also Maddy on Believing the Axioms.

• In my paper, Categorical large cardinals and the tension between categoricity and set-theoretic reflection, J.D. Hamkins and H.R. Solberg, we discuss the inherent tension between reflection and categoricity in set-theoretic foundations. Mathematicians often prefer categorical accounts of their basic mathematical structures, and yet also prefer reflection, which is at heart anti-categorical.

• A cardinal $\kappa$ is $\Sigma_n$-correct if all $\Sigma_n$ assertions are absolute between $V$ and $V_\kappa$. These are the cardinals to which assertions of that complexity are reflected. There is a closed unbounded class $C^{(n)}$ of $\Sigma_n$ correct cardinals, also known as the $C^{(n)}$-cardinals. But beware, one cannot necessarily intersect these class clubs to form $\bigcap_n C^{(n)}$, the class of fully correct cardinals, since the definition of being $\Sigma_n$ is not uniform in $n$, but increasingly complex as $n$ increases.

• Reflection is equivalent to the axiom of replacement over the other axioms of set theory. This is because if $A$ is a set and you have witness objects $b$ for every element $a\in A$, then by reflection there is some $V_\alpha$ in which those witnesses reside and for which the witnessing property is absolute. So now the set of witnesses can be picked out via separation as a subset of $V_\alpha$.

• Reflection is also equivalent to the principle of transfinite recursion. I explain on my blog.

• Philosophical ideas based on reflection are often mentioned, by some of the authors you mention in your post, as providing philosophical justification for the existence of large cardinals. For example, one early idea is that if we accept reflection as a fundamental principle, then ZFC is saying essentially that the class of ordinals is a regular strong limit, and so we should expect there to be some rank initial segment $V_\kappa$ in which this is true, making $\kappa$ an inaccessible cardinal. Further arguments continue with this to much larger large cardinals. See work of Reinhardt, and also Maddy on Believing the Axioms.

• In my paper, Categorical large cardinals and the tension between categoricity and set-theoretic reflection, J.D. Hamkins and H.R. Solberg, we discuss the inherent tension between reflection and categoricity in set-theoretic foundations. Mathematicians often prefer categorical accounts of their basic mathematical structures, and yet also prefer reflection, which is at heart anti-categorical.

• A cardinal $\kappa$ is $\Sigma_n$-correct if all $\Sigma_n$ assertions are absolute between $V$ and $V_\kappa$. There is a closed unbounded class $C_n$ of $\Sigma_n$ correct cardinals, also known as the $C_n$-cardinals. But beware, one cannot necessarily intersect these class clubs to form $\bigcap_n C_n$, the class of fully correct cardinals, since the definition of being $\Sigma_n$ is not uniform in $n$, but increasingly complex as $n$ increases.

• Reflection is equivalent to the axiom of replacement over the other axioms of set theory. This is because if $A$ is a set and you have witness objects $b$ for every element $a\in A$, then by reflection there is some $V_\alpha$ in which those witnesses reside and for which the witnessing property is absolute. So now the set of witnesses can be picked out via separation as a subset of $V_\alpha$.

• Reflection is also equivalent to the principle of transfinite recursion. I explain on my blog.

• Philosophical ideas based on reflection are often mentioned, by some of the authors you mention in your post, as providing philosophical justification for the existence of large cardinals. For example, one early idea is that if we accept reflection as a fundamental principle, then ZFC is saying essentially that the class of ordinals is a regular strong limit, and so we should expect there to be some rank initial segment $V_\kappa$ in which this is true, making $\kappa$ an inaccessible cardinal. Further arguments continue with this to much larger large cardinals. See work of Reinhardt, and also Maddy on Believing the Axioms.

• In my paper, Categorical large cardinals and the tension between categoricity and set-theoretic reflection, J.D. Hamkins and H.R. Solberg, we discuss the inherent tension between reflection and categoricity in set-theoretic foundations. Mathematicians often prefer categorical accounts of their basic mathematical structures, and yet also prefer reflection, which is at heart anti-categorical.

• A cardinal $\kappa$ is $\Sigma_n$-correct if all $\Sigma_n$ assertions are absolute between $V$ and $V_\kappa$. These are the cardinals to which assertions of that complexity are reflected. There is a closed unbounded class $C_n$ of $\Sigma_n$ correct cardinals, also known as the $C^{(n)}$-cardinals. But beware, one cannot necessarily intersect these class clubs to form $\bigcap_n C^{(n)}$, the class of fully correct cardinals, since the definition of being $\Sigma_n$ is not uniform in $n$, but increasingly complex as $n$ increases.

