Easy Applications of Model Theory
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The question is inspired by this MathOverflow post and this post on MathSE.
The applications mentioned are usually pretty complicated (except for Ax-Grothendieck, but it seems to be a rare occurrence). Aside from these examples, are there easy "applications of model theory"? For example, can we use Morley's categorecity theorem to determine that a class of structures of interest "outside of model theory" has only unique models in higher cardinalities, where the actual proof of the fact is fairly complicated. I know that this works for $mathrm{ACF}_0$ for example, but it seems this can be established with the same amount of effort (i.e. proving Morley's theorem would be the same level of difficulty).
Also are there examples of nice categorizations? For example (this may be false but something in the spirit of) "omega-stable countable groups have a nice classification as products of some class of groups".
To sum up, I'm looking for "simple" applications that follow from basic results in, say Marker's text on Model Theory for example.
reference-request model-theory applications
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
asked Aug 6 '16 at 21:07
dav11
955512
|
show 6 more comments
up vote
8
down vote
favorite
The question is inspired by this MathOverflow post and this post on MathSE.
The applications mentioned are usually pretty complicated (except for Ax-Grothendieck, but it seems to be a rare occurrence). Aside from these examples, are there easy "applications of model theory"? For example, can we use Morley's categorecity theorem to determine that a class of structures of interest "outside of model theory" has only unique models in higher cardinalities, where the actual proof of the fact is fairly complicated. I know that this works for $mathrm{ACF}_0$ for example, but it seems this can be established with the same amount of effort (i.e. proving Morley's theorem would be the same level of difficulty).
Also are there examples of nice categorizations? For example (this may be false but something in the spirit of) "omega-stable countable groups have a nice classification as products of some class of groups".
To sum up, I'm looking for "simple" applications that follow from basic results in, say Marker's text on Model Theory for example.
reference-request model-theory applications
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
asked Aug 6 '16 at 21:07
dav11
955512
5
The infinite version of Ramsey's Theorem follows from the finite version via compactness.
– vhspdfg
Aug 6 '16 at 23:36
3
What about the simple consequence of the elimination of quantifiers for the fields $mathbb{C}$ and $mathbb{R}$: the projection of a semi-algebraic set is again semi-algebraic. (In the case of $mathbb{R}$, this is called the Tarski-Seidenberg theorem.)
– russoo
Aug 9 '16 at 16:54
2
Moreover, there is an easy generalization of the four color theorem to infinite graphs which uses the theorem for the finite case and compactness.
– russoo
Aug 9 '16 at 16:59
1
@dav11 Exercise. Consider an infinite planar graph and an appropriate language and theory such that, if that graph did not admit a four-coloring, then by compactness there would be some finite subset of the theory (which we arrange to correspond to a finite subset of the graph) which does not admit a four-coloring.
– vhspdfg
Aug 14 '16 at 16:45
1
It occurs that we can show the uniqueness of a countable atomless Boolean algebra by $omega$-categoricity if we wished to avoid back-and-forth. We can also show the existence of an algebraic closure of a field by compactness.
– vhspdfg
Aug 14 '16 at 16:46
|
show 6 more comments
up vote
8
down vote
favorite
up vote
8
down vote
favorite
The question is inspired by this MathOverflow post and this post on MathSE.
The applications mentioned are usually pretty complicated (except for Ax-Grothendieck, but it seems to be a rare occurrence). Aside from these examples, are there easy "applications of model theory"? For example, can we use Morley's categorecity theorem to determine that a class of structures of interest "outside of model theory" has only unique models in higher cardinalities, where the actual proof of the fact is fairly complicated. I know that this works for $mathrm{ACF}_0$ for example, but it seems this can be established with the same amount of effort (i.e. proving Morley's theorem would be the same level of difficulty).
Also are there examples of nice categorizations? For example (this may be false but something in the spirit of) "omega-stable countable groups have a nice classification as products of some class of groups".
