Does naturality imply independence from choice?
$begingroup$
My question concerns the notion of naturality defined in Section 12.3 of Hodge's (longer) Model Theory. See the addenda for the definition.
Hodges proves a result that if an algebraic construction of $B$ from $A$ is not natural, then (if Pincus's transfer principle from ZFA to ZF is applicable) there is a model of ZF with $A$ but with nothing isomorphic to $B$. An example of such a construction is one of the algebraic closure $B$ from a field $A$. One is naturally interested in whether its converse obtains in certain circumstances.
The textbook cites two papers by the author himself and Shelah (Naturality and definability, I and II) which study a similar question but concern global choice principles. The papers study the converse as well, but I do not believe that they answer the question above.
Is there a paper that address the question whether Hodges's notion of naturality implies that the construction can be done without AC, at least in some cases?
Addendum. The following is Hodges's definition of naturality. The setting is that $B$ has extra nonlogical symbols that $A$ doesn't have, and a unary predicate symbol $P$ is among the extra symbols. The subset $P^B$ is assumed to induce a substructure of $B$ that happens to be $A$. Moreover, we assume that every automorphism on $A$ extends to one on $B$, i.e., there is a (set-theoretic) map (called a section) $mathrm{Aut}(A) to mathrm{Aut}(B)$. Under all these assumptions, $B$ is natural over $A$ if the section is a group homomorphism, i.e., a splitting.
Addendum II. I guess that Hodge's notion of naturality translates to that in category theory as follows. Let $C(M)$ stand for the category of structures isomorphic to a structure $M$ whose morphisms are the isomorphisms. The construction in question can be thought of as a functor $F : C(A) to C(B).$ In this setting, the construction is natural in the sense of Hodges iff the inclusion maps $eta_A : A to F(A)$ forms a natural transformation from the identity functor on $C(A)$ to $F$.
logic reference-request set-theory model-theory
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
$endgroup$
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show 10 more comments
$begingroup$
My question concerns the notion of naturality defined in Section 12.3 of Hodge's (longer) Model Theory. See the addenda for the definition.
Hodges proves a result that if an algebraic construction of $B$ from $A$ is not natural, then (if Pincus's transfer principle from ZFA to ZF is applicable) there is a model of ZF with $A$ but with nothing isomorphic to $B$. An example of such a construction is one of the algebraic closure $B$ from a field $A$. One is naturally interested in whether its converse obtains in certain circumstances.
The textbook cites two papers by the author himself and Shelah (Naturality and definability, I and II) which study a similar question but concern global choice principles. The papers study the converse as well, but I do not believe that they answer the question above.
Is there a paper that address the question whether Hodges's notion of naturality implies that the construction can be done without AC, at least in some cases?
Addendum. The following is Hodges's definition of naturality. The setting is that $B$ has extra nonlogical symbols that $A$ doesn't have, and a unary predicate symbol $P$ is among the extra symbols. The subset $P^B$ is assumed to induce a substructure of $B$ that happens to be $A$. Moreover, we assume that every automorphism on $A$ extends to one on $B$, i.e., there is a (set-theoretic) map (called a section) $mathrm{Aut}(A) to mathrm{Aut}(B)$. Under all these assumptions, $B$ is natural over $A$ if the section is a group homomorphism, i.e., a splitting.
Addendum II. I guess that Hodge's notion of naturality translates to that in category theory as follows. Let $C(M)$ stand for the category of structures isomorphic to a structure $M$ whose morphisms are the isomorphisms. The construction in question can be thought of as a functor $F : C(A) to C(B).$ In this setting, the construction is natural in the sense of Hodges iff the inclusion maps $eta_A : A to F(A)$ forms a natural transformation from the identity functor on $C(A)$ to $F$.
logic reference-request set-theory model-theory
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
$endgroup$
$begingroup$
It's maybe too long, but also 12.3 in the book is too long for me to search for the definition. How about giving an accurate full reference (page number, etc.)?