• Reflection is equivalent to the axiom of replacement over the other axioms of set theory. This is because if $A$ is a set and you have witness objects $b$ for every element $a\in A$, then by reflection there is some $V_\alpha$ in which those witnesses reside and for which the witnessing property is absolute. So now the set of witnesses can be picked out via separation as a subset of $V_\alpha$.

• Reflection is also equivalent to the principle of transfinite recursion. I explain on my blog.

• Philosophical ideas based on reflection are often mentioned, by some of the authors you mention in your post, as providing philosophical justification for the existence of large cardinals. For example, one early idea is that if we accept reflection as a fundamental principle, then ZFC is saying essentially that the class of ordinals is a regular strong limit, and so we should expect there to be some rank initial segment $V_\kappa$ in which this is true, making $\kappa$ an inaccessible cardinal. Further arguments continue with this to much larger large cardinals. See work of Reinhardt, and also Maddy on Believing the Axioms.

• In my paper, Categorical large cardinals and the tension between categoricity and set-theoretic reflection, J.D. Hamkins and H.R. Solberg, we discuss the inherent tension between reflection and categoricity in set-theoretic foundations. Mathematicians often prefer categorical accounts of their basic mathematical structures, and yet also prefer reflection, which is at heart anti-categorical.

• Reflection is equivalent to the axiom of replacement over the other axioms of set theory. This is because if $A$ is a set and you have witness objects $b$ for every element $a\in A$, then by reflection there is some $V_\alpha$ in which those witnesses reside and for which the witnessing property is absolute. So now the set of witnesses can be picked out via separation as a subset of $V_\alpha$.

• Reflection is also equivalent to the principle of transfinite recursion. I explain on my blog.

• Philosophical ideas based on reflection are often mentioned, by some of the authors you mention in your post, as providing philosophical justification for the existence of large cardinals. For example, one early idea is that if we accept reflection as a fundamental principle, then ZFC is saying essentially that the class of ordinals is a regular strong limit, and so we should expect there to be some rank initial segment $V_\kappa$ in which this is true, making $\kappa$ an inaccessible cardinal. Further arguments continue with this to much larger large cardinals. See work of Reinhardt, and also Maddy on Believing the Axioms.

• In my paper, Categorical large cardinals and the tension between categoricity and set-theoretic reflection, J.D. Hamkins and H.R. Solberg, we discuss the inherent tension between reflection and categoricity in set-theoretic foundations. Mathematicians often prefer categorical accounts of their basic mathematical structures, and yet also prefer reflection, which is at heart anti-categorical.

In our joint paper, Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal, we prove that this princi

• A cardinal $\kappa$ is $\Sigma_n$-correct if all $\Sigma_n$ assertions are absolute between $V$ and $V_\kappa$. There is a closed unbounded class $C_n$ of $\Sigma_n$ correct cardinals, also known as the $C_n$-cardinals. But beware, one cannot necessarily intersect these class clubs to form $\bigcap_n C_n$, the class of fully correct cardinals, since the definition of being $\Sigma_n$ is not uniform in $n$, but increasingly complex as $n$ increases.

• Reflection is equivalent to the axiom of replacement over the other axioms of set theory. This is because if $A$ is a set and you have witness objects $b$ for every element $a\in A$, then by reflection there is some $V_\alpha$ in which those witnesses reside and for which the witnessing property is absolute. So now the set of witnesses can be picked out via separation as a subset of $V_\alpha$.

• Reflection is also equivalent to the principle of transfinite recursion. I explain on my blog.

• Philosophical ideas based on reflection are often mentioned, by some of the authors you mention in your post, as providing philosophical justification for the existence of large cardinals. For example, one early idea is that if we accept reflection as a fundamental principle, then ZFC is saying essentially that the class of ordinals is a regular strong limit, and so we should expect there to be some rank initial segment $V_\kappa$ in which this is true, making $\kappa$ an inaccessible cardinal. Further arguments continue with this to much larger large cardinals. See work of Reinhardt, and also Maddy on Believing the Axioms.

• In my paper, Categorical large cardinals and the tension between categoricity and set-theoretic reflection, J.D. Hamkins and H.R. Solberg, we discuss the inherent tension between reflection and categoricity in set-theoretic foundations. Mathematicians often prefer categorical accounts of their basic mathematical structures, and yet also prefer reflection, which is at heart anti-categorical.

In fact, we identify a collection of bi-interpretation results between these urelement theories and pure set theories.