Is Reflection consistent with Resemblance?
$begingroup$
The following theory is a class theory that combines two principles that of reflection and resemblance, informally it says that the class $V$ of all sets resembles a set $W$ that stands as a sub-world in the sense that $W$ has the same internal structure describable in the pure language of class theory (i.e. not using $W$) that $V$ has, and in addtion $W$ would reflect inside it all existential properties of sets describable in the pure language of class theory from parameters in $W$
Two questions:
Is this theory consistent?
If 1. is positively answered, then is Resemblance schema independent of the other axioms?
Formally speaking the theory is formulated in mono-sorted first order predicate logic with primitive symbols of $``=, in, W"$, standing for "identity, membership, and Sub-world", where the last is a constant symbol.
Axioms are those of identity theory +
Extensionaltiy: $forall x (x in a leftrightarrow x in b) to a=b$
Define: $set(x) iff exists y (x in y)$
Class Comprehension: if $varphi(y)$ is a formula in which $x$ is not free, then: $[exists x forall y (y in x leftrightarrow set(y) wedge varphi(y))]$ is an axiom.
Let $V$ be the class of all sets.
Subworld: $W in V$
Reflection: if $varphi$ is a formula in $L(=,in)$, in which $x$ occurs free, with parameters $vec{p}$, then: $$vec{p} in W to [exists x in V (varphi) to exists x in W (varphi)]$$ is an axiom.
Resemblance: if $varphi^X$ is a formula in prenex normal form, whose matrix is in $L(=,in)$, and each variable in its prefix must appear as bounded either $in X$ or $subseteq X$, then: $$varphi^V leftrightarrow varphi^W $$, is an axiom.
Size limitation: $forall x (x in V leftrightarrow |x|<|V|)$
Foundation over all classes.
I personally tend to think that if this theory is consistent, then the model spoken about in the answer to this question, would satisfy this theory. But I'm not sure of that.
first-order-logic large-cardinals alternative-set-theories
asked Jan 15 at 21:17
ZuhairZuhair
363212
$endgroup$
add a comment |
$begingroup$
The following theory is a class theory that combines two principles that of reflection and resemblance, informally it says that the class $V$ of all sets resembles a set $W$ that stands as a sub-world in the sense that $W$ has the same internal structure describable in the pure language of class theory (i.e. not using $W$) that $V$ has, and in addtion $W$ would reflect inside it all existential properties of sets describable in the pure language of class theory from parameters in $W$
Two questions:
Is this theory consistent?
If 1. is positively answered, then is Resemblance schema independent of the other axioms?
Formally speaking the theory is formulated in mono-sorted first order predicate logic with primitive symbols of $``=, in, W"$, standing for "identity, membership, and Sub-world", where the last is a constant symbol.
Axioms are those of identity theory +
Extensionaltiy: $forall x (x in a leftrightarrow x in b) to a=b$
Define: $set(x) iff exists y (x in y)$
Class Comprehension: if $varphi(y)$ is a formula in which $x$ is not free, then: $[exists x forall y (y in x leftrightarrow set(y) wedge varphi(y))]$ is an axiom.
Let $V$ be the class of all sets.
Subworld: $W in V$
Reflection: if $varphi$ is a formula in $L(=,in)$, in which $x$ occurs free, with parameters $vec{p}$, then: $$vec{p} in W to [exists x in V (varphi) to exists x in W (varphi)]$$ is an axiom.
Resemblance: if $varphi^X$ is a formula in prenex normal form, whose matrix is in $L(=,in)$, and each variable in its prefix must appear as bounded either $in X$ or $subseteq X$, then: $$varphi^V leftrightarrow varphi^W $$, is an axiom.
Size limitation: $forall x (x in V leftrightarrow |x|<|V|)$
Foundation over all classes.
I personally tend to think that if this theory is consistent, then the model spoken about in the answer to this question, would satisfy this theory. But I'm not sure of that.
first-order-logic large-cardinals alternative-set-theories
asked Jan 15 at 21:17
ZuhairZuhair
363212
$endgroup$
add a comment |
$begingroup$
The following theory is a class theory that combines two principles that of reflection and resemblance, informally it says that the class $V$ of all sets resembles a set $W$ that stands as a sub-world in the sense that $W$ has the same internal structure describable in the pure language of class theory (i.e. not using $W$) that $V$ has, and in addtion $W$ would reflect inside it all existential properties of sets describable in the pure language of class theory from parameters in $W$
Two questions:
Is this theory consistent?