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:47
$begingroup$
Also if you provide the location of where he has the discussion about Pincus' theorem, it would be great.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:50
$begingroup$
@AsafKaragila The definition is on the very first page of the section. The discussion about Pincus's theorem is in the very last subsection starting on page 630. Theorem 12.3.7 on page 631 has the result about ZFA.
$endgroup$
– Pteromys
Dec 11 '18 at 12:54
$begingroup$
Okay. Having spent some time thinking about this whole thing. It's an interesting question and I will pursue it further. But what is your actual question? I mean, Hodges prove that under reasonable conditions, not natural requires choice. And we know that something requiring choice is generally not natural. So natural things, in reasonable scenarios are those things that do not require any choices.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 13:14
1
$begingroup$
@Pteromys Do you have any evidence that Hodges's notion of naturality "is meant to be equivalent, at least in spirit, and perhaps technically in some cases, to the notion of the same name in category theory." I don't see any connection to the notion of natural transformation here; I think it's more likely that Hodges just picked a "natural" name, which happens to also have a meaning in category theory.
$endgroup$
– Alex Kruckman
Dec 11 '18 at 15:10
|
show 10 more comments
$begingroup$
My question concerns the notion of naturality defined in Section 12.3 of Hodge's (longer) Model Theory. See the addenda for the definition.
Hodges proves a result that if an algebraic construction of $B$ from $A$ is not natural, then (if Pincus's transfer principle from ZFA to ZF is applicable) there is a model of ZF with $A$ but with nothing isomorphic to $B$. An example of such a construction is one of the algebraic closure $B$ from a field $A$. One is naturally interested in whether its converse obtains in certain circumstances.
The textbook cites two papers by the author himself and Shelah (Naturality and definability, I and II) which study a similar question but concern global choice principles. The papers study the converse as well, but I do not believe that they answer the question above.
Is there a paper that address the question whether Hodges's notion of naturality implies that the construction can be done without AC, at least in some cases?
Addendum. The following is Hodges's definition of naturality. The setting is that $B$ has extra nonlogical symbols that $A$ doesn't have, and a unary predicate symbol $P$ is among the extra symbols. The subset $P^B$ is assumed to induce a substructure of $B$ that happens to be $A$. Moreover, we assume that every automorphism on $A$ extends to one on $B$, i.e., there is a (set-theoretic) map (called a section) $mathrm{Aut}(A) to mathrm{Aut}(B)$. Under all these assumptions, $B$ is natural over $A$ if the section is a group homomorphism, i.e., a splitting.
Addendum II. I guess that Hodge's notion of naturality translates to that in category theory as follows. Let $C(M)$ stand for the category of structures isomorphic to a structure $M$ whose morphisms are the isomorphisms. The construction in question can be thought of as a functor $F : C(A) to C(B).$ In this setting, the construction is natural in the sense of Hodges iff the inclusion maps $eta_A : A to F(A)$ forms a natural transformation from the identity functor on $C(A)$ to $F$.
logic reference-request set-theory model-theory
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
$endgroup$
My question concerns the notion of naturality defined in Section 12.3 of Hodge's (longer) Model Theory. See the addenda for the definition.
Hodges proves a result that if an algebraic construction of $B$ from $A$ is not natural, then (if Pincus's transfer principle from ZFA to ZF is applicable) there is a model of ZF with $A$ but with nothing isomorphic to $B$. An example of such a construction is one of the algebraic closure $B$ from a field $A$. One is naturally interested in whether its converse obtains in certain circumstances.
The textbook cites two papers by the author himself and Shelah (Naturality and definability, I and II) which study a similar question but concern global choice principles. The papers study the converse as well, but I do not believe that they answer the question above.
Is there a paper that address the question whether Hodges's notion of naturality implies that the construction can be done without AC, at least in some cases?
