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I pondered this question some time ago. As pointed out by Stefan Geschke, infinity and power set are indispensable. Moreover:

• The replacement or collection schema is indispensable: $V_{\omega+\omega}$ is a model of all other axioms of ZFC (including separation).

• The sum set axiom is indispensable: $H_{\beth_\omega}$ is a model of other axioms of ZFC.

• The collection schema (if included in your axiomatization of ZFC) or extensionality is indispensable. One can build a model of other axioms by essentially taking $V_{\omega+\omega}$, and blowing it up to have a lot of automorphisms. Namely, put $M_0=\varnothing$, $M_{\alpha+1}=2\times\mathcal P(M_\alpha)$, and $M_\gamma=\bigcup_{\alpha<\gamma}M_\alpha$ for limit $\gamma$. Then take the model $\langle M,\in^M\rangle$, where $M=M_{\omega+\omega}$, and $\in^M$ is defined as follows: $x\in^M\langle i,y\rangle$ iff $x\in y$. The only nontrivial axiom to check is the replacement schema. It follows from the following property: if $M\models\exists!x\,\phi(x,\vec a)$, where $\vec a\in M_\alpha$, $\alpha<\omega+\omega$, then the unique witness for $x$ also belongs to $M_\alpha$. This in turn holds because there exists an automorphism $f$ of $M$ which is identical on $M_\alpha$, but which moves all elements of $M-M_\alpha$, namely

  $f(\langle i,x\rangle)=\begin{cases}\langle i,x\rangle&\text{if }\langle i,x\rangle\in M_\alpha,\\\\ \langle 1-i,x\rangle&\text{otherwise.}\end{cases}$

On the other hand:

• Infinity, sum set, power set, extensionality and replacement interpret ZFC. That's just ZF without foundation: it interprets ZF in the well-founded kernel $WF=\bigcup_\alpha V_\alpha$, and ZF interprets ZFC in $L$.

• Infinity, sum set, power set, separation (or replacement) and collection interpret ZFC. This requires some work, but here's at least the definition: the domain of interpretation consists of triples $\langle a,e,x\rangle$ such that $e$ is a well-founded extensional binary relation on $a$, and $x\in a$. Equality is interpreted by the relation $\langle a,e,x\rangle=^*\langle a',e',x'\rangle$ defined to hold iff there exists a partial embedding $f$ of the relational structure $\langle a,e\rangle$ to $\langle a',e'\rangle$ with $e$-transitive domain, $e'$-transitive range, and such that $f(x)=x'$. The elementhood predicate is interpreted by $\langle a,e,x\rangle\in^*\langle a',e',x'\rangle$ iff there exists $y\in a'$ such that $\langle y,x'\rangle\in e'$ and $\langle a,e,x\rangle=^*\langle a',e',y\rangle$. One can check that this is an interpretation of all axioms of ZF without foundation, which in turn interprets ZFC by the previous point.

EDIT: I realized that I'm using here a nonstandard definition of $H_\kappa$ which is equivalent to the usual one for regular $\kappa$, but not for singular. The $H_{\beth_\omega}$ above is meant to denote the set of all sets $x$ such that every set in the transitive closure of $\{x\}$ has cardinality (strictly) less than $\beth_\omega$ (but the transitive closure itself can have cardinality $\beth_\omega$).

I pondered this question some time ago. As pointed out by Stefan Geschke, infinity and power set are indispensable. Moreover:

• The replacement or collection schema is indispensable: $V_{\omega+\omega}$ is a model of all other axioms of ZFC (including separation).

• The sum set axiom is indispensable: $H_{\beth_\omega}$ is a model of other axioms of ZFC.

