Modern algebra and set theory: ZFC vs. NBG
$begingroup$
This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little:
Is it not more natural consider NBG set theory as the foundation for modern algebra as opposed to traditional ZFC? To me, ZF has always seemed sort of hacky, for lack of a better word, as if it has been patched and patched over the years; kind of how windows vista would look today if it were still in use. It is no doubt an extremely powerful theory, but the point is that in modern application ZF tends to be somewhat inadequate, seemingly always requiring a work around; thus, hacky. On the other hand, NBG deals with classes directly, and is just for all intents and purposes more accessible from the algebraic viewpoint, especially from the point of view of lattice and order theory, all the way to class field theory. NBG is just better equipped for the job.
I guess an easier way to say all of this is that while ZF is more concerned with objects, NBG is designed to exploit the relationships between objects, which, in my opinion is more fundamental to not only mathematics, but to logic itself. NBG is implemented naturally to exhibit the abilities of comparison and deduction, which can be argued to form the basis for the concept of logic, in and of itself.
Am I crazy, or has anyone else ever felt this way?
abstract-algebra soft-question set-theory foundations
edited Dec 14 '18 at 13:56
rr01
1158
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
$endgroup$
|
show 5 more comments
$begingroup$
This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little:
Is it not more natural consider NBG set theory as the foundation for modern algebra as opposed to traditional ZFC? To me, ZF has always seemed sort of hacky, for lack of a better word, as if it has been patched and patched over the years; kind of how windows vista would look today if it were still in use. It is no doubt an extremely powerful theory, but the point is that in modern application ZF tends to be somewhat inadequate, seemingly always requiring a work around; thus, hacky. On the other hand, NBG deals with classes directly, and is just for all intents and purposes more accessible from the algebraic viewpoint, especially from the point of view of lattice and order theory, all the way to class field theory. NBG is just better equipped for the job.
I guess an easier way to say all of this is that while ZF is more concerned with objects, NBG is designed to exploit the relationships between objects, which, in my opinion is more fundamental to not only mathematics, but to logic itself. NBG is implemented naturally to exhibit the abilities of comparison and deduction, which can be argued to form the basis for the concept of logic, in and of itself.
Am I crazy, or has anyone else ever felt this way?
abstract-algebra soft-question set-theory foundations
edited Dec 14 '18 at 13:56
rr01
1158
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
$endgroup$
$begingroup$
You should switch to linux and SEAR.
$endgroup$
– k.stm
Jan 3 '15 at 21:17
$begingroup$
@k.stm: How is that Linux? If anything, that's DEC10.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:18
$begingroup$
Lol, I do use Linux, it was just an analogy.
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:21
$begingroup$
@AsafKaragila Funnily, the page I linked to even indulges in that metaphor itself (which I didn’t know before): “Or, using an alternate metaphor, ZFC is like Windows, ETCS is like UNIX, and SEAR is like OS X (or maybe Ubuntu).” And maybe it’s not Ubuntu, but Debian or Arch or something.
$endgroup$
– k.stm
Jan 3 '15 at 21:22
$begingroup$
@k.stm: As someone who's been doing set theory for a while now, and been working with Linux for a while longer, and with Windows even longer than that, the comparison is bad. The comparison should be between CPU architectures, not operating systems.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:25
|
show 5 more comments
$begingroup$
This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little:
Is it not more natural consider NBG set theory as the foundation for modern algebra as opposed to traditional ZFC? To me, ZF has always seemed sort of hacky, for lack of a better word, as if it has been patched and patched over the years; kind of how windows vista would look today if it were still in use. It is no doubt an extremely powerful theory, but the point is that in modern application ZF tends to be somewhat inadequate, seemingly always requiring a work around; thus, hacky. On the other hand, NBG deals with classes directly, and is just for all intents and purposes more accessible from the algebraic viewpoint, especially from the point of view of lattice and order theory, all the way to class field theory. NBG is just better equipped for the job.