If 1. is positively answered, then is Resemblance schema independent of the other axioms?
Formally speaking the theory is formulated in mono-sorted first order predicate logic with primitive symbols of $``=, in, W"$, standing for "identity, membership, and Sub-world", where the last is a constant symbol.
Axioms are those of identity theory +
Extensionaltiy: $forall x (x in a leftrightarrow x in b) to a=b$
Define: $set(x) iff exists y (x in y)$
Class Comprehension: if $varphi(y)$ is a formula in which $x$ is not free, then: $[exists x forall y (y in x leftrightarrow set(y) wedge varphi(y))]$ is an axiom.
Let $V$ be the class of all sets.
Subworld: $W in V$
Reflection: if $varphi$ is a formula in $L(=,in)$, in which $x$ occurs free, with parameters $vec{p}$, then: $$vec{p} in W to [exists x in V (varphi) to exists x in W (varphi)]$$ is an axiom.
Resemblance: if $varphi^X$ is a formula in prenex normal form, whose matrix is in $L(=,in)$, and each variable in its prefix must appear as bounded either $in X$ or $subseteq X$, then: $$varphi^V leftrightarrow varphi^W $$, is an axiom.
Size limitation: $forall x (x in V leftrightarrow |x|<|V|)$
Foundation over all classes.
I personally tend to think that if this theory is consistent, then the model spoken about in the answer to this question, would satisfy this theory. But I'm not sure of that.
first-order-logic large-cardinals alternative-set-theories
asked Jan 15 at 21:17
ZuhairZuhair
363212
$endgroup$
The following theory is a class theory that combines two principles that of reflection and resemblance, informally it says that the class $V$ of all sets resembles a set $W$ that stands as a sub-world in the sense that $W$ has the same internal structure describable in the pure language of class theory (i.e. not using $W$) that $V$ has, and in addtion $W$ would reflect inside it all existential properties of sets describable in the pure language of class theory from parameters in $W$
Two questions:
Is this theory consistent?
If 1. is positively answered, then is Resemblance schema independent of the other axioms?
Formally speaking the theory is formulated in mono-sorted first order predicate logic with primitive symbols of $``=, in, W"$, standing for "identity, membership, and Sub-world", where the last is a constant symbol.
Axioms are those of identity theory +
Extensionaltiy: $forall x (x in a leftrightarrow x in b) to a=b$
Define: $set(x) iff exists y (x in y)$
Class Comprehension: if $varphi(y)$ is a formula in which $x$ is not free, then: $[exists x forall y (y in x leftrightarrow set(y) wedge varphi(y))]$ is an axiom.
Let $V$ be the class of all sets.
Subworld: $W in V$
Reflection: if $varphi$ is a formula in $L(=,in)$, in which $x$ occurs free, with parameters $vec{p}$, then: $$vec{p} in W to [exists x in V (varphi) to exists x in W (varphi)]$$ is an axiom.
Resemblance: if $varphi^X$ is a formula in prenex normal form, whose matrix is in $L(=,in)$, and each variable in its prefix must appear as bounded either $in X$ or $subseteq X$, then: $$varphi^V leftrightarrow varphi^W $$, is an axiom.
Size limitation: $forall x (x in V leftrightarrow |x|<|V|)$
Foundation over all classes.
I personally tend to think that if this theory is consistent, then the model spoken about in the answer to this question, would satisfy this theory. But I'm not sure of that.
first-order-logic large-cardinals alternative-set-theories
first-order-logic large-cardinals alternative-set-theories
asked Jan 15 at 21:17
ZuhairZuhair
363212
asked Jan 15 at 21:17
ZuhairZuhair
363212
edited Jan 28 at 4:41
Zuhair
asked Jan 15 at 21:17
ZuhairZuhair
363212
asked Jan 15 at 21:17
ZuhairZuhair
363212
asked Jan 15 at 21:17
ZuhairZuhair
363212
363212
add a comment |
add a comment |
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