• The collection schema (if included in your axiomatization of ZFC) or extensionality is indispensable. One can build a model of other axioms by essentially taking $V_{\omega+\omega}$, and blowing it up to have a lot of automorphisms. Namely, put $M_0=\varnothing$, $M_{\alpha+1}=2\times\mathcal P(M_\alpha)$, and $M_\gamma=\bigcup_{\alpha<\gamma}M_\alpha$ for limit $\gamma$. Then take the model $\langle M,\in^M\rangle$, where $M=M_{\omega+\omega}$, and $\in^M$ is defined as follows: $x\in^M\langle i,y\rangle$ iff $x\in y$. The only nontrivial axiom to check is the replacement schema. It follows from the following property: if $M\models\exists!x\,\phi(x,\vec a)$, where $\vec a\in M_\alpha$, $\alpha<\omega+\omega$, then the unique witness for $x$ also belongs to $M_\alpha$. This in turn holds because there exists an automorphism $f$ of $M$ which is identical on $M_\alpha$, but which moves all elements of $M-M_\alpha$, namely

  $f(\langle i,x\rangle)=\begin{cases}\langle i,x\rangle&\text{if }\langle i,x\rangle\in M_\alpha,\\\\ \langle 1-i,x\rangle&\text{otherwise.}\end{cases}$

On the other hand:

• Infinity, sum set, power set, extensionality and replacement interpret ZFC. That's just ZF without foundation: it interprets ZF in the well-founded kernel $WF=\bigcup_\alpha V_\alpha$, and ZF interprets ZFC in $L$.

• Infinity, sum set, power set, separation (or replacement) and collection interpret ZFC. This requires some work, but here's at least the definition: the domain of interpretation consists of triples $\langle a,e,x\rangle$ such that $e$ is a well-founded extensional binary relation on $a$, and $x\in a$. Equality is interpreted by the relation $\langle a,e,x\rangle =^*\langle a',e',x'\rangle$ defined to hold iff there exists a partial embedding $f$ of the relational structure $\langle a,e\rangle$ to $\langle a',e'\rangle$ with $e$-transitive domain, $e'$-transitive range, and such that $f(x)=x'$. The elementhood predicate is interpreted by $\langle a,e,x\rangle\in^*\langle a',e',x'\rangle$ iff there exists $y\in a'$ such that $\langle y,x'\rangle\in e'$ and $\langle a,e,x\rangle=^*\langle a',e',y\rangle$. One can check that this is an interpretation of all axioms of ZF without foundation, which in turn interprets ZFC by the previous point.

EDIT: I realized that I'm using here a nonstandard definition of $H_\kappa$ which is equivalent to the usual one for regular $\kappa$, but not for singular. The $H_{\beth_\omega}$ above is meant to denote the set of all sets $x$ such that every set in the transitive closure of $\{x\}$ has cardinality (strictly) less than $\beth_\omega$ (but the transitive closure itself can have cardinality $\beth_\omega$).

I pondered this question some time ago. As pointed out by Stefan Geschke, infinity and power set are indispensable. Moreover:

• The replacement or collection schema is indispensable: $V_{\omega+\omega}$ is a model of all other axioms of ZFC (including separation).

• The sum set axiom is indispensable: $H_{\beth_\omega}$ is a model of other axioms of ZFC.

• The collection schema (if included in your axiomatization of ZFC) or extensionality is indispensable. One can build a model of other axioms by essentially taking $V_{\omega+\omega}$, and blowing it up to have a lot of automorphisms. Namely, put $M_0=\varnothing$, $M_{\alpha+1}=2\times\mathcal P(M_\alpha)$, and $M_\gamma=\bigcup_{\alpha<\gamma}M_\alpha$ for limit $\gamma$. Then take the model $\langle M,\in^M\rangle$, where $M=M_{\omega+\omega}$, and $\in^M$ is defined as follows: $x\in^M\langle i,y\rangle$ iff $x\in y$. The only nontrivial axiom to check is the replacement schema. It follows from the following property: if $M\models\exists!x\,\phi(x,\vec a)$, where $\vec a\in M_\alpha$, $\alpha<\omega+\omega$, then the unique witness for $x$ also belongs to $M_\alpha$. This in turn holds because there exists an automorphism $f$ of $M$ which is identical on $M_\alpha$, but which moves all elements of $M-M_\alpha$, namely

