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The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength really.

Define: $set(X) \iff \exists Y ( X \in Y)$

Extensionality: $\forall A \forall B: \forall Z (Z \in A \Leftrightarrow Z \in B) \implies A=B$

Comprehension: $\exists X \forall Y (Y \in X \iff set(Y) \land \phi)$; where $X$ not free in $\phi$.

Limitation of Size: $X \in V \iff \not \exists f:V \hookrightarrow X $; $``\hookrightarrow"$ for "injections"; $V$ is the class of all sets.

Foundation: $X \neq \emptyset \implies \exists Y \in X (Y \cap X = \emptyset)$

Reflection: if $\phi_1,..,\phi_n$ are pure set theoretic formulas in which $Y$ is free, and their parameters among symbols $A,B$; and if $\pi_i$ is the following formula (with omission of subformulas with un-used parameters) $$``\forall A,B \ are \ H_{<W} \\ [\forall Y(\phi_i \Rightarrow H_{<W}(Y)) \to \exists X < W \forall Y(Y \in X \Leftrightarrow\phi_i) ]"$$

then: $$\exists W: \pi_1 \land ... \land \pi_n $$

Where $H_{<W}$ is the predicate "hereditarily strictly subnumerous to $W$" defined by the predicated classes being strictly subnumerous to $W$ and every element of their transitive closures being also strictly subnumerous to $W$; and $<$ stands for strict subnumerousity defined in the usual sense. A pure set theoretic formula means a bounded $``\in V"$ formula whose predicates are $\in,=$ and its terms are variables, except $V$ which only appears in bounding all quantified variables in it.

I claim that this theory can interpret Ackermann and Morse-Kelley set theory [with modified size limitation axiom of every class equinumerous to a set is a set], and also Muller's extension of Ackermann's. Also, I think it interprets Tarski–Grothendieck set theory (TG). I'm not really sure if it reaches up to the consistency of $ORD$ is Mahlo. It does all of that in the usual language of set theory!

Is ORD is Mahlo, the exact consistency strength of this theory?

The reason why I'm asking that is because adding limitation of size principle on top of Ackermann will blow it up to ORD is Mahlo. But here we are proving finite instances of that reflection principle of Ackermann, on top of size limitation, so this might prove to be much weaker possibly at the level of just interpreting TG, however I'm not so sure of that, hence my question.

The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength really.

Define: $set(X) \iff \exists Y ( X \in Y)$

Extensionality: $\forall A \forall B: \forall Z (Z \in A \Leftrightarrow Z \in B) \implies A=B$

Comprehension: $\exists X \forall Y (Y \in X \iff set(Y) \land \phi)$; where $X$ not free in $\phi$.

Limitation of Size: $X \in V \iff \not \exists f:V \hookrightarrow X $; $``\hookrightarrow"$ for "injections"; $V$ is the class of all sets.

Foundation: $X \neq \emptyset \implies \exists Y \in X (Y \cap X = \emptyset)$

Reflection: if $\phi_1,..,\phi_n$ are pure set theoretic formulas in which $Y$ is free, and their parameters among symbols $A,B$; and if $\pi_i$ is the following formula (or the maximal subformula of it after omissions of unused parameters) $$``\forall A,B \ are \ H_{<W} \\ [\forall Y(\phi_i \Rightarrow H_{<W}(Y)) \to \exists X < W \forall Y(Y \in X \Leftrightarrow\phi_i) ]"$$

then: $$\exists W: \pi_1 \land ... \land \pi_n $$

Where $H_{<W}$ is the predicate "hereditarily strictly subnumerous to $W$" defined by the predicated classes being strictly subnumerous to $W$ and every element of their transitive closures being also strictly subnumerous to $W$; and $<$ stands for strict subnumerousity defined in the usual sense. A pure set theoretic formula means a bounded $``\in V"$ formula whose predicates are $\in,=$ and its terms are variables, except $V$ which only appears in bounding all quantified variables in it.

I claim that this theory can interpret Ackermann and Morse-Kelley set theory [with modified size limitation axiom of every class equinumerous to a set is a set], and also Muller's extension of Ackermann's. Also, I think it interprets Tarski–Grothendieck set theory (TG). I'm not really sure if it reaches up to the consistency of $ORD$ is Mahlo. It does all of that in the usual language of set theory!

Is ORD is Mahlo, the exact consistency strength of this theory?

The reason why I'm asking that is because adding limitation of size principle on top of Ackermann will blow it up to ORD is Mahlo. But here we are proving finite instances of that reflection principle of Ackermann, on top of size limitation, so this might prove to be much weaker possibly at the level of just interpreting TG, however I'm not so sure of that, hence my question.

