Mixing
Is there such a thing as ordering algebraic theories and, secondarily, their theorems?
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Often, after learning a new definition, I find myself wondering what the "simplest" thing I can say now is, and the next "simplest" and so on. I do the same for structures as well. It seems like it must be possible to design measures on these things and partially order them. Like to me, obviously magma $ le $ monoid due to what is required to express either (in standard foundations I guess). Is the arithmetic hierarchy something that can be used for this, both for theorems and for theories? If not, is there such a tool or project already? In either case, where can I read more about this?
abstract-algebra first-order-logic meta-math
edited Jan 21 at 4:00
user549397
1,5061418
asked Jan 21 at 3:48
AnthonyAnthony
1064
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add a comment |
$begingroup$
Often, after learning a new definition, I find myself wondering what the "simplest" thing I can say now is, and the next "simplest" and so on. I do the same for structures as well. It seems like it must be possible to design measures on these things and partially order them. Like to me, obviously magma $ le $ monoid due to what is required to express either (in standard foundations I guess). Is the arithmetic hierarchy something that can be used for this, both for theorems and for theories? If not, is there such a tool or project already? In either case, where can I read more about this?
abstract-algebra first-order-logic meta-math
edited Jan 21 at 4:00
user549397
1,5061418
asked Jan 21 at 3:48
AnthonyAnthony
1064
$endgroup$
$begingroup$
Some hierarchical structures of mathematical concepts have been proposed, but these change as our understanding of the concepts changes.
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– David G. Stork
Jan 21 at 4:00
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Ok, could you name some? What if we're more specific and say in ZFC or in category theory?
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– Anthony
Jan 21 at 4:02
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In the example you gave, you are just (for a set with a binary operation) ordering by inclusion on the axioms you're imposing. This certainly is a logical thing to do, at least, within certain boundaries. I wouldn't advis e attempting to order the definition of "group" with "triangle" using a scheme like this. But I would definitely order the definitions of convex quadrilaterals this way, for example.
$endgroup$
– rschwieb
Jan 21 at 15:35
$begingroup$
en.wikipedia.org/wiki/Arithmetical_hierarchy en.wikipedia.org/wiki/Theory_(mathematical_logic)
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– Anthony
Jan 21 at 18:55
$begingroup$
If your definition of magma and monoid use the same binary symbol, you could say all magmas are monoids. In general, you can order theories by the subset relation on their instances.
$endgroup$
– PyRulez
Feb 9 at 22:23
add a comment |
$begingroup$
Often, after learning a new definition, I find myself wondering what the "simplest" thing I can say now is, and the next "simplest" and so on. I do the same for structures as well. It seems like it must be possible to design measures on these things and partially order them. Like to me, obviously magma $ le $ monoid due to what is required to express either (in standard foundations I guess). Is the arithmetic hierarchy something that can be used for this, both for theorems and for theories? If not, is there such a tool or project already? In either case, where can I read more about this?
abstract-algebra first-order-logic meta-math
edited Jan 21 at 4:00
user549397
1,5061418
asked Jan 21 at 3:48
AnthonyAnthony
1064
$endgroup$
Often, after learning a new definition, I find myself wondering what the "simplest" thing I can say now is, and the next "simplest" and so on. I do the same for structures as well. It seems like it must be possible to design measures on these things and partially order them. Like to me, obviously magma $ le $ monoid due to what is required to express either (in standard foundations I guess). Is the arithmetic hierarchy something that can be used for this, both for theorems and for theories? If not, is there such a tool or project already? In either case, where can I read more about this?
abstract-algebra first-order-logic meta-math
abstract-algebra first-order-logic meta-math
edited Jan 21 at 4:00
user549397
1,5061418
asked Jan 21 at 3:48
AnthonyAnthony
1064
edited Jan 21 at 4:00
user549397
1,5061418
asked Jan 21 at 3:48
AnthonyAnthony
1064
edited Jan 21 at 4:00
user549397
1,5061418
edited Jan 21 at 4:00
user549397
1,5061418
edited Jan 21 at 4:00
user549397
1,5061418
1,5061418
asked Jan 21 at 3:48
AnthonyAnthony
1064
asked Jan 21 at 3:48
AnthonyAnthony
1064
asked Jan 21 at 3:48
AnthonyAnthony
1064
1064
$begingroup$
Some hierarchical structures of mathematical concepts have been proposed, but these change as our understanding of the concepts changes.
$endgroup$
– David G. Stork
Jan 21 at 4:00
$begingroup$
Ok, could you name some? What if we're more specific and say in ZFC or in category theory?
$endgroup$
– Anthony
Jan 21 at 4:02
$begingroup$
In the example you gave, you are just (for a set with a binary operation) ordering by inclusion on the axioms you're imposing. This certainly is a logical thing to do, at least, within certain boundaries. I wouldn't advis e attempting to order the definition of "group" with "triangle" using a scheme like this. But I would definitely order the definitions of convex quadrilaterals this way, for example.
