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Zuhair Al-Johar
Zuhair Al-Johar
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Can $\sf NBG$ class theory prove the foundation scheme:

Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing every occurrence of "$X$" in $\Phi(X)$ by "$Y$"; then: $$\forall \vec{P} \, \bigl(\exists X \, (\Phi(X)) \to \exists X: \Phi(X) \land \forall Y \, (\Phi(X/Y) \to Y \not \in X)\bigr)$$

This would prove second order transfinite recursion$\in$-induction I suppose.

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$? Would it be equi-consistent with $\sf MK$?

Can $\sf NBG$ class theory prove the foundation scheme:

Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing every occurrence of "$X$" in $\Phi(X)$ by "$Y$"; then: $$\forall \vec{P} \, \bigl(\exists X \, (\Phi(X)) \to \exists X: \Phi(X) \land \forall Y \, (\Phi(X/Y) \to Y \not \in X)\bigr)$$

This would prove transfinite recursion I suppose.

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$? Would it be equi-consistent with $\sf MK$?

Can $\sf NBG$ class theory prove the foundation scheme:

Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing every occurrence of "$X$" in $\Phi(X)$ by "$Y$"; then: $$\forall \vec{P} \, \bigl(\exists X \, (\Phi(X)) \to \exists X: \Phi(X) \land \forall Y \, (\Phi(X/Y) \to Y \not \in X)\bigr)$$

This would prove second order transfinite $\in$-induction I suppose.

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$? Would it be equi-consistent with $\sf MK$?

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Zuhair Al-Johar
Zuhair Al-Johar
  • 11.3k
  • 1
  • 13
  • 47

Can $\sf NBG$ class theory prove the foundation scheme:

Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing every occurrence of "$X$" in $\Phi(X)$ by "$Y$"; then: $$\forall \vec{P} \, \bigl(\exists X \, (\Phi(X)) \to \exists X: \Phi(X) \land \forall Y \, (\Phi(X/Y) \to Y \not \in X)\bigr)$$

This would prove transfinite recursion I suppose.

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$, and thus be equi-consistent with $\sf MK$?

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$? Would it be equi-consistent with $\sf MK$?

Can $\sf NBG$ class theory prove the foundation scheme:

Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing every occurrence of "$X$" in $\Phi(X)$ by "$Y$"; then: $$\forall \vec{P} \, \bigl(\exists X \, (\Phi(X)) \to \exists X: \Phi(X) \land \forall Y \, (\Phi(X/Y) \to Y \not \in X)\bigr)$$

This would prove transfinite recursion I suppose.

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$, and thus be equi-consistent with $\sf MK$?

Can $\sf NBG$ class theory prove the foundation scheme:

Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing every occurrence of "$X$" in $\Phi(X)$ by "$Y$"; then: $$\forall \vec{P} \, \bigl(\exists X \, (\Phi(X)) \to \exists X: \Phi(X) \land \forall Y \, (\Phi(X/Y) \to Y \not \in X)\bigr)$$

This would prove transfinite recursion I suppose.

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$? Would it be equi-consistent with $\sf MK$?

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Zuhair Al-Johar
Zuhair Al-Johar
  • 11.3k
  • 1
  • 13
  • 47

Is strengthening Foundation in NBG sufficient to make it prove Con(ZFC)?

Can $\sf NBG$ class theory prove the foundation scheme:

Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing every occurrence of "$X$" in $\Phi(X)$ by "$Y$"; then: $$\forall \vec{P} \, \bigl(\exists X \, (\Phi(X)) \to \exists X: \Phi(X) \land \forall Y \, (\Phi(X/Y) \to Y \not \in X)\bigr)$$

This would prove transfinite recursion I suppose.

If $\sf NBG$ cannot prove this schema, then would replacing Foundation axiom of $\sf NBG$ by this scheme makes the resulting theory prove the existence of a model of $\sf ZFC$, and thus be equi-consistent with $\sf MK$?