Naming Set Theory "NST".
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From: zuhair on 4 Oct 2009 00:44 Hi all, This topic is a continuation of an earlier two topics post to this usenet see links: http://groups.google.com/group/sci.logic/browse_thread/thread/735538d... http://groups.google.com.jm/group/sci.logic/browse_thread/thread/0180... Here I will represent the basic idea of this set theory which I will call it NST for "Naming Set Theory". The basic idea is to have a "defined" membership relation symbolized as "in", this is defined by a formula using two primitive binary relations that of "name" and epsilon membership "e" in the following manner: Define(in):- y in x iff Exist $ ( $ is the name of y & $ e x ) The main idea was arrived after I had some observations that help avoid known paradoxes of set theory, wereby if we use certain kinds of comprehension schemata and restrict the formula in the comprehension schemata from using the relation "name" in it then I noticed that the known paradoxes would be blocked(see links above for details). One can imagine the matter in the following manner take the formula of Naive comprehension which can be rewrittin here in the following manner: Exist a set x for all y ( y in x iff y is a set:Phi ) NOTE:"set" is defined as a named object, while "ur-element" or even better termed as "sub-element" is defined as an unnamed object. now "in" here is a defined membership relation, the trick is to apply a restriction on Phi such that we cannot have phi use the defined relation "in"; a rather harsh way to acheive that is by limiting Phi from using the relation "name" altogether ( a lesser restriction might be a one which only forbids using "both" binary relations "e" and "name" at the same time in Phi). This would block the known paradoxes of Naive comprehension because we cannot have the defined membership relation "in" used by Phi. Also there is another comprehension axiom schema in this theory which is related to ur-elements Exist a set x for all y ( y e x iff y is a ur-element:Phi ) also here Phi is forbidden from using the relation "name", and I observed that this blocks the known paradoxes. From these observations I made this theory; in addition other goals were in my mind also, and these are mainly two goals, one is that this theory should premit the existence of Large sets, like the set of all sets, Fregian cardinals and ordinals, etc.. Second to prove Infinity, or better phrased to prove the existence of the set of all natural numbers. ONTOLOGY: This theory has only "sets" and "sub-elements" the latter is also refered to as "ur-elements" although they don't share all the properties ur-elements has in other theories. "sets" are collections (with respect to "e") or ur-elements i.e. only ur-elements have the relation "e" to a set. ur-elements are epsilom members of sets, but they are themselfs not sets, no object has the relation "e" nor "in" to a ur-elements, however in this theory a ur- element cannot be "in" a set, only sets are "in" sets. There are no proper classes in this theory, although actually a ur-element can be considered a proper object in the sense that is not contained by the defined membership relation "in" in any set. This theory contain the set of all sets V, and it has absolute complements, finite intersections and finite unions so it is Boolean. EXPOSITION Language: First order logic with identity "=", epsilon membership "e" , and the following additional primitives: "name" which is a binary relation symbol. ordered pair "<>" which is a two place function symbol. ordered pair projections t1,t2 which are binary relation symbols. Definitions: x is a sub-element iff ~ Exist y ( y is the name of x ) x is a set iff Exist y ( y is the name of x ) y in x iff Exist $ ( $ is the name of y & y e x ) Axioms: 1)Extensionality1: for all sets x,y (for all z ( z e x <-> z e y) -> x=y) 2)Extensionality2: for all #,$,a,b ((# is the name of a & $ is the name of b) -> (# =$ <-> a=b)) 3) Ordered pairs: for all a,b,c,d (<a,b>=<c,d> <-> (a=c & b=d)) 4) Projections: for all x,a,b ((x t1 <a,b> -> x=a)& (x t2 <a,b> -> x=b)) 5) Naming: For all x,y (x is the name of y -> x is a ur-element) 6) Sets: For all x For all y ( y e x -> y is a ur-element ) 7) Comprehension1: If Phi is a formula in which x is not free, and which do not use the relation "name", then all closures of Exist a set x for all y ( y e x iff y is a ur-element:Phi ) are axioms. Define: x=[y|Phi] iff for all y ( y e x iff y is a ur-element:Phi ) 8) Comprehension2: If Phi is a formula in which neither x is not free, and that doesn't use the primitive binary relation "name", then all closures of Exist a set x for all y ( y in x iff y is a set:Phi ) are axioms. Define: x={y|Phi} iff for all y ( y in x iff y is a set:Phi ) 9) Pair coding1: For all #,$,a,b ( (# is the name of a & $ is the name of b) -> <#,$> is the name of {{a},{a,b}} ) 10)Pair coding2: For all sets x,y ( <x,y> is the name of {{x},{x,y}} ). 