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I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality, in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. 1961.

Let the language be the language of set theory with identity, and let set(x) abbreviate $\exists y(x\in y)$. All sets are classes, but not vice versa.

Let $a\overset{e}{=}b$ signify that $a$ and $b$ are coextensional.

Let a first order condition A(x,y) be extent functional just if $\forall x, y, z(A(x,y)\wedge A(x,z)\to y\overset{e}{=}z)$.

Suppose $\mathbb{I}$ is the least fixed point class so that $$\forall x(x\in\mathbb{I}\leftrightarrow Set(x)\wedge x\subset\mathbb{I})$$

for the following version of $ST$:

Strong infinity says that the following class, with binders restricted to sets when need be, is as well a set: $$\{x|\forall v([\forall w(\forall x(x\notin w)\to w\in v)\wedge \forall w\forall x(w\in v\wedge\forall y(y\in x\leftrightarrow\forall z(z\in y\to z\in w))\to x\in v)]\to x\in v)\}$$

Finite union says that $\{x|x\in a\vee x\in b\}$ is a set if $a$ and $b$ are sets.

Infinite union says that $\{x|\exists y(x\in y\wedge y\in a\}$ is a set if $a$ is a set.

Power restricted to sets: $\{x|Set(x)\wedge x\subset a\}$ is a set if $a$ is a set.

Extent-replacement says that $\{y|\exists x(x\in a\wedge A(x,y)\}$ is a set if $a$ is a set, and $A(x,y)$ is an extent-functional first order condition on $x$ and $y$.

Specification follows from Extent-replacement

The principle of extensionality is not an axiom.

(Scott, 1961) showed that his version of ST with an unresricted power set, interprets $ZF$ minus regularity. Given the relative consistency result for the axiom of foundation by (von Neumann, 1929), Scott showed that his version of $ST$ interprets $ZF$, and so $ZFC$ via (Gödel, 1938 & 1940).

Am I right in thinking that by the results of (Gödel, 1940) and (Cohen, 1963), and others, the minimality of $\mathbb{I}$ entails that regularity and $AC$ and $CH$ hold in the Scott-induced interpretation of $ZF$ via $\mathbb{I}$?

I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality, in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. 1961.

Let the language be the language of set theory with identity, and let set(x) abbreviate $\exists y(x\in y)$. All sets are classes, but not vice versa.

Let $a\overset{e}{=}b$ signify that $a$ and $b$ are coextensional.

Let a first order condition A(x,y) be extent functional just if $\forall x, y, z(A(x,y)\wedge A(x,z)\to y\overset{e}{=}z)$.

Suppose $\mathbb{I}$ is the least fixed point class $$\forall x(x\in\mathbb{I}\leftrightarrow Set(x)\wedge x\subset\mathbb{I})$$

for the following version of $ST$:

Strong infinity which says that the following class, with binders restricted to sets when need be, is as well a set: $$\{x|\forall v([\forall w(\forall x(x\notin w)\to w\in v)\wedge \forall w\forall x(w\in v\wedge\forall y(y\in x\leftrightarrow\forall z(z\in y\to z\in w))\to x\in v)]\to x\in v)\}$$

Finite union which says that $\{x|x\in a\vee x\in b\}$ is a set if $a$ and $b$ are sets.

Infinite union which says that $\{x|\exists y(x\in y\wedge y\in a\}$ is a set if $a$ is a set.

Power restricted to sets, which says that $\{x|Set(x)\wedge x\subset a\}$ is a set if $a$ is a set.

Extent-replacement says that $\{y|\exists x(x\in a\wedge A(x,y)\}$ is a set if $a$ is a set, and $A(x,y)$ is an extent-functional first order condition on $x$ and $y$.

Specification follows from Extent-replacement

The principle of extensionality is not an axiom.

(Scott, 1961) showed that his version of ST with an unresricted power set, interprets $ZF$ minus regularity. Given the relative consistency result for the axiom of foundation by (von Neumann, 1929), Scott showed that his version of $ST$ interprets $ZF$, and so $ZFC$ via (Gödel, 1938 & 1940).

Am I right in thinking that by the results of (Gödel, 1940) and (Cohen, 1963), and others, the minimality of $\mathbb{I}$ entails that regularity and $AC$ and $CH$ hold in the Scott-induced interpretation of $ZF$ via $\mathbb{I}$?

I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality, in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. 1961.

Let the language be the language of set theory with identity, and let set(x) abbreviate $\exists y(x\in y)$. All sets are classes, but not vice versa.

Let $a\overset{e}{=}b$ signify that $a$ and $b$ are coextensional.

Let a first order condition A(x,y) be extent functional just if $\forall x, y, z(A(x,y)\wedge A(x,z)\to y\overset{e}{=}z)$.