The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength really.

Define: $set(X) \iff \exists Y ( X \in Y)$

Extensionality: $\forall A \forall B: \forall Z (Z \in A \Leftrightarrow Z \in B) \implies A=B$

Comprehension: $\exists X \forall Y (Y \in X \iff set(Y) \land \phi)$; where $X$ not free in $\phi$.

Limitation of Size: $X \in V \iff \not \exists f:V \hookrightarrow X $; $``\hookrightarrow"$ for "injections"; $V$ is the class of all sets.

Foundation: $X \neq \emptyset \implies \exists Y \in X (Y \cap X = \emptyset)$

Reflection: if $\phi_1,..,\phi_n$ are pure set theoretic formulas in which $Y$ is free, and their parameters among symbols $A,B$; and if $\pi_i$ is the formula $$``\forall A,B \ are \ H_{<W} \\ [\forall Y(\phi_i \Rightarrow H_{<W}(Y)) \to \exists X < W \forall Y(Y \in X \Leftrightarrow\phi_i) ]"$$

then: $$\exists W: \pi_1 \land ... \land \pi_n $$

Where $H_{<W}$ is the predicate "hereditarily strictly subnumerous to $W$" defined by the predicated classes being strictly subnumerous to $W$ and every element of their transitive closures being also strictly subnumerous to $W$; and $<$ stands for strict subnumerousity defined in the usual sense. A pure set theoretic formula means a bounded $``\in V"$ formula whose predicates are $\in,=$ and its terms are variables, except $V$ which only appears in bounding all quantified variables in it.

I claim that this theory can interpret Ackermann and Morse-Kelley set theory [with modified size limitation axiom of every class equinumerous to a set is a set], and also Muller's extension of Ackermann's. Also, I think it interprets Tarski–Grothendieck set theory (TG). I'm not really sure if it reaches up to the consistency of $ORD$ is Mahlo. It does all of that in the usual language of set theory!

Is ORD is Mahlo, the exact consistency strength of this theory?

The reason why I'm asking that is because adding limitation of size principle on top of Ackermann will blow it up to ORD is Mahlo. But here we are proving finite instances of that reflection principle of Ackermann, on top of size limitation, so this might prove to be much weaker possibly at the level of just interpreting TG, however I'm not so sure of that, hence my question.

The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength really.

Define: $set(X) \iff \exists Y ( X \in Y)$

Extensionality: $\forall A \forall B: \forall Z (Z \in A \Leftrightarrow Z \in B) \implies A=B$

Comprehension: $\exists X \forall Y (Y \in X \iff set(Y) \land \phi)$; where $X$ not free in $\phi$.

Limitation of Size: $X \in V \iff \not \exists f:V \hookrightarrow X $; $``\hookrightarrow"$ for "injections"; $V$ is the class of all sets.

Foundation: $X \neq \emptyset \implies \exists Y \in X (Y \cap X = \emptyset)$

Reflection: if $\phi_1,..,\phi_n$ are pure set theoretic formulas in which $Y$ is free, and their parameters among symbols $A,B$; and if $\pi_i$ is the following formula (with omission of subformulas with un-used parameters) $$``\forall A,B \ are \ H_{<W} \\ [\forall Y(\phi_i \Rightarrow H_{<W}(Y)) \to \exists X < W \forall Y(Y \in X \Leftrightarrow\phi_i) ]"$$

then: $$\exists W: \pi_1 \land ... \land \pi_n $$

Where $H_{<W}$ is the predicate "hereditarily strictly subnumerous to $W$" defined by the predicated classes being strictly subnumerous to $W$ and every element of their transitive closures being also strictly subnumerous to $W$; and $<$ stands for strict subnumerousity defined in the usual sense. A pure set theoretic formula means a bounded $``\in V"$ formula whose predicates are $\in,=$ and its terms are variables, except $V$ which only appears in bounding all quantified variables in it.

I claim that this theory can interpret Ackermann and Morse-Kelley set theory [with modified size limitation axiom of every class equinumerous to a set is a set], and also Muller's extension of Ackermann's. Also, I think it interprets Tarski–Grothendieck set theory (TG). I'm not really sure if it reaches up to the consistency of $ORD$ is Mahlo. It does all of that in the usual language of set theory!

Is ORD is Mahlo, the exact consistency strength of this theory?

The reason why I'm asking that is because adding limitation of size principle on top of Ackermann will blow it up to ORD is Mahlo. But here we are proving finite instances of that reflection principle of Ackermann, on top of size limitation, so this might prove to be much weaker possibly at the level of just interpreting TG, however I'm not so sure of that, hence my question.