11)Pair coding3: For all #,a,y ( # is the name of a & y is a set -> <#,y> is the name of {{a},{a,y}} & <y,#> is the name of {{y},{y,a}} ) 12) Union: For all c Exist x for all y (y in x <-> Exist z ( z in c & y in z )) Theory definition finished/ I don't like axiom 12 since it is against the general philosophy of this theory, however if one want this theory to prove NF and its related systems, then we'll need union. If this theory is consistent then it interprets NFU. This theory proves Infinity, define Large sets, and can derive theorems about these large sets, if consistent I think it is a useful theory, it can provide a simple ulternative to stratification in dealing with large sets. Zuhair
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From: hagman on 4 Oct 2009 06:39 On 4 Okt., 06:44, zuhair <zaljo...(a)gmail.com> wrote: > Hi all, > > This topic is a continuation of an earlier two topics post to this > usenet > > see links: > > http://groups.google.com/group/sci.logic/browse_thread/thread/735538d... > > http://groups.google.com.jm/group/sci.logic/browse_thread/thread/0180... > > Here I will represent the basic idea of this set theory which I will > call it NST for "Naming Set Theory". > > The basic idea is to have a "defined" membership relation symbolized > as "in", this is defined by a formula using two primitive binary > relations that of "name" and epsilon membership "e" in the following > manner: > > Define(in):- y in x iff Exist $ ( $ is the name of y & $ e x ) > > The main idea was arrived after I had some observations that help > avoid known paradoxes of set theory, wereby if we use certain kinds of > comprehension schemata and restrict the formula in the comprehension > schemata from using the relation "name" in it then I noticed that the > known paradoxes would be blocked(see links above for details). > > One can imagine the matter in the following manner > take the formula of Naive comprehension which can be rewrittin here in > the following manner: > > Exist a set x for all y ( y in x iff y is a set:Phi ) > > NOTE:"set" is defined as a named object, while "ur-element" or even > better termed as "sub-element" is defined as an unnamed object. > > now "in" here is a defined membership relation, the trick is to apply > a restriction on Phi such that we cannot have phi use the defined > relation "in"; a rather harsh way to acheive that is by limiting Phi > from using the relation "name" altogether ( a lesser restriction might > be a one which only forbids using "both" binary relations "e" and > "name" at the same time in Phi). This would block the known paradoxes > of Naive comprehension because we cannot have the defined membership > relation "in" used by Phi. > > Also there is another comprehension axiom schema in this theory which > is related to ur-elements > > Exist a set x for all y ( y e x iff y is a ur-element:Phi ) > > also here Phi is forbidden from using the relation "name", and I > observed that this blocks the known paradoxes. > > From these observations I made this theory; in addition other goals > were in my mind also, and these are mainly two goals, one is that this > theory should premit the existence of Large sets, like the set of all > sets, Fregian cardinals and ordinals, etc.. > Second to prove Infinity, or better phrased to prove the existence of > the set of all natural numbers. > > ONTOLOGY: This theory has only "sets" and "sub-elements" the latter is > also refered to as "ur-elements" although they don't share all the > properties ur-elements has in other theories. "sets" are collections > (with respect to "e") or ur-elements i.e. only > ur-elements have the relation "e" to a set. ur-elements are epsilom > members of sets, but they are themselfs not sets, no object has the > relation "e" nor "in" to a ur-elements, however in this theory a ur- > element cannot be "in" a set, only sets are "in" sets. > There are no proper classes in this theory, although actually > a