  $f(\langle i,x\rangle)=\begin{cases}\langle i,x\rangle&\text{if }\langle i,x\rangle\in M_\alpha,\\\\ \langle 1-i,x\rangle&\text{otherwise.}\end{cases}$

On the other hand:

• Infinity, sum set, power set, extensionality and replacement interpret ZFC. That's just ZF without foundation: it interprets ZF in the well-founded kernel $WF=\bigcup_\alpha V_\alpha$, and ZF interprets ZFC in $L$.

• Infinity, sum set, power set, separation (or replacement) and collection interpret ZFC. This requires some work, but here's at least the definition: the domain of interpretation consists of triples $\langle a,e,x\rangle$ such that $e$ is a well-founded extensional binary relation on $a$, and $x\in a$. Equality is interpreted by the relation $\langle a,e,x\rangle=^*\langle a',e',x'\rangle$ defined to hold iff there exists a partial embedding $f$ of the relational structure $\langle a,e\rangle$ to $\langle a',e'\rangle$ with $e$-transitive domain, $e'$-transitive range, and such that $f(x)=x'$. The elementhood predicate is interpreted by $\langle a,e,x\rangle\in^*\langle a',e',x'\rangle$ iff there exists $y\in a'$ such that $\langle y,x'\rangle\in e'$ and $\langle a,e,x\rangle=^*\langle a',e',y\rangle$. One can check that this is an interpretation of all axioms of ZF without foundation, which in turn interprets ZFC by the previous point.

EDIT: I realized that I'm using here a nonstandard definition of $H_\kappa$ which is equivalent to the usual one for regular $\kappa$, but not for singular. The $H_{\beth_\omega}$ above is meant to denote the set of all sets $x$ such that every set in the transitive closure of $\{x\}$ has cardinality (strictly) less than $\beth_\omega$ (but the transitive closure itself can have larger cardinality).

I pondered this question some time ago. As pointed out by Stefan Geschke, infinity and power set are indispensable. Moreover:

• The replacement or collection schema is indispensable: $V_{\omega+\omega}$ is a model of all other axioms of ZFC (including separation).

• The sum set axiom is indispensable: $H_{\beth_\omega}$ is a model of other axioms of ZFC.

• The collection schema (if included in your axiomatization of ZFC) or extensionality is indispensable. One can build a model of other axioms by essentially taking $V_{\omega+\omega}$, and blowing it up to have a lot of automorphisms. Namely, put $M_0=\varnothing$, $M_{\alpha+1}=2\times\mathcal P(M_\alpha)$, and $M_\gamma=\bigcup_{\alpha<\gamma}M_\alpha$ for limit $\gamma$. Then take the model $\langle M,\in^M\rangle$, where $M=M_{\omega+\omega}$, and $\in^M$ is defined as follows: $x\in^M\langle i,y\rangle$ iff $x\in y$. The only nontrivial axiom to check is the replacement schema. It follows from the following property: if $M\models\exists!x\,\phi(x,\vec a)$, where $\vec a\in M_\alpha$, $\alpha<\omega+\omega$, then the unique witness for $x$ also belongs to $M_\alpha$. This in turn holds because there exists an automorphism $f$ of $M$ which is identical on $M_\alpha$, but which moves all elements of $M-M_\alpha$, namely

  $f(\langle i,x\rangle)=\begin{cases}\langle i,x\rangle&\text{if }\langle i,x\rangle\in M_\alpha,\\\\ \langle 1-i,x\rangle&\text{otherwise.}\end{cases}$

On the other hand:

• Infinity, sum set, power set, extensionality and replacement interpret ZFC. That's just ZF without foundation: it interprets ZF in the well-founded kernel $WF=\bigcup_\alpha V_\alpha$, and ZF interprets ZFC in $L$.