A nice development of the theory presented at the prior posting to Mathoverflow, is that one can dispense with the notion of smallness altogether! And can work solely in the language of set theory, so no need to add any primitive notion over the predicates of equality and membership. To re-iterate the axioms:

Define: $set(X) \iff \exists Y ( X \in Y)$

Extensionality: $\forall A \forall B: \forall Z (Z \in A \Leftrightarrow Z \in B) \implies A=B$

Comprehension: $\exists X \forall Y (Y \in X \iff set(Y) \land \phi)$; where $X$ not free in $\phi$.

Limitation of Size: $X \in V \iff \not \exists f:V \hookrightarrow X $; $``\hookrightarrow"$ for "injections"; $V$ is the class of all sets.

Foundation: $X \neq \emptyset \implies \exists Y \in X (Y \cap X = \emptyset)$

Reflection: if $\phi_1,..,\phi_n$ are pure set theoretic formulas in which $Y$ is free, and their parameters among symbols $A,B$; and if $\pi_i$ is the formula $$``\forall A,B \ are \ H_{<W} \\ [\forall Y(\phi_i \Rightarrow H_{<W}(Y)) \to \exists X < W \forall Y(Y \in X \Leftrightarrow\phi_i) ]"$$

then: $$\exists W: \pi_1 \land ... \land \pi_n $$

Where $H_{<W}$ is the predicate "hereditarily strictly subnumerous to $W$" defined by the predicated classes being strictly subnumerous to $W$ and every element of their transitive closures being also strictly subnumerous to $W$; and $<$ stands for strict subnumerousity defined in the usual sense. A pure set theoretic formula means a bounded $``\in V"$ formula whose predicates are $\in,=$ and its terms are variables, except $V$ which only appears in bounding all quantified variables in it.

I claim that this theory can interpret Ackermann and Morse-Kelley set theory [with modified size limitation axiom of every class equinumerous to a set is a set], and also Muller's extension of Ackermann's. Also, I think it interprets Tarski–Grothendieck set theory (TG). I'm not really sure if it reaches up to the consistency of $ORD$ is Mahlo. It does all of that in the usual language of set theory!

Is ORD is Mahlo, the exact consistency strength of this theory?

The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength really.

Define: $set(X) \iff \exists Y ( X \in Y)$

Extensionality: $\forall A \forall B: \forall Z (Z \in A \Leftrightarrow Z \in B) \implies A=B$

Comprehension: $\exists X \forall Y (Y \in X \iff set(Y) \land \phi)$; where $X$ not free in $\phi$.

Limitation of Size: $X \in V \iff \not \exists f:V \hookrightarrow X $; $``\hookrightarrow"$ for "injections"; $V$ is the class of all sets.

Foundation: $X \neq \emptyset \implies \exists Y \in X (Y \cap X = \emptyset)$

Reflection: if $\phi_1,..,\phi_n$ are pure set theoretic formulas in which $Y$ is free, and their parameters among symbols $A,B$; and if $\pi_i$ is the formula $$``\forall A,B \ are \ H_{<W} \\ [\forall Y(\phi_i \Rightarrow H_{<W}(Y)) \to \exists X < W \forall Y(Y \in X \Leftrightarrow\phi_i) ]"$$

then: $$\exists W: \pi_1 \land ... \land \pi_n $$

Where $H_{<W}$ is the predicate "hereditarily strictly subnumerous to $W$" defined by the predicated classes being strictly subnumerous to $W$ and every element of their transitive closures being also strictly subnumerous to $W$; and $<$ stands for strict subnumerousity defined in the usual sense. A pure set theoretic formula means a bounded $``\in V"$ formula whose predicates are $\in,=$ and its terms are variables, except $V$ which only appears in bounding all quantified variables in it.

I claim that this theory can interpret Ackermann and Morse-Kelley set theory [with modified size limitation axiom of every class equinumerous to a set is a set], and also Muller's extension of Ackermann's. Also, I think it interprets Tarski–Grothendieck set theory (TG). I'm not really sure if it reaches up to the consistency of $ORD$ is Mahlo. It does all of that in the usual language of set theory!

Is ORD is Mahlo, the exact consistency strength of this theory?

The reason why I'm asking that is because adding limitation of size principle on top of Ackermann will blow it up to ORD is Mahlo. But here we are proving finite instances of that reflection principle of Ackermann, on top of size limitation, so this might prove to be much weaker possibly at the level of just interpreting TG, however I'm not so sure of that, hence my question.