Mixing
“Natural†orders on universes
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I'm a student and I'm thinking about the sensibility of the notion that there might be some intrinsic order to mathematical objects in general, one that is "natural" (free of arbitrary choice) to every extent that the objects can be free from context. What is the right way to think about this?
Say I would like to define a total order $preceq$ on $V$, the class of all objects formalized in a formal system $F$.
A total order $le$ on language $L$ of $F$ together with the following can specify a total order $preceq$ on the class $V$ of all objects formalized in $F$.
For all $x_1,x_2 in V$, define $x_1preceq x_2$ to mean that for any string $s_2 in L$ denoting $x_2$, there exists a string $s_1 in L$ denoting $x_1$ such that $s_1 leq s_2$.
For example, such an ordering on $L$ could be obtained from any given Gödel numbering.
Orders defined like this:
- Imply a lower bound
- Depend on a poorly motivated choice (to use a Gödel numbering in this case)
- Depend on the means of expression (what $F$ is, in this case)
I would like to understand if it is even sensible to consider a specific "universal order" without any or all of these three points.
order-theory philosophy foundations
asked Jul 17 at 21:28
googlefailedme
12
add a comment |Â
up vote
-1
down vote
favorite
I'm a student and I'm thinking about the sensibility of the notion that there might be some intrinsic order to mathematical objects in general, one that is "natural" (free of arbitrary choice) to every extent that the objects can be free from context. What is the right way to think about this?
Say I would like to define a total order $preceq$ on $V$, the class of all objects formalized in a formal system $F$.
A total order $le$ on language $L$ of $F$ together with the following can specify a total order $preceq$ on the class $V$ of all objects formalized in $F$.
For all $x_1,x_2 in V$, define $x_1preceq x_2$ to mean that for any string $s_2 in L$ denoting $x_2$, there exists a string $s_1 in L$ denoting $x_1$ such that $s_1 leq s_2$.
For example, such an ordering on $L$ could be obtained from any given Gödel numbering.
Orders defined like this:
- Imply a lower bound
- Depend on a poorly motivated choice (to use a Gödel numbering in this case)
- Depend on the means of expression (what $F$ is, in this case)
I would like to understand if it is even sensible to consider a specific "universal order" without any or all of these three points.
order-theory philosophy foundations
asked Jul 17 at 21:28
googlefailedme
12
1
The immediate problem with your idea is the following: There may not be a string (rather formula) denoting a given element. And even if there is (which is possible), the function mapping an element to such a string will not be accessible to your universe (as otherwise it would know that it is countable).
– Stefan Mesken
Jul 17 at 21:38
Building off of Stefan's point, in $L$ every object is definable with ordinal parameters. Since the ordinals themselves come with a well-ordering, this is enough to give a (class) well-ordering of $L$, but in general there will be objects not even definable from ordinal parameters. Indeed, if the axiom of choice fails there may even be sets which cannot be linearly ordered at all, let alone in some "definable" way.
– Noah Schweber
Jul 17 at 23:06
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I'm a student and I'm thinking about the sensibility of the notion that there might be some intrinsic order to mathematical objects in general, one that is "natural" (free of arbitrary choice) to every extent that the objects can be free from context. What is the right way to think about this?
Say I would like to define a total order $preceq$ on $V$, the class of all objects formalized in a formal system $F$.
A total order $le$ on language $L$ of $F$ together with the following can specify a total order $preceq$ on the class $V$ of all objects formalized in $F$.
For all $x_1,x_2 in V$, define $x_1preceq x_2$ to mean that for any string $s_2 in L$ denoting $x_2$, there exists a string $s_1 in L$ denoting $x_1$ such that $s_1 leq s_2$.
For example, such an ordering on $L$ could be obtained from any given Gödel numbering.
Orders defined like this:
- Imply a lower bound
- Depend on a poorly motivated choice (to use a Gödel numbering in this case)
- Depend on the means of expression (what $F$ is, in this case)
I would like to understand if it is even sensible to consider a specific "universal order" without any or all of these three points.
order-theory philosophy foundations
asked Jul 17 at 21:28
googlefailedme
12
I'm a student and I'm thinking about the sensibility of the notion that there might be some intrinsic order to mathematical objects in general, one that is "natural" (free of arbitrary choice) to every extent that the objects can be free from context. What is the right way to think about this?
Say I would like to define a total order $preceq$ on $V$, the class of all objects formalized in a formal system $F$.