ur-element can be considered a proper object in the sense that is > not contained by the defined membership relation "in" in any set. > > This theory contain the set of all sets V, and it has absolute > complements, finite intersections and finite unions so it is Boolean. > > EXPOSITION > > Language: First order logic with identity "=", epsilon membership > "e" , and the following additional primitives: > > "name" which is a binary relation symbol. > > ordered pair "<>" which is a two place function symbol. > > ordered pair projections t1,t2 which are binary relation symbols. > > Definitions: > > x is a sub-element iff ~ Exist y ( y is the name of x ) > > x is a set iff Exist y ( y is the name of x ) > > y in x iff Exist $ ( $ is the name of y & y e x ) In other words y in x iff y is a set & y e x > > Axioms: > > 1)Extensionality1: > > for all sets x,y (for all z ( z e x <-> z e y) -> x=y) > > 2)Extensionality2: > > for all #,$,a,b ((# is the name of a & $ is the name of b) -> > (# =$ <-> a=b)) So names are, well, names - unique identifiers. > > 3) Ordered pairs: for all a,b,c,d (<a,b>=<c,d> <-> (a=c & b=d)) > > 4) Projections: for all x,a,b ((x t1 <a,b> -> x=a)& > (x t2 <a,b> -> x=b)) Don't you also want forall a,b ( a t1 <a,b> & b t2 <a,b>) ? > > 5) Naming: For all x,y (x is the name of y -> x is a ur-element) Above you said that in your theory ur-elemnts and sub-elements ar the same. So this just says there are no "names of names"? > > 6) Sets: For all x For all y ( y e x -> y is a ur-element ) > > 7) Comprehension1: If Phi is a formula in which x is not free, and > which do not use the relation "name", then all closures of > > Exist a set x for all y ( y e x iff y is a ur-element:Phi ) > > are axioms. > > Define: x=[y|Phi] iff for all y ( y e x iff y is a ur-element:Phi ) > > 8) Comprehension2: If Phi is a formula in which neither x is not free, > and that doesn't use the primitive binary relation "name", then all > closures of > > Exist a set x for all y ( y in x iff y is a set:Phi ) > > are axioms. > > Define: x={y|Phi} iff for all y ( y in x iff y is a set:Phi ) > > 9) Pair coding1: > For all #,$,a,b ( (# is the name of a & $ is the name of b) > -> > <#,$> is the name of {{a},{a,b}} ) Should I infer that forall a,b exists c forall z: (z e c <-> (z=a or z=b)) Or is {a,b} a shorthand for one of {z|z=a v z=b} or [z|z=a v z=b] > 10)Pair coding2: > For all sets x,y ( <x,y> is the name of {{x},{x,y}} ). > > 11)Pair coding3: > For all #,a,y ( # is the name of a & y is a set > -> > <#,y> is the name of {{a},{a,y}} & > <y,#> is the name of {{y},{y,a}} ) So if # is the name of x and $ is the name of y, we conclude that x,y are sets and <x,y> is the name of {{x},{x,y}} // Pair Coding 2 <#,y> is the name of {{x},{x,y}} // Pair Coding 3 <#,$> is the name of {{x},{x,y}} // Pair Coding 1 <x,$> is the name of {{x},{x,y}} // Pair Coding 3 Thus by Extensionality2 <x,y> = <#,y> etc. Following your axioms strictly, t1 might be the empty relation. However, if I assume you also want forall a,b ( a t1 <a,b> & b t2 <a,b>) to hold, then we find x = # which contradicts Axiom 5. > > 12) Union: > > For all c Exist x for all y (y in x <-> Exist z ( z in c & y in z )) > > Theory definition finished/ > > I don't like axiom 12 since it is against the general philosophy of > this theory, however if one want this theory to prove NF and its > related systems, then we'll need union. > > If this theory is consistent then it interprets NFU. > > This theory proves Infinity, define Large sets, and can derive > theorems about these large sets, if consistent I think it is a useful > theory, it can provide a simple ulternative to stratification in > dealing with large sets. > > Zuhair
From: zuhair on 4 Oct 2009 07:57 On Oct 4, 5:39 am, hagman <goo...(a)von-eitzen.de> wrote: > On 4 Okt., 06:44, zuhair <zaljo...