I guess an easier way to say all of this is that while ZF is more concerned with objects, NBG is designed to exploit the relationships between objects, which, in my opinion is more fundamental to not only mathematics, but to logic itself. NBG is implemented naturally to exhibit the abilities of comparison and deduction, which can be argued to form the basis for the concept of logic, in and of itself.
Am I crazy, or has anyone else ever felt this way?
abstract-algebra soft-question set-theory foundations
edited Dec 14 '18 at 13:56
rr01
1158
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
$endgroup$
This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little:
Is it not more natural consider NBG set theory as the foundation for modern algebra as opposed to traditional ZFC? To me, ZF has always seemed sort of hacky, for lack of a better word, as if it has been patched and patched over the years; kind of how windows vista would look today if it were still in use. It is no doubt an extremely powerful theory, but the point is that in modern application ZF tends to be somewhat inadequate, seemingly always requiring a work around; thus, hacky. On the other hand, NBG deals with classes directly, and is just for all intents and purposes more accessible from the algebraic viewpoint, especially from the point of view of lattice and order theory, all the way to class field theory. NBG is just better equipped for the job.
I guess an easier way to say all of this is that while ZF is more concerned with objects, NBG is designed to exploit the relationships between objects, which, in my opinion is more fundamental to not only mathematics, but to logic itself. NBG is implemented naturally to exhibit the abilities of comparison and deduction, which can be argued to form the basis for the concept of logic, in and of itself.
Am I crazy, or has anyone else ever felt this way?
abstract-algebra soft-question set-theory foundations
abstract-algebra soft-question set-theory foundations
edited Dec 14 '18 at 13:56
rr01
1158
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
edited Dec 14 '18 at 13:56
rr01
1158
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
edited Dec 14 '18 at 13:56
rr01
1158
edited Dec 14 '18 at 13:56
rr01
1158
edited Dec 14 '18 at 13:56
rr01
1158
1158
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
asked Jan 3 '15 at 21:12
justsomeguy90justsomeguy90
887
887
$begingroup$
You should switch to linux and SEAR.
$endgroup$
– k.stm
Jan 3 '15 at 21:17
$begingroup$
@k.stm: How is that Linux? If anything, that's DEC10.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:18
$begingroup$
Lol, I do use Linux, it was just an analogy.
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:21
$begingroup$
@AsafKaragila Funnily, the page I linked to even indulges in that metaphor itself (which I didn’t know before): “Or, using an alternate metaphor, ZFC is like Windows, ETCS is like UNIX, and SEAR is like OS X (or maybe Ubuntu).” And maybe it’s not Ubuntu, but Debian or Arch or something.
$endgroup$
– k.stm
Jan 3 '15 at 21:22
$begingroup$
@k.stm: As someone who's been doing set theory for a while now, and been working with Linux for a while longer, and with Windows even longer than that, the comparison is bad. The comparison should be between CPU architectures, not operating systems.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:25
|
show 5 more comments
$begingroup$
You should switch to linux and SEAR.
$endgroup$
– k.stm
Jan 3 '15 at 21:17
$begingroup$
@k.stm: How is that Linux? If anything, that's DEC10.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:18
$begingroup$
Lol, I do use Linux, it was just an analogy.
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:21
$begingroup$
@AsafKaragila Funnily, the page I linked to even indulges in that metaphor itself (which I didn’t know before): “Or, using an alternate metaphor, ZFC is like Windows, ETCS is like UNIX, and SEAR is like OS X (or maybe Ubuntu).” And maybe it’s not Ubuntu, but Debian or Arch or something.
$endgroup$
– k.stm
Jan 3 '15 at 21:22
$begingroup$
@k.stm: As someone who's been doing set theory for a while now, and been working with Linux for a while longer, and with Windows even longer than that, the comparison is bad. The comparison should be between CPU architectures, not operating systems.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:25
$begingroup$
You should switch to linux and SEAR.