Let $WF(x)$ signify that x is well founded, and suppose $\mathbb{I}$ is the least fixed point class so that $$\forall x(x\in\mathbb{I}\leftrightarrow Set(x)\wedge WF(x)\wedge x\subset\mathbb{I})$$

for the following version of $ST$:

Strong infinity says that the following class, with binders restricted to sets when need be, is as well a set: $$\{x|\forall v([\forall w(\forall x(x\notin w)\to w\in v)\wedge \forall w\forall x(w\in v\wedge\forall y(y\in x\leftrightarrow\forall z(z\in y\to z\in w))\to x\in v)]\to x\in v)\}$$

Finite union says that $\{x|x\in a\vee x\in b\}$ is a set if $a$ and $b$ are sets.

Infinite union says that $\{x|\exists y(x\in y\wedge y\in a\}$ is a set if $a$ is a set.

Power restricted to sets: $\{x|Set(x)\wedge x\subset a\}$ is a set if $a$ is a set.

Extent-replacement says that $\{y|\exists x(x\in a\wedge A(x,y)\}$ is a set if $a$ is a set, and $A(x,y)$ is an extent-functional first order condition on $x$ and $y$.

Specification follows from Extent-replacement

We include regularity: $\exists x(x\in a)\to\exists x(x\in a\wedge\lnot\exists y(y\in x\wedge y\in a))$

(Scott, 1961) showed that his version of ST without regularity, and an unresricted power set, interprets $ZF$ minus regularity. Given the relative consistency result for the axiom of foundation by (von Neumann, 1929), Scott showed that his version of $ST$ interprets $ZF$, and so $ZFC$ via (Gödel, 1938 & 1940).

I here included regularity as I may obtain the minimal fixed point $\mathbb{I}$ in a particular setting, and it seems to me that it does not block Scott's interpretation of the rest of $ZF$. The inclusion of regularity gives a direct intake to the question I articulate. Nevertheless, my inclusion of regularity may be dispensed with if there are important objections, and if so I would rephrase my question with a detour via (von Neumann, 1929) as indicated with (Scott, 1961).

Am I right in thinking that by the results of (Gödel, 1940) and (Cohen, 1963), and others, the minimality of $\mathbb{I}$ entails that $AC$ and $CH$ hold in the Scott-induced interpretation of $ZF$ via $\mathbb{I}$?

I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality, in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. 1961.

Let the language be the language of set theory with identity, and let set(x) abbreviate $\exists y(x\in y)$. All sets are classes, but not vice versa.

Let $a\overset{e}{=}b$ signify that $a$ and $b$ are coextensional.

Let a first order condition A(x,y) be extent functional just if $\forall x, y, z(A(x,y)\wedge A(x,z)\to y\overset{e}{=}z)$.

Suppose $\mathbb{I}$ is the least fixed point class so that $$\forall x(x\in\mathbb{I}\leftrightarrow Set(x)\wedge x\subset\mathbb{I})$$

for the following version of $ST$:

Strong infinity says that the following class, with binders restricted to sets when need be, is as well a set: $$\{x|\forall v([\forall w(\forall x(x\notin w)\to w\in v)\wedge \forall w\forall x(w\in v\wedge\forall y(y\in x\leftrightarrow\forall z(z\in y\to z\in w))\to x\in v)]\to x\in v)\}$$

Finite union says that $\{x|x\in a\vee x\in b\}$ is a set if $a$ and $b$ are sets.

Infinite union says that $\{x|\exists y(x\in y\wedge y\in a\}$ is a set if $a$ is a set.

Power restricted to sets: $\{x|Set(x)\wedge x\subset a\}$ is a set if $a$ is a set.

Extent-replacement says that $\{y|\exists x(x\in a\wedge A(x,y)\}$ is a set if $a$ is a set, and $A(x,y)$ is an extent-functional first order condition on $x$ and $y$.

Specification follows from Extent-replacement

The principle of extensionality is not an axiom.

(Scott, 1961) showed that his version of ST with an unresricted power set, interprets $ZF$ minus regularity. Given the relative consistency result for the axiom of foundation by (von Neumann, 1929), Scott showed that his version of $ST$ interprets $ZF$, and so $ZFC$ via (Gödel, 1938 & 1940).

Am I right in thinking that by the results of (Gödel, 1940) and (Cohen, 1963), and others, the minimality of $\mathbb{I}$ entails that regularity and $AC$ and $CH$ hold in the Scott-induced interpretation of $ZF$ via $\mathbb{I}$?

I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality, in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. 1961.

Let the language be the language of set theory with identity, and let set(x) abbreviate $\exists y(x\in y)$. All sets are classes, but not vice versa.

Let $a\overset{e}{=}b$ signify that $a$ and $b$ are coextensional.

Let a first order condition A(x,y) be extent functional just if $\forall x, y, z(A(x,y)\wedge A(x,z)\to y\overset{e}{=}z)$.