(a)gmail.com> wrote: > > > > > > > Hi all, > > > This topic is a continuation of an earlier two topics post to this > > usenet > > > see links: > > >http://groups.google.com/group/sci.logic/browse_thread/thread/735538d... > > >http://groups.google.com.jm/group/sci.logic/browse_thread/thread/0180... > > > Here I will represent the basic idea of this set theory which I will > > call it NST for "Naming Set Theory". > > > The basic idea is to have a "defined" membership relation symbolized > > as "in", this is defined by a formula using two primitive binary > > relations that of "name" and epsilon membership "e" in the following > > manner: > > > Define(in):- y in x iff Exist $ ( $ is the name of y & $ e x ) > > > The main idea was arrived after I had some observations that help > > avoid known paradoxes of set theory, wereby if we use certain kinds of > > comprehension schemata and restrict the formula in the comprehension > > schemata from using the relation "name" in it then I noticed that the > > known paradoxes would be blocked(see links above for details). > > > One can imagine the matter in the following manner > > take the formula of Naive comprehension which can be rewrittin here in > > the following manner: > > > Exist a set x for all y ( y in x iff y is a set:Phi ) > > > NOTE:"set" is defined as a named object, while "ur-element" or even > > better termed as "sub-element" is defined as an unnamed object. > > > now "in" here is a defined membership relation, the trick is to apply > > a restriction on Phi such that we cannot have phi use the defined > > relation "in"; a rather harsh way to acheive that is by limiting Phi > > from using the relation "name" altogether ( a lesser restriction might > > be a one which only forbids using "both" binary relations "e" and > > "name" at the same time in Phi). This would block the known paradoxes > > of Naive comprehension because we cannot have the defined membership > > relation "in" used by Phi. > > > Also there is another comprehension axiom schema in this theory which > > is related to ur-elements > > > Exist a set x for all y ( y e x iff y is a ur-element:Phi ) > > > also here Phi is forbidden from using the relation "name", and I > > observed that this blocks the known paradoxes. > > > From these observations I made this theory; in addition other goals > > were in my mind also, and these are mainly two goals, one is that this > > theory should premit the existence of Large sets, like the set of all > > sets, Fregian cardinals and ordinals, etc.. > > Second to prove Infinity, or better phrased to prove the existence of > > the set of all natural numbers. > > > ONTOLOGY: This theory has only "sets" and "sub-elements" the latter is > > also refered to as "ur-elements" although they don't share all the > > properties ur-elements has in other theories. "sets" are collections > > (with respect to "e") or ur-elements i.e. only > > ur-elements have the relation "e" to a set. ur-elements are epsilom > > members of sets, but they are themselfs not sets, no object has the > > relation "e" nor "in" to a ur-elements, however in this theory a ur- > > element cannot be "in" a set, only sets are "in" sets. > > There are no proper classes in this theory, although actually > > a ur-element can be considered a proper object in the sense that is > > not contained by the defined membership relation "in" in any set. > > > This theory contain the set of all sets V, and it has absolute > > complements, finite intersections and finite unions so it is Boolean. > > > EXPOSITION > > > Language: First order logic with identity "=", epsilon membership > > "e" , and the following additional primitives: > > > "name" which is a binary relation symbol. > > > ordered pair "<>" which is a two place function symbol. > > > ordered pair projections t1,t2 which are binary relation symbols. > > > Definitions: > > > x is a sub-element iff ~ Exist y ( y is the name of x ) > > > x is a set iff Exist y ( y is the name of x ) > > > y in x iff Exist $ ( $ is the name of y & y e x ) > > In other words > y in x iff y is a set & y e x Oops that was a typo, the definition was given earlier in a correct manner: Define(in):- y in x iff Exist $ ( $ is the name of y & $ e x ) > > > > > Axioms: > > > 1)Extensionality1: > > > for all sets x,y (for all z ( z e x <-> z e y) -> x=y) > > > 2)Extensionality2: > > > for all #,$,a,b ((# is the name of a & $ is the name of b) -> > > (# =$ <-> a=b)) > > So names are, well, names - unique identifiers. > > > > > 3) Ordered pairs: for all a,b,c,d (<a,b>=<c,d> <-> (a=c & b=d)) > > > 4) Projections: for all x,a,b ((x t1 <a,b> -> x=a)& > > (x t2 <a,b> -> x=b)) > > Don't you also want > forall a,b ( a t1 <a,b> & b t2 <a,b>) > ? Yes, this is actually what I wanted. > > > > > 5) Naming: For all x,y (x is the name of y -> x is a ur-element) > > Above you said that in your theory ur-elemnts and sub-elements ar the > same. > So this just says there are no "names of names"? > > > > > > > 6) Sets: For all x For all y ( y e x -> y is a ur-element ) > > > 7) Comprehension1: If Phi is a formula in which x is not