$endgroup$
– k.stm
Jan 3 '15 at 21:17
$begingroup$
You should switch to linux and SEAR.
$endgroup$
– k.stm
Jan 3 '15 at 21:17
$begingroup$
@k.stm: How is that Linux? If anything, that's DEC10.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:18
$begingroup$
@k.stm: How is that Linux? If anything, that's DEC10.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:18
$begingroup$
Lol, I do use Linux, it was just an analogy.
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:21
$begingroup$
Lol, I do use Linux, it was just an analogy.
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:21
$begingroup$
@AsafKaragila Funnily, the page I linked to even indulges in that metaphor itself (which I didn’t know before): “Or, using an alternate metaphor, ZFC is like Windows, ETCS is like UNIX, and SEAR is like OS X (or maybe Ubuntu).” And maybe it’s not Ubuntu, but Debian or Arch or something.
$endgroup$
– k.stm
Jan 3 '15 at 21:22
$begingroup$
@AsafKaragila Funnily, the page I linked to even indulges in that metaphor itself (which I didn’t know before): “Or, using an alternate metaphor, ZFC is like Windows, ETCS is like UNIX, and SEAR is like OS X (or maybe Ubuntu).” And maybe it’s not Ubuntu, but Debian or Arch or something.
$endgroup$
– k.stm
Jan 3 '15 at 21:22
$begingroup$
@k.stm: As someone who's been doing set theory for a while now, and been working with Linux for a while longer, and with Windows even longer than that, the comparison is bad. The comparison should be between CPU architectures, not operating systems.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:25
$begingroup$
@k.stm: As someone who's been doing set theory for a while now, and been working with Linux for a while longer, and with Windows even longer than that, the comparison is bad. The comparison should be between CPU architectures, not operating systems.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:25
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Many people have felt like that.
This is why you often hear people in algebra moaning about set theory, or ignoring set theoretic issues (they usually can point out where these issues arise, and that someone out there knows how to solve them). This is also why there are people who are very enthusiastic about algebraic set theories like $sf ETCS$, or type theories like $sf SEAR$ and $sf HTT$, which may or may not prove to be a better foundation for algebra.
But switching from $sf ZFC$ to $sf NBG$ only pushes the problem "one step further". Sure, now you have the class of all groups as an actual object. But what about the category of all small categories? That's not a class anymore, since only sets are allowed to be elements of other classes, and small categories are not necessarily classes.
This is why working with universes is easier here. They allow you to jump "one level up" without any consequences. Each time you just extend the definition of what it means to be a set, and include more things as sets.
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
$endgroup$
$begingroup$
I see your point, however, I was under the impression that NBG could handle such discussion. Is a universe not a class?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:40
$begingroup$
Actually, I guess since we are talking about the categories of all categories, we would simply allow for impredicative comprehension, right?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:43
$begingroup$
The universe is a class. But classes are not objects of other classes. So a category whose objects are small categories is a collection whose objects are themselves classes, and for this collection to be "in the universe" you need to either allow 2-classes (thus extend again), or that all the categories you worked with were actually sets to begin with. What NBG gives you is the ability to quantify over proper classes, which makes dealing with small categories (and to a very very minor degree, for making statements about arbitrary small categories) much easier.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:44
$begingroup$
Altenatively, couldn't we simply a priori give a formal definition of say the "logical scope" of a variable in a logical formula and allow for modification to a particular subject matter, thus making any logical formula "scope-dependent". I.e. we could stop defining scope-independent variables, so that when we choose to generalize, paradoxical "spillover" of quantified variables are in some sense modded out by restriction to the new scope.
$endgroup$
– justsomeguy90
Jan 3 '15 at 22:07
$begingroup$
That sounds dangerously close to type theory. :-)
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 22:07
|
show 1 more comment
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$begingroup$
Many people have felt like that.