• Infinity, sum set, power set, separation (or replacement) and collection interpret ZFC. This requires some work, but here's at least the definition: the domain of interpretation consists of triples $\langle a,e,x\rangle$ such that $e$ is a well-founded extensional binary relation on $a$, and $x\in a$. Equality is interpreted by the relation $\langle a,e,x\rangle=^*\langle a',e',x'\rangle$ defined to hold iff there exists a partial embedding $f$ of the relational structure $\langle a,e\rangle$ to $\langle a',e'\rangle$ with $e$-transitive domain, $e'$-transitive range, and such that $f(x)=x'$. The elementhood predicate is interpreted by $\langle a,e,x\rangle\in^*\langle a',e',x'\rangle$ iff there exists $y\in a'$ such that $\langle y,x'\rangle\in e'$ and $\langle a,e,x\rangle=^*\langle a',e',y\rangle$. One can check that this is an interpretation of all axioms of ZF without foundation, which in turn interprets ZFC by the previous point.

EDIT: I realized that I'm using here a nonstandard definition of $H_\kappa$ which is equivalent to the usual one for regular $\kappa$, but not for singular. The $H_{\beth_\omega}$ above is meant to denote the set of all sets $x$ such that every set in the transitive closure of $\{x\}$ has cardinality (strictly) less than $\beth_\omega$ (but the transitive closure itself can have cardinality $\beth_\omega$).

I pondered this question some time ago. As pointed out by Stefan Geschke, infinity and power set are indispensable. Moreover:

• The replacement or collection schema is indispensable: $V_{\omega+\omega}$ is a model of all other axioms of ZFC (including separation).

• The sum set axiom is indispensable: $H_{\beth_\omega}$ is a model of other axioms of ZFC.

• The collection schema (if included in your axiomatization of ZFC) or extensionality is indispensable. One can build a model of other axioms by essentially taking $V_{\omega+\omega}$, and blowing it up to have a lot of automorphisms. Namely, put $M_0=\varnothing$, $M_{\alpha+1}=2\times\mathcal P(M_\alpha)$, and $M_\gamma=\bigcup_{\alpha<\gamma}M_\alpha$ for limit $\gamma$. Then take the model $\langle M,\in^M\rangle$, where $M=M_{\omega+\omega}$, and $\in^M$ is defined as follows: $x\in^M\langle i,y\rangle$ iff $x\in y$. The only nontrivial axiom to check is the replacement schema. It follows from the following property: if $M\models\exists!x\,\phi(x,\vec a)$, where $\vec a\in M_\alpha$, $\alpha<\omega+\omega$, then the unique witness for $x$ also belongs to $M_\alpha$. This in turn holds because there exists an automorphism $f$ of $M$ which is identical on $M_\alpha$, but which moves all elements of $M-M_\alpha$, namely

  $f(\langle i,x\rangle)=\begin{cases}\langle i,x\rangle&\text{if }\langle i,x\rangle\in M_\alpha,\\\\ \langle 1-i,x\rangle&\text{otherwise.}\end{cases}$

On the other hand:

• Infinity, sum set, extensionality and replacement interpret ZFC. That's just ZF without foundation: it interprets ZF in the well-founded kernel $WF=\bigcup_\alpha V_\alpha$, and ZF interprets ZFC in $L$.

• Infinity, sum set, separation (or replacement) and collection interpret ZFC. This requires some work, but here's at least the definition: the domain of interpretation consists of triples $\langle a,e,x\rangle$ such that $e$ is a well-founded extensional binary relation on $a$, and $x\in a$. Equality is interpreted by the relation $\langle a,e,x\rangle=^*\langle a',e',x'\rangle$ defined to hold iff there exists a partial embedding $f$ of the relational structure $\langle a,e\rangle$ to $\langle a',e'\rangle$ with $e$-transitive domain, $e'$-transitive range, and such that $f(x)=x'$. The elementhood predicate is interpreted by $\langle a,e,x\rangle\in^*\langle a',e',x'\rangle$ iff there exists $y\in a'$ such that $\langle y,x'\rangle\in e'$ and $\langle a,e,x\rangle=^*\langle a',e',y\rangle$. One can check that this is an interpretation of all axioms of ZF without foundation, which in turn interprets ZFC by the previous point.