$endgroup$
– rschwieb
Jan 21 at 15:35
$begingroup$
en.wikipedia.org/wiki/Arithmetical_hierarchy en.wikipedia.org/wiki/Theory_(mathematical_logic)
$endgroup$
– Anthony
Jan 21 at 18:55
$begingroup$
If your definition of magma and monoid use the same binary symbol, you could say all magmas are monoids. In general, you can order theories by the subset relation on their instances.
$endgroup$
– PyRulez
Feb 9 at 22:23
add a comment |
$begingroup$
Some hierarchical structures of mathematical concepts have been proposed, but these change as our understanding of the concepts changes.
$endgroup$
– David G. Stork
Jan 21 at 4:00
$begingroup$
Ok, could you name some? What if we're more specific and say in ZFC or in category theory?
$endgroup$
– Anthony
Jan 21 at 4:02
$begingroup$
In the example you gave, you are just (for a set with a binary operation) ordering by inclusion on the axioms you're imposing. This certainly is a logical thing to do, at least, within certain boundaries. I wouldn't advis e attempting to order the definition of "group" with "triangle" using a scheme like this. But I would definitely order the definitions of convex quadrilaterals this way, for example.
$endgroup$
– rschwieb
Jan 21 at 15:35
$begingroup$
en.wikipedia.org/wiki/Arithmetical_hierarchy en.wikipedia.org/wiki/Theory_(mathematical_logic)
$endgroup$
– Anthony
Jan 21 at 18:55
$begingroup$
If your definition of magma and monoid use the same binary symbol, you could say all magmas are monoids. In general, you can order theories by the subset relation on their instances.
$endgroup$
– PyRulez
Feb 9 at 22:23
$begingroup$
Some hierarchical structures of mathematical concepts have been proposed, but these change as our understanding of the concepts changes.
$endgroup$
– David G. Stork
Jan 21 at 4:00
$begingroup$
Some hierarchical structures of mathematical concepts have been proposed, but these change as our understanding of the concepts changes.
$endgroup$
– David G. Stork
Jan 21 at 4:00
$begingroup$
Ok, could you name some? What if we're more specific and say in ZFC or in category theory?
$endgroup$
– Anthony
Jan 21 at 4:02
$begingroup$
Ok, could you name some? What if we're more specific and say in ZFC or in category theory?
$endgroup$
– Anthony
Jan 21 at 4:02
$begingroup$
In the example you gave, you are just (for a set with a binary operation) ordering by inclusion on the axioms you're imposing. This certainly is a logical thing to do, at least, within certain boundaries. I wouldn't advis e attempting to order the definition of "group" with "triangle" using a scheme like this. But I would definitely order the definitions of convex quadrilaterals this way, for example.
$endgroup$
– rschwieb
Jan 21 at 15:35
$begingroup$
In the example you gave, you are just (for a set with a binary operation) ordering by inclusion on the axioms you're imposing. This certainly is a logical thing to do, at least, within certain boundaries. I wouldn't advis e attempting to order the definition of "group" with "triangle" using a scheme like this. But I would definitely order the definitions of convex quadrilaterals this way, for example.
$endgroup$
– rschwieb
Jan 21 at 15:35
$begingroup$
en.wikipedia.org/wiki/Arithmetical_hierarchy en.wikipedia.org/wiki/Theory_(mathematical_logic)
$endgroup$
– Anthony
Jan 21 at 18:55
$begingroup$
en.wikipedia.org/wiki/Arithmetical_hierarchy en.wikipedia.org/wiki/Theory_(mathematical_logic)
$endgroup$
– Anthony
Jan 21 at 18:55
$begingroup$
If your definition of magma and monoid use the same binary symbol, you could say all magmas are monoids. In general, you can order theories by the subset relation on their instances.
$endgroup$
– PyRulez
Feb 9 at 22:23
$begingroup$
If your definition of magma and monoid use the same binary symbol, you could say all magmas are monoids. In general, you can order theories by the subset relation on their instances.
$endgroup$
– PyRulez
Feb 9 at 22:23
add a comment |
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$begingroup$
Some hierarchical structures of mathematical concepts have been proposed, but these change as our understanding of the concepts changes.
$endgroup$
– David G. Stork
Jan 21 at 4:00
$begingroup$
Ok, could you name some? What if we're more specific and say in ZFC or in category theory?
$endgroup$
– Anthony
Jan 21 at 4:02
$begingroup$
In the example you gave, you are just (for a set with a binary operation) ordering by inclusion on the axioms you're imposing. This certainly is a logical thing to do, at least, within certain boundaries. I wouldn't advis e attempting to order the definition of "group" with "triangle" using a scheme like this. But I would definitely order the definitions of convex quadrilaterals this way, for example.
$endgroup$
– rschwieb
Jan 21 at 15:35
$begingroup$
en.wikipedia.org/wiki/Arithmetical_hierarchy en.wikipedia.org/wiki/Theory_(mathematical_logic)
$endgroup$
– Anthony
Jan 21 at 18:55
$begingroup$
If your definition of magma and monoid use the same binary symbol, you could say all magmas are monoids. In general, you can order theories by the subset relation on their instances.
$endgroup$
– PyRulez
Feb 9 at 22:23