This is why you often hear people in algebra moaning about set theory, or ignoring set theoretic issues (they usually can point out where these issues arise, and that someone out there knows how to solve them). This is also why there are people who are very enthusiastic about algebraic set theories like $sf ETCS$, or type theories like $sf SEAR$ and $sf HTT$, which may or may not prove to be a better foundation for algebra.
But switching from $sf ZFC$ to $sf NBG$ only pushes the problem "one step further". Sure, now you have the class of all groups as an actual object. But what about the category of all small categories? That's not a class anymore, since only sets are allowed to be elements of other classes, and small categories are not necessarily classes.
This is why working with universes is easier here. They allow you to jump "one level up" without any consequences. Each time you just extend the definition of what it means to be a set, and include more things as sets.
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
$endgroup$
$begingroup$
I see your point, however, I was under the impression that NBG could handle such discussion. Is a universe not a class?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:40
$begingroup$
Actually, I guess since we are talking about the categories of all categories, we would simply allow for impredicative comprehension, right?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:43
$begingroup$
The universe is a class. But classes are not objects of other classes. So a category whose objects are small categories is a collection whose objects are themselves classes, and for this collection to be "in the universe" you need to either allow 2-classes (thus extend again), or that all the categories you worked with were actually sets to begin with. What NBG gives you is the ability to quantify over proper classes, which makes dealing with small categories (and to a very very minor degree, for making statements about arbitrary small categories) much easier.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:44
$begingroup$
Altenatively, couldn't we simply a priori give a formal definition of say the "logical scope" of a variable in a logical formula and allow for modification to a particular subject matter, thus making any logical formula "scope-dependent". I.e. we could stop defining scope-independent variables, so that when we choose to generalize, paradoxical "spillover" of quantified variables are in some sense modded out by restriction to the new scope.
$endgroup$
– justsomeguy90
Jan 3 '15 at 22:07
$begingroup$
That sounds dangerously close to type theory. :-)
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 22:07
|
show 1 more comment
$begingroup$
Many people have felt like that.
This is why you often hear people in algebra moaning about set theory, or ignoring set theoretic issues (they usually can point out where these issues arise, and that someone out there knows how to solve them). This is also why there are people who are very enthusiastic about algebraic set theories like $sf ETCS$, or type theories like $sf SEAR$ and $sf HTT$, which may or may not prove to be a better foundation for algebra.
But switching from $sf ZFC$ to $sf NBG$ only pushes the problem "one step further". Sure, now you have the class of all groups as an actual object. But what about the category of all small categories? That's not a class anymore, since only sets are allowed to be elements of other classes, and small categories are not necessarily classes.
This is why working with universes is easier here. They allow you to jump "one level up" without any consequences. Each time you just extend the definition of what it means to be a set, and include more things as sets.
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
$endgroup$
$begingroup$
I see your point, however, I was under the impression that NBG could handle such discussion. Is a universe not a class?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:40
$begingroup$
Actually, I guess since we are talking about the categories of all categories, we would simply allow for impredicative comprehension, right?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:43
$begingroup$
The universe is a class. But classes are not objects of other classes. So a category whose objects are small categories is a collection whose objects are themselves classes, and for this collection to be "in the universe" you need to either allow 2-classes (thus extend again), or that all the categories you worked with were actually sets to begin with. What NBG gives you is the ability to quantify over proper classes, which makes dealing with small categories (and to a very very minor degree, for making statements about arbitrary small categories) much easier.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:44
$begingroup$
Altenatively, couldn't we simply a priori give a formal definition of say the "logical scope" of a variable in a logical formula and allow for modification to a particular subject matter, thus making any logical formula "scope-dependent". I.e. we could stop defining scope-independent variables, so that when we choose to generalize, paradoxical "spillover" of quantified variables are in some sense modded out by restriction to the new scope.
$endgroup$
– justsomeguy90
Jan 3 '15 at 22:07
$begingroup$
That sounds dangerously close to type theory. :-)
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 22:07
|
show 1 more comment
$begingroup$
Many people have felt like that.