free, and > > which do not use the relation "name", then all closures of > > > Exist a set x for all y ( y e x iff y is a ur-element:Phi ) > > > are axioms. > > > Define: x=[y|Phi] iff for all y ( y e x iff y is a ur-element:Phi ) > > > 8) Comprehension2: If Phi is a formula in which neither x is not free, > > and that doesn't use the primitive binary relation "name", then all > > closures of > > > Exist a set x for all y ( y in x iff y is a set:Phi ) > > > are axioms. > > > Define: x={y|Phi} iff for all y ( y in x iff y is a set:Phi ) > > > 9) Pair coding1: > > For all #,$,a,b ( (# is the name of a & $ is the name of b) > > -> > > <#,$> is the name of {{a},{a,b}} ) > > Should I infer that > forall a,b exists c forall z: (z e c <-> (z=a or z=b)) > Or is {a,b} a shorthand for one of {z|z=a v z=b} or [z|z=a v z=b] > > > 10)Pair coding2: > > For all sets x,y ( <x,y> is the name of {{x},{x,y}} ). > > > 11)Pair coding3: > > For all #,a,y ( # is the name of a & y is a set > > -> > > <#,y> is the name of {{a},{a,y}} & > > <y,#> is the name of {{y},{y,a}} ) > > So if # is the name of x and $ is the name of y, we conclude that x,y > are sets and > <x,y> is the name of {{x},{x,y}} // Pair Coding 2 > <#,y> is the name of {{x},{x,y}} // Pair Coding 3 > <#,$> is the name of {{x},{x,y}} // Pair Coding 1 > <x,$> is the name of {{x},{x,y}} // Pair Coding 3 > Thus by Extensionality2 > <x,y> = <#,y> etc. > Following your axioms strictly, t1 might be the empty relation. > However, if I assume you also want > forall a,b ( a t1 <a,b> & b t2 <a,b>) > to hold, then we find > x = # > which contradicts Axiom 5. Yes, you are right, I should modify that. Thanks hagman. > > > > > > > 12) Union: > > > For all c Exist x for all y (y in x <-> Exist z ( z in c & y in z )) > > > Theory definition finished/ > > > I don't like axiom 12 since it is against the general philosophy of > > this theory, however if one want this theory to prove NF and its > > related systems, then we'll need union. > > > If this theory is consistent then it interprets NFU. > > > This theory proves Infinity, define Large sets, and can derive > > theorems about these large sets, if consistent I think it is a useful > > theory, it can provide a simple ulternative to stratification in > > dealing with large sets. > > > Zuhair
From: zuhair on 5 Oct 2009 18:02
On second look, I see that there is no need to code things in the matters that I've done. Since we have "ordered pairs" then we can have the following axiom For all a,b ( a is the name of x & b is the name of y -> <a,b> is the name of <x,y> ) That will be quite enough. Also regarding relations t1 and t2 we actually don't need them to be primitive concepts they can be defined in the following manner: Define(t1):- a t1 x iff Exist b ( x=<a,b> ) Define(t2):- b t2 x iff Exist a ( x=<a,b> ) So the theory can have only four primitives and nine axioms. Let me rewrite it: EXPOSITION of NST. Language: First order logic with identity "=", epsilon membership "e" , and the following additional primitives: "name" which is a binary relation symbol. ordered pair "<,>" which is a two place function symbol. Definitions: x is a ur-element iff ~ Exist y ( y is the name of x ) x is a set iff Exist y ( y is the name of x ) y in x iff Exist # ( # is the name of y and # e x ) a t1 x iff Exist b ( x=<a,b> ) b t2 x iff Exist a ( x=<a,b> ) AXIOMS: 1)Extensionality1: for all sets x,y (for all z ( z e x <-> z e y) -> x=y) 2)Extensionality2: for all #,$,a,b ((# is the name of a & $ is the name of b) -> (# =$ <-> a=b)) 3) Ordered pairs: for all a,b,c,d (<a,b>=<c,d> <-> (a=c & b=d)) 4) Naming: For all x,y (x is the name of y -> x is a ur-element) 5) Sets: For all x For all y ( y e x -> y is a ur-element ) 6) Pair coding: For all a,b ((a is the name of x & b is the name of y) -> <a,b> is the name of <x,y>) 7)Ur-Comprehension: If Phi is a formula in which x is not free, and which do not use the relation "name", then all closures of Exist a set x for all y ( y e x iff y is a ur-element:Phi ) are axioms. Define: x=[y|Phi] iff for all y ( y e x iff y is a ur-element:Phi ) 8)Set-Comprehension: If Phi is a formula in which neither x is not free, and that doesn't use the primitive binary relation "name", then all closures of Exist a set x for all y ( y in x iff y is a set:Phi ) are axioms. Define: x={y|Phi} iff for all y ( y in x iff y is a set:Phi ) 9) Union: For all c Exist x for all y (y in x <-> Exist z ( z in c & y in z )) Theory definition finished/ |