Addendum. The following is Hodges's definition of naturality. The setting is that $B$ has extra nonlogical symbols that $A$ doesn't have, and a unary predicate symbol $P$ is among the extra symbols. The subset $P^B$ is assumed to induce a substructure of $B$ that happens to be $A$. Moreover, we assume that every automorphism on $A$ extends to one on $B$, i.e., there is a (set-theoretic) map (called a section) $mathrm{Aut}(A) to mathrm{Aut}(B)$. Under all these assumptions, $B$ is natural over $A$ if the section is a group homomorphism, i.e., a splitting.
Addendum II. I guess that Hodge's notion of naturality translates to that in category theory as follows. Let $C(M)$ stand for the category of structures isomorphic to a structure $M$ whose morphisms are the isomorphisms. The construction in question can be thought of as a functor $F : C(A) to C(B).$ In this setting, the construction is natural in the sense of Hodges iff the inclusion maps $eta_A : A to F(A)$ forms a natural transformation from the identity functor on $C(A)$ to $F$.
logic reference-request set-theory model-theory
logic reference-request set-theory model-theory
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
edited Dec 15 '18 at 6:22
Pteromys
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
asked Dec 11 '18 at 12:41
PteromysPteromys
2,46121745
2,46121745
$begingroup$
It's maybe too long, but also 12.3 in the book is too long for me to search for the definition. How about giving an accurate full reference (page number, etc.)?
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:47
$begingroup$
Also if you provide the location of where he has the discussion about Pincus' theorem, it would be great.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:50
$begingroup$
@AsafKaragila The definition is on the very first page of the section. The discussion about Pincus's theorem is in the very last subsection starting on page 630. Theorem 12.3.7 on page 631 has the result about ZFA.
$endgroup$
– Pteromys
Dec 11 '18 at 12:54
$begingroup$
Okay. Having spent some time thinking about this whole thing. It's an interesting question and I will pursue it further. But what is your actual question? I mean, Hodges prove that under reasonable conditions, not natural requires choice. And we know that something requiring choice is generally not natural. So natural things, in reasonable scenarios are those things that do not require any choices.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 13:14
1
$begingroup$
@Pteromys Do you have any evidence that Hodges's notion of naturality "is meant to be equivalent, at least in spirit, and perhaps technically in some cases, to the notion of the same name in category theory." I don't see any connection to the notion of natural transformation here; I think it's more likely that Hodges just picked a "natural" name, which happens to also have a meaning in category theory.
$endgroup$
– Alex Kruckman
Dec 11 '18 at 15:10
|
show 10 more comments
$begingroup$
It's maybe too long, but also 12.3 in the book is too long for me to search for the definition. How about giving an accurate full reference (page number, etc.)?
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:47
$begingroup$
Also if you provide the location of where he has the discussion about Pincus' theorem, it would be great.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:50
$begingroup$
@AsafKaragila The definition is on the very first page of the section. The discussion about Pincus's theorem is in the very last subsection starting on page 630. Theorem 12.3.7 on page 631 has the result about ZFA.
$endgroup$
– Pteromys
Dec 11 '18 at 12:54
$begingroup$
Okay. Having spent some time thinking about this whole thing. It's an interesting question and I will pursue it further. But what is your actual question? I mean, Hodges prove that under reasonable conditions, not natural requires choice. And we know that something requiring choice is generally not natural. So natural things, in reasonable scenarios are those things that do not require any choices.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 13:14
1
$begingroup$
@Pteromys Do you have any evidence that Hodges's notion of naturality "is meant to be equivalent, at least in spirit, and perhaps technically in some cases, to the notion of the same name in category theory." I don't see any connection to the notion of natural transformation here; I think it's more likely that Hodges just picked a "natural" name, which happens to also have a meaning in category theory.
$endgroup$
– Alex Kruckman
Dec 11 '18 at 15:10
$begingroup$
It's maybe too long, but also 12.3 in the book is too long for me to search for the definition. How about giving an accurate full reference (page number, etc.)?