This is why you often hear people in algebra moaning about set theory, or ignoring set theoretic issues (they usually can point out where these issues arise, and that someone out there knows how to solve them). This is also why there are people who are very enthusiastic about algebraic set theories like $sf ETCS$, or type theories like $sf SEAR$ and $sf HTT$, which may or may not prove to be a better foundation for algebra.
But switching from $sf ZFC$ to $sf NBG$ only pushes the problem "one step further". Sure, now you have the class of all groups as an actual object. But what about the category of all small categories? That's not a class anymore, since only sets are allowed to be elements of other classes, and small categories are not necessarily classes.
This is why working with universes is easier here. They allow you to jump "one level up" without any consequences. Each time you just extend the definition of what it means to be a set, and include more things as sets.
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
$endgroup$
Many people have felt like that.
This is why you often hear people in algebra moaning about set theory, or ignoring set theoretic issues (they usually can point out where these issues arise, and that someone out there knows how to solve them). This is also why there are people who are very enthusiastic about algebraic set theories like $sf ETCS$, or type theories like $sf SEAR$ and $sf HTT$, which may or may not prove to be a better foundation for algebra.
But switching from $sf ZFC$ to $sf NBG$ only pushes the problem "one step further". Sure, now you have the class of all groups as an actual object. But what about the category of all small categories? That's not a class anymore, since only sets are allowed to be elements of other classes, and small categories are not necessarily classes.
This is why working with universes is easier here. They allow you to jump "one level up" without any consequences. Each time you just extend the definition of what it means to be a set, and include more things as sets.
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
answered Jan 3 '15 at 21:23
Asaf Karagila♦Asaf Karagila
305k33435766
305k33435766
$begingroup$
I see your point, however, I was under the impression that NBG could handle such discussion. Is a universe not a class?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:40
$begingroup$
Actually, I guess since we are talking about the categories of all categories, we would simply allow for impredicative comprehension, right?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:43
$begingroup$
The universe is a class. But classes are not objects of other classes. So a category whose objects are small categories is a collection whose objects are themselves classes, and for this collection to be "in the universe" you need to either allow 2-classes (thus extend again), or that all the categories you worked with were actually sets to begin with. What NBG gives you is the ability to quantify over proper classes, which makes dealing with small categories (and to a very very minor degree, for making statements about arbitrary small categories) much easier.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:44
$begingroup$
Altenatively, couldn't we simply a priori give a formal definition of say the "logical scope" of a variable in a logical formula and allow for modification to a particular subject matter, thus making any logical formula "scope-dependent". I.e. we could stop defining scope-independent variables, so that when we choose to generalize, paradoxical "spillover" of quantified variables are in some sense modded out by restriction to the new scope.
$endgroup$
– justsomeguy90
Jan 3 '15 at 22:07
$begingroup$
That sounds dangerously close to type theory. :-)
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 22:07
|
show 1 more comment
$begingroup$
I see your point, however, I was under the impression that NBG could handle such discussion. Is a universe not a class?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:40
$begingroup$
Actually, I guess since we are talking about the categories of all categories, we would simply allow for impredicative comprehension, right?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:43
$begingroup$
The universe is a class. But classes are not objects of other classes. So a category whose objects are small categories is a collection whose objects are themselves classes, and for this collection to be "in the universe" you need to either allow 2-classes (thus extend again), or that all the categories you worked with were actually sets to begin with. What NBG gives you is the ability to quantify over proper classes, which makes dealing with small categories (and to a very very minor degree, for making statements about arbitrary small categories) much easier.
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 21:44
$begingroup$
Altenatively, couldn't we simply a priori give a formal definition of say the "logical scope" of a variable in a logical formula and allow for modification to a particular subject matter, thus making any logical formula "scope-dependent". I.e. we could stop defining scope-independent variables, so that when we choose to generalize, paradoxical "spillover" of quantified variables are in some sense modded out by restriction to the new scope.