EDIT: I realized that I'm using here a nonstandard definition of $H_\kappa$ which is equivalent to the usual one for regular $\kappa$, but not for singular. The $H_{\beth_\omega}$ above is meant to denote the set of all sets $x$ such that every set in the transitive closure of $\{x\}$ has cardinality (strictly) less than $\beth_\omega$ (but the transitive closure itself can have larger cardinality).

I pondered this question some time ago. As pointed out by Stefan Geschke, infinity and power set are indispensable. Moreover:

• The replacement or collection schema is indispensable: $V_{\omega+\omega}$ is a model of all other axioms of ZFC (including separation).

• The sum set axiom is indispensable: $H_{\beth_\omega}$ is a model of other axioms of ZFC.

• The collection schema (if included in your axiomatization of ZFC) or extensionality is indispensable. One can build a model of other axioms by essentially taking $V_{\omega+\omega}$, and blowing it up to have a lot of automorphisms. Namely, put $M_0=\varnothing$, $M_{\alpha+1}=2\times\mathcal P(M_\alpha)$, and $M_\gamma=\bigcup_{\alpha<\gamma}M_\alpha$ for limit $\gamma$. Then take the model $\langle M,\in^M\rangle$, where $M=M_{\omega+\omega}$, and $\in^M$ is defined as follows: $x\in^M\langle i,y\rangle$ iff $x\in y$. The only nontrivial axiom to check is the replacement schema. It follows from the following property: if $M\models\exists!x\,\phi(x,\vec a)$, where $\vec a\in M_\alpha$, $\alpha<\omega+\omega$, then the unique witness for $x$ also belongs to $M_\alpha$. This in turn holds because there exists an automorphism $f$ of $M$ which is identical on $M_\alpha$, but which moves all elements of $M-M_\alpha$, namely

  $f(\langle i,x\rangle)=\begin{cases}\langle i,x\rangle&\text{if }\langle i,x\rangle\in M_\alpha,\\\\ \langle 1-i,x\rangle&\text{otherwise.}\end{cases}$

On the other hand:

• Infinity, sum set, power set, extensionality and replacement interpret ZFC. That's just ZF without foundation: it interprets ZF in the well-founded kernel $WF=\bigcup_\alpha V_\alpha$, and ZF interprets ZFC in $L$.

• Infinity, sum set, power set, separation (or replacement) and collection interpret ZFC. This requires some work, but here's at least the definition: the domain of interpretation consists of triples $\langle a,e,x\rangle$ such that $e$ is a well-founded extensional binary relation on $a$, and $x\in a$. Equality is interpreted by the relation $\langle a,e,x\rangle=^*\langle a',e',x'\rangle$ defined to hold iff there exists a partial embedding $f$ of the relational structure $\langle a,e\rangle$ to $\langle a',e'\rangle$ with $e$-transitive domain, $e'$-transitive range, and such that $f(x)=x'$. The elementhood predicate is interpreted by $\langle a,e,x\rangle\in^*\langle a',e',x'\rangle$ iff there exists $y\in a'$ such that $\langle y,x'\rangle\in e'$ and $\langle a,e,x\rangle=^*\langle a',e',y\rangle$. One can check that this is an interpretation of all axioms of ZF without foundation, which in turn interprets ZFC by the previous point.

EDIT: I realized that I'm using here a nonstandard definition of $H_\kappa$ which is equivalent to the usual one for regular $\kappa$, but not for singular. The $H_{\beth_\omega}$ above is meant to denote the set of all sets $x$ such that every set in the transitive closure of $\{x\}$ has cardinality (strictly) less than $\beth_\omega$ (but the transitive closure itself can have larger cardinality).