To sum up, I'm looking for "simple" applications that follow from basic results in, say Marker's text on Model Theory for example.
reference-request model-theory applications
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
asked Aug 6 '16 at 21:07
dav11
955512
The question is inspired by this MathOverflow post and this post on MathSE.
The applications mentioned are usually pretty complicated (except for Ax-Grothendieck, but it seems to be a rare occurrence). Aside from these examples, are there easy "applications of model theory"? For example, can we use Morley's categorecity theorem to determine that a class of structures of interest "outside of model theory" has only unique models in higher cardinalities, where the actual proof of the fact is fairly complicated. I know that this works for $mathrm{ACF}_0$ for example, but it seems this can be established with the same amount of effort (i.e. proving Morley's theorem would be the same level of difficulty).
Also are there examples of nice categorizations? For example (this may be false but something in the spirit of) "omega-stable countable groups have a nice classification as products of some class of groups".
To sum up, I'm looking for "simple" applications that follow from basic results in, say Marker's text on Model Theory for example.
reference-request model-theory applications
reference-request model-theory applications
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
asked Aug 6 '16 at 21:07
dav11
955512
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
asked Aug 6 '16 at 21:07
dav11
955512
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
edited Nov 23 at 17:55
Martin Sleziak
44.6k7115270
44.6k7115270
asked Aug 6 '16 at 21:07
dav11
955512
asked Aug 6 '16 at 21:07
dav11
955512
asked Aug 6 '16 at 21:07
dav11
955512
955512
5
The infinite version of Ramsey's Theorem follows from the finite version via compactness.
– vhspdfg
Aug 6 '16 at 23:36
3
What about the simple consequence of the elimination of quantifiers for the fields $mathbb{C}$ and $mathbb{R}$: the projection of a semi-algebraic set is again semi-algebraic. (In the case of $mathbb{R}$, this is called the Tarski-Seidenberg theorem.)
– russoo
Aug 9 '16 at 16:54
2
Moreover, there is an easy generalization of the four color theorem to infinite graphs which uses the theorem for the finite case and compactness.
– russoo
Aug 9 '16 at 16:59
1
@dav11 Exercise. Consider an infinite planar graph and an appropriate language and theory such that, if that graph did not admit a four-coloring, then by compactness there would be some finite subset of the theory (which we arrange to correspond to a finite subset of the graph) which does not admit a four-coloring.
– vhspdfg
Aug 14 '16 at 16:45
1
It occurs that we can show the uniqueness of a countable atomless Boolean algebra by $omega$-categoricity if we wished to avoid back-and-forth. We can also show the existence of an algebraic closure of a field by compactness.
– vhspdfg
Aug 14 '16 at 16:46
|
show 6 more comments
5
The infinite version of Ramsey's Theorem follows from the finite version via compactness.
– vhspdfg
Aug 6 '16 at 23:36
3
What about the simple consequence of the elimination of quantifiers for the fields $mathbb{C}$ and $mathbb{R}$: the projection of a semi-algebraic set is again semi-algebraic. (In the case of $mathbb{R}$, this is called the Tarski-Seidenberg theorem.)
– russoo
Aug 9 '16 at 16:54
2
Moreover, there is an easy generalization of the four color theorem to infinite graphs which uses the theorem for the finite case and compactness.
– russoo
Aug 9 '16 at 16:59
1
@dav11 Exercise. Consider an infinite planar graph and an appropriate language and theory such that, if that graph did not admit a four-coloring, then by compactness there would be some finite subset of the theory (which we arrange to correspond to a finite subset of the graph) which does not admit a four-coloring.
– vhspdfg
Aug 14 '16 at 16:45
1
It occurs that we can show the uniqueness of a countable atomless Boolean algebra by $omega$-categoricity if we wished to avoid back-and-forth. We can also show the existence of an algebraic closure of a field by compactness.
– vhspdfg
Aug 14 '16 at 16:46
5
5
The infinite version of Ramsey's Theorem follows from the finite version via compactness.
– vhspdfg
Aug 6 '16 at 23