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The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrace$$

is provable. (Please forgive my blunt use of the set-builder notation $\lbrace y : \phi(y)\rbrace$. I know there's the rub.)

The same holds for (unrestricted) definitions with parameters: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y,z_1,\dots,z_n)$ such that $x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$ for some $z_1,\dots,z_n$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$$

is provable.

But the notion of ordinal-definable sets is expressible in the language of set theory (see Jech, p. 194), i.e. there is a formula $\delta(x)$ such that - bluntly said -

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \mathsf{ON}(z_i) $$

is provable.

My questions are:

1. Are there other restrictions on the parameters like $\mathsf{ON}(z_i)$ (the formula that states that $z_i$ is an ordinal) that give rise to a definable notion of definability, i.e. a class of accordingly definable sets?

2. Can these restrictions been characterized? What do they have in common, eventually? What is essential?

3. Are there restrictions $\Psi(\phi)$ on the formulas that comparably give rise to a definable notion of definability, i.e. there would be a formula $\delta(x)$ such that $$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \Psi(\phi) $$ is provable.

4. Or maybe some mixture $\Psi(\phi,z_1,\dots,z_n)$?

Question 3 includes: Can we - in a Goedelian way - define for a formula (as a set), that some of its arguments must fulfill some other formula?

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrace$$

is provable. (Please forgive my blunt use of the set-builder notation $\lbrace y : \phi(y)\rbrace$. I know there's the rub.)

The same holds for (unrestricted) definitions with parameters: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y,z_1,\dots,z_n)$ such that $x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$ for some $z_1,\dots,z_n$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$$

is provable.

But the notion of ordinal-definable sets is expressible in the language of set theory (see Jech, p. 194), i.e. there is a formula $\delta(x)$ such that - bluntly said -

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \mathsf{ON}(z_i) $$

is provable.

My questions are:

1. Are there other restrictions on the parameters like $\mathsf{ON}(z_i)$ (the formula that states that $z_i$ is an ordinal) that give rise to a definable notion of definability, i.e. a class of accordingly definable sets?

2. Can these restrictions been characterized? What do they have in common, eventually? What is essential?

3. Are there restrictions $\Psi(\phi)$ on the formulas that comparably give rise to a definable notion of definability, i.e. there would be a formula $\delta(x)$ such that $$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \Psi(\phi) $$ is provable.

4. Or maybe some mixture $\Psi(\phi,z_1,\dots,z_n)$?

Question 3 includes: Can we - in a Goedelian way - define for a formula (as a set), that some of its arguments must fulfill some other formula?

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrace$$

is provable. (Please forgive my blunt use of the set-builder notation $\lbrace y : \phi(y)\rbrace$. I know there's the rub.)

The same holds for (unrestricted) definitions with parameters: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y,z_1,\dots,z_n)$ such that $x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$ for some $z_1,\dots,z_n$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$$

is provable.

But the notion of ordinal-definable sets is expressible in the language of set theory (see Jech, p. 194), i.e. there is a formula $\delta(x)$ such that

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \mathsf{ON}(z_i) $$

is provable.

My questions are:

1. Are there other restrictions on the parameters like $\mathsf{ON}(z_i)$ (the formula that states that $z_i$ is an ordinal) that give rise to a definable notion of definability, i.e. a class of accordingly definable sets?

2. Can these restrictions been characterized? What do they have in common, eventually? What is essential?

3. Are there restrictions $\Psi(\phi)$ on the formulas that comparably give rise to a definable notion of definability, i.e. there would be a formula $\delta(x)$ such that $$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \Psi(\phi) $$ is provable.

4. Or maybe some mixture $\Psi(\phi,z_1,\dots,z_n)$?

Question 3 includes: Can we - in a Goedelian way - define for a formula (as a set), that some of its arguments must fulfill some other formula?

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrace$$

is provable. (Please forgive my blunt use of the set-builder notation $\lbrace y : \phi(y)\rbrace$. I know there's the rub.)

The same holds for (unrestricted) definitions with parameters: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y,z_1,\dots,z_n)$ such that $x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$ for some $z_1,\dots,z_n$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$$

is provable.

But the notion of ordinal-definable sets is expressible in the language of set theory (see Jech, p. 194), i.e. there is a formula $\delta(x)$ such that - bluntly said -

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \mathsf{ON}(z_i) $$

is provable.

My questions are:

1. Are there other restrictions on the parameters like $\mathsf{ON}(z_i)$ (the formula that states that $z_i$ is an ordinal) that give rise to a definable notion of definability, i.e. a class of accordingly definable sets?

2. Can these restrictions been characterized? What do they have in common, eventually? What is essential?

3. Are there restrictions $\Psi(\phi)$ on the formulas that comparably give rise to a definable notion of definability, i.e. there would be a formula $\delta(x)$ such that $$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \Psi(\phi) $$ is provable.

4. Or maybe some mixture $\Psi(\phi,z_1,\dots,z_n)$?

Question 3 includes: Can we - in a Goedelian way - define for a formula (as a set), that some of its arguments must fulfill some other formula?

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrace$$

is provable. (Please forgive my blunt use of the set-builder notation $\lbrace y : \phi(y)\rbrace$. I know there's the rub.)

The same holds for (unrestricted) definitions with parameters: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y,z_1,\dots,z_n)$ such that $x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$ for some $z_1,\dots,z_n$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$$

is provable.

But the notion of ordinal-definable sets is expressible in the language of set theory (see Jech, p. 194), i.e. there is a formula $\delta(x)$ such that

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \mathsf{ON}(z_i) $$

is provable.

My questions are:

1. Are there other restrictions on the parameters like $\mathsf{ON}(z_i)$ (the formula that states that $z_i$ is an ordinal) that give rise to a definable notion of definability, i.e. a class of accordingly definable sets?

2. Can these restrictions been characterized? What do they have in common, eventually? What is essential?

3. Are there restrictions on the formulas that comparably give rise to a definable notion of definability, i.e. there would be a formula $\delta(x)$ such that $$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \Psi(\phi) $$ is provable.

4. Or maybe some mixture $\Psi(\phi,z_1,\dots,z_n)$?

Question 3 includes: Can we - in a Goedelian way - define for a formula (as a set), that some of its arguments must fulfill some other formula?

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrace$$

is provable. (Please forgive my blunt use of the set-builder notation $\lbrace y : \phi(y)\rbrace$. I know there's the rub.)

The same holds for (unrestricted) definitions with parameters: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y,z_1,\dots,z_n)$ such that $x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$ for some $z_1,\dots,z_n$, i.e. for which

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$$

is provable.

But the notion of ordinal-definable sets is expressible in the language of set theory (see Jech, p. 194), i.e. there is a formula $\delta(x)$ such that

$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \mathsf{ON}(z_i) $$

is provable.

My questions are:

1. Are there other restrictions on the parameters like $\mathsf{ON}(z_i)$ (the formula that states that $z_i$ is an ordinal) that give rise to a definable notion of definability, i.e. a class of accordingly definable sets?

2. Can these restrictions been characterized? What do they have in common, eventually? What is essential?

3. Are there restrictions $\Psi(\phi)$ on the formulas that comparably give rise to a definable notion of definability, i.e. there would be a formula $\delta(x)$ such that $$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \Psi(\phi) $$ is provable.

4. Or maybe some mixture $\Psi(\phi,z_1,\dots,z_n)$?

Question 3 includes: Can we - in a Goedelian way - define for a formula (as a set), that some of its arguments must fulfill some other formula?