A total order $le$ on language $L$ of $F$ together with the following can specify a total order $preceq$ on the class $V$ of all objects formalized in $F$.
For all $x_1,x_2 in V$, define $x_1preceq x_2$ to mean that for any string $s_2 in L$ denoting $x_2$, there exists a string $s_1 in L$ denoting $x_1$ such that $s_1 leq s_2$.
For example, such an ordering on $L$ could be obtained from any given Gödel numbering.
Orders defined like this:
- Imply a lower bound
- Depend on a poorly motivated choice (to use a Gödel numbering in this case)
- Depend on the means of expression (what $F$ is, in this case)
I would like to understand if it is even sensible to consider a specific "universal order" without any or all of these three points.
order-theory philosophy foundations
asked Jul 17 at 21:28
googlefailedme
12
edited Jul 17 at 21:30
asked Jul 17 at 21:28
googlefailedme
12
asked Jul 17 at 21:28
googlefailedme
12
asked Jul 17 at 21:28
googlefailedme
12
12
1
The immediate problem with your idea is the following: There may not be a string (rather formula) denoting a given element. And even if there is (which is possible), the function mapping an element to such a string will not be accessible to your universe (as otherwise it would know that it is countable).
– Stefan Mesken
Jul 17 at 21:38
Building off of Stefan's point, in $L$ every object is definable with ordinal parameters. Since the ordinals themselves come with a well-ordering, this is enough to give a (class) well-ordering of $L$, but in general there will be objects not even definable from ordinal parameters. Indeed, if the axiom of choice fails there may even be sets which cannot be linearly ordered at all, let alone in some "definable" way.
– Noah Schweber
Jul 17 at 23:06
add a comment |Â
1
The immediate problem with your idea is the following: There may not be a string (rather formula) denoting a given element. And even if there is (which is possible), the function mapping an element to such a string will not be accessible to your universe (as otherwise it would know that it is countable).
– Stefan Mesken
Jul 17 at 21:38
Building off of Stefan's point, in $L$ every object is definable with ordinal parameters. Since the ordinals themselves come with a well-ordering, this is enough to give a (class) well-ordering of $L$, but in general there will be objects not even definable from ordinal parameters. Indeed, if the axiom of choice fails there may even be sets which cannot be linearly ordered at all, let alone in some "definable" way.
– Noah Schweber
Jul 17 at 23:06
1
1
The immediate problem with your idea is the following: There may not be a string (rather formula) denoting a given element. And even if there is (which is possible), the function mapping an element to such a string will not be accessible to your universe (as otherwise it would know that it is countable).
– Stefan Mesken
Jul 17 at 21:38
The immediate problem with your idea is the following: There may not be a string (rather formula) denoting a given element. And even if there is (which is possible), the function mapping an element to such a string will not be accessible to your universe (as otherwise it would know that it is countable).
– Stefan Mesken
Jul 17 at 21:38
Building off of Stefan's point, in $L$ every object is definable with ordinal parameters. Since the ordinals themselves come with a well-ordering, this is enough to give a (class) well-ordering of $L$, but in general there will be objects not even definable from ordinal parameters. Indeed, if the axiom of choice fails there may even be sets which cannot be linearly ordered at all, let alone in some "definable" way.
– Noah Schweber
Jul 17 at 23:06
Building off of Stefan's point, in $L$ every object is definable with ordinal parameters. Since the ordinals themselves come with a well-ordering, this is enough to give a (class) well-ordering of $L$, but in general there will be objects not even definable from ordinal parameters. Indeed, if the axiom of choice fails there may even be sets which cannot be linearly ordered at all, let alone in some "definable" way.
– Noah Schweber
Jul 17 at 23:06
add a comment |Â
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1
The immediate problem with your idea is the following: There may not be a string (rather formula) denoting a given element. And even if there is (which is possible), the function mapping an element to such a string will not be accessible to your universe (as otherwise it would know that it is countable).
– Stefan Mesken
Jul 17 at 21:38
Building off of Stefan's point, in $L$ every object is definable with ordinal parameters. Since the ordinals themselves come with a well-ordering, this is enough to give a (class) well-ordering of $L$, but in general there will be objects not even definable from ordinal parameters. Indeed, if the axiom of choice fails there may even be sets which cannot be linearly ordered at all, let alone in some "definable" way.
– Noah Schweber
Jul 17 at 23:06