Let $WF(x)$ signify that x is well founded, and suppose $\mathbb{I}$ is the least fixed point class so that $$\forall x(x\in\mathbb{I}\leftrightarrow Set(x)\wedge WF(x)\wedge x\subset\mathbb{I})$$

for the following version of $ST$:

Strong infinity says that the following class, with binders restricted to sets when need be, is as well a set: $$\{x|\forall v([\forall w(\forall x(x\notin w)\to w\in v)\wedge \forall w\forall x(w\in v\wedge\forall y(y\in x\leftrightarrow\forall z(z\in y\to z\in w))\to x\in v)]\to x\in v)\}$$

Finite union says that $\{x|x\in a\vee x\in b\}$ is a set if $a$ and $b$ are sets.

Infinite union says that $\{x|\exists y(x\in y\wedge y\in a\}$ is a set if $a$ is a set.

Power restricted to sets: $\{x|Set(x)\wedge x\subset a\}$ is a set if $a$ is a set.

Extent-replacement says that $\{y|\exists x(x\in a\wedge A(x,y)\}$ is a set if $a$ is a set, and $A(x,y)$ is an extent-functional first order condition on $x$ and $y$.

Specification follows from Extent-replacement

We include regularity: $\exists x(x\in a)\to\exists x(x\in a\wedge\lnot\exists y(y\in x\wedge y\in a))$

(Scott, 1961) showed that his version of ST without regularity, and an unresricted power set, interprets $ZF$ minus regularity. Given the relative consistency result for the axiom of foundation by (von Neumann, 1929), Scott showed that his version of $ST$ interprets $ZF$, and so $ZFC$ via (Gödel, 1938 & 1940).

I here included regularity as I may obtain the minimal fixed point $\mathbb{I}$ in a particular setting, and it seems to me that it does not block Scott's interpretation of the rest of $ZF$. The inclusion of regularity gives a direct intake to the question I articulate. Nevertheless, my inclusion of regularity may be dispensed with if there are important objections, and if so I would rephrase my question with a detour via (von Neumann, 1929) as indicated with (Scott, 1961).

Am I right in thinking that by the results of (Gödel, 1940) and (Cohen, 1963), and others, the minimality of $\mathbb{I}$ entails that $AC$ and $CH$ hold in $\mathbb{I}$?

I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality, in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. 1961.

Let the language be the language of set theory with identity, and let set(x) abbreviate $\exists y(x\in y)$. All sets are classes, but not vice versa.

Let $a\overset{e}{=}b$ signify that $a$ and $b$ are coextensional.

Let a first order condition A(x,y) be extent functional just if $\forall x, y, z(A(x,y)\wedge A(x,z)\to y\overset{e}{=}z)$.

Let $WF(x)$ signify that x is well founded, and suppose $\mathbb{I}$ is the least fixed point class so that $$\forall x(x\in\mathbb{I}\leftrightarrow Set(x)\wedge WF(x)\wedge x\subset\mathbb{I})$$

for the following version of $ST$:

Strong infinity says that the following class, with binders restricted to sets when need be, is as well a set: $$\{x|\forall v([\forall w(\forall x(x\notin w)\to w\in v)\wedge \forall w\forall x(w\in v\wedge\forall y(y\in x\leftrightarrow\forall z(z\in y\to z\in w))\to x\in v)]\to x\in v)\}$$

Finite union says that $\{x|x\in a\vee x\in b\}$ is a set if $a$ and $b$ are sets.

Infinite union says that $\{x|\exists y(x\in y\wedge y\in a\}$ is a set if $a$ is a set.

Power restricted to sets: $\{x|Set(x)\wedge x\subset a\}$ is a set if $a$ is a set.

Extent-replacement says that $\{y|\exists x(x\in a\wedge A(x,y)\}$ is a set if $a$ is a set, and $A(x,y)$ is an extent-functional first order condition on $x$ and $y$.

Specification follows from Extent-replacement

We include regularity: $\exists x(x\in a)\to\exists x(x\in a\wedge\lnot\exists y(y\in x\wedge y\in a))$

(Scott, 1961) showed that his version of ST without regularity, and an unresricted power set, interprets $ZF$ minus regularity. Given the relative consistency result for the axiom of foundation by (von Neumann, 1929), Scott showed that his version of $ST$ interprets $ZF$, and so $ZFC$ via (Gödel, 1938 & 1940).

I here included regularity as I may obtain the minimal fixed point $\mathbb{I}$ in a particular setting, and it seems to me that it does not block Scott's interpretation of the rest of $ZF$. The inclusion of regularity gives a direct intake to the question I articulate. Nevertheless, my inclusion of regularity may be dispensed with if there are important objections, and if so I would rephrase my question with a detour via (von Neumann, 1929) as indicated with (Scott, 1961).

Am I right in thinking that by the results of (Gödel, 1940) and (Cohen, 1963), and others, the minimality of $\mathbb{I}$ entails that $AC$ and $CH$ hold in the Scott-induced interpretation of $ZF$ via $\mathbb{I}$?