$endgroup$
– justsomeguy90
Jan 3 '15 at 22:07
$begingroup$
That sounds dangerously close to type theory. :-)
$endgroup$
– Asaf Karagila♦
Jan 3 '15 at 22:07
$begingroup$
I see your point, however, I was under the impression that NBG could handle such discussion. Is a universe not a class?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:40
$begingroup$
I see your point, however, I was under the impression that NBG could handle such discussion. Is a universe not a class?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:40
$begingroup$
Actually, I guess since we are talking about the categories of all categories, we would simply allow for impredicative comprehension, right?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:43
$begingroup$
Actually, I guess since we are talking about the categories of all categories, we would simply allow for impredicative comprehension, right?
$endgroup$
– justsomeguy90
Jan 3 '15 at 21:43
$begingroup$
The universe is a class. But classes are not objects of other classes. So a category whose objects are small categories is a collection whose objects are themselves classes, and for this collection to be "in the universe" you need to either allow 2-classes (thus extend again), or that all the categories you worked with were actually sets to begin with. What NBG gives you is the ability to quantify over proper classes, which makes dealing with small categories (and to a very very minor degree, for making statements about arbitrary small categories) much easier.
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– Asaf Karagila♦
Jan 3 '15 at 21:44
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The universe is a class. But classes are not objects of other classes. So a category whose objects are small categories is a collection whose objects are themselves classes, and for this collection to be "in the universe" you need to either allow 2-classes (thus extend again), or that all the categories you worked with were actually sets to begin with. What NBG gives you is the ability to quantify over proper classes, which makes dealing with small categories (and to a very very minor degree, for making statements about arbitrary small categories) much easier.
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– Asaf Karagila♦
Jan 3 '15 at 21:44
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Altenatively, couldn't we simply a priori give a formal definition of say the "logical scope" of a variable in a logical formula and allow for modification to a particular subject matter, thus making any logical formula "scope-dependent". I.e. we could stop defining scope-independent variables, so that when we choose to generalize, paradoxical "spillover" of quantified variables are in some sense modded out by restriction to the new scope.
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– justsomeguy90
Jan 3 '15 at 22:07
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Altenatively, couldn't we simply a priori give a formal definition of say the "logical scope" of a variable in a logical formula and allow for modification to a particular subject matter, thus making any logical formula "scope-dependent". I.e. we could stop defining scope-independent variables, so that when we choose to generalize, paradoxical "spillover" of quantified variables are in some sense modded out by restriction to the new scope.
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– justsomeguy90
Jan 3 '15 at 22:07
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That sounds dangerously close to type theory. :-)
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– Asaf Karagila♦
Jan 3 '15 at 22:07
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That sounds dangerously close to type theory. :-)
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– Asaf Karagila♦
Jan 3 '15 at 22:07
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You should switch to linux and SEAR.
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– k.stm
Jan 3 '15 at 21:17
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@k.stm: How is that Linux? If anything, that's DEC10.
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– Asaf Karagila♦
Jan 3 '15 at 21:18
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Lol, I do use Linux, it was just an analogy.
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– justsomeguy90
Jan 3 '15 at 21:21
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@AsafKaragila Funnily, the page I linked to even indulges in that metaphor itself (which I didn’t know before): “Or, using an alternate metaphor, ZFC is like Windows, ETCS is like UNIX, and SEAR is like OS X (or maybe Ubuntu).” And maybe it’s not Ubuntu, but Debian or Arch or something.
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– k.stm
Jan 3 '15 at 21:22
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@k.stm: As someone who's been doing set theory for a while now, and been working with Linux for a while longer, and with Windows even longer than that, the comparison is bad. The comparison should be between CPU architectures, not operating systems.
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– Asaf Karagila♦
Jan 3 '15 at 21:25