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:47
$begingroup$
It's maybe too long, but also 12.3 in the book is too long for me to search for the definition. How about giving an accurate full reference (page number, etc.)?
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:47
$begingroup$
Also if you provide the location of where he has the discussion about Pincus' theorem, it would be great.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:50
$begingroup$
Also if you provide the location of where he has the discussion about Pincus' theorem, it would be great.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:50
$begingroup$
@AsafKaragila The definition is on the very first page of the section. The discussion about Pincus's theorem is in the very last subsection starting on page 630. Theorem 12.3.7 on page 631 has the result about ZFA.
$endgroup$
– Pteromys
Dec 11 '18 at 12:54
$begingroup$
@AsafKaragila The definition is on the very first page of the section. The discussion about Pincus's theorem is in the very last subsection starting on page 630. Theorem 12.3.7 on page 631 has the result about ZFA.
$endgroup$
– Pteromys
Dec 11 '18 at 12:54
$begingroup$
Okay. Having spent some time thinking about this whole thing. It's an interesting question and I will pursue it further. But what is your actual question? I mean, Hodges prove that under reasonable conditions, not natural requires choice. And we know that something requiring choice is generally not natural. So natural things, in reasonable scenarios are those things that do not require any choices.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 13:14
$begingroup$
Okay. Having spent some time thinking about this whole thing. It's an interesting question and I will pursue it further. But what is your actual question? I mean, Hodges prove that under reasonable conditions, not natural requires choice. And we know that something requiring choice is generally not natural. So natural things, in reasonable scenarios are those things that do not require any choices.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 13:14
1
1
$begingroup$
@Pteromys Do you have any evidence that Hodges's notion of naturality "is meant to be equivalent, at least in spirit, and perhaps technically in some cases, to the notion of the same name in category theory." I don't see any connection to the notion of natural transformation here; I think it's more likely that Hodges just picked a "natural" name, which happens to also have a meaning in category theory.
$endgroup$
– Alex Kruckman
Dec 11 '18 at 15:10
$begingroup$
@Pteromys Do you have any evidence that Hodges's notion of naturality "is meant to be equivalent, at least in spirit, and perhaps technically in some cases, to the notion of the same name in category theory." I don't see any connection to the notion of natural transformation here; I think it's more likely that Hodges just picked a "natural" name, which happens to also have a meaning in category theory.
$endgroup$
– Alex Kruckman
Dec 11 '18 at 15:10
|
show 10 more comments
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$begingroup$
It's maybe too long, but also 12.3 in the book is too long for me to search for the definition. How about giving an accurate full reference (page number, etc.)?
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:47
$begingroup$
Also if you provide the location of where he has the discussion about Pincus' theorem, it would be great.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 12:50
$begingroup$
@AsafKaragila The definition is on the very first page of the section. The discussion about Pincus's theorem is in the very last subsection starting on page 630. Theorem 12.3.7 on page 631 has the result about ZFA.
$endgroup$
– Pteromys
Dec 11 '18 at 12:54
$begingroup$
Okay. Having spent some time thinking about this whole thing. It's an interesting question and I will pursue it further. But what is your actual question? I mean, Hodges prove that under reasonable conditions, not natural requires choice. And we know that something requiring choice is generally not natural. So natural things, in reasonable scenarios are those things that do not require any choices.
$endgroup$
– Asaf Karagila♦
Dec 11 '18 at 13:14
1
$begingroup$
@Pteromys Do you have any evidence that Hodges's notion of naturality "is meant to be equivalent, at least in spirit, and perhaps technically in some cases, to the notion of the same name in category theory." I don't see any connection to the notion of natural transformation here; I think it's more likely that Hodges just picked a "natural" name, which happens to also have a meaning in category theory.
$endgroup$
– Alex Kruckman
Dec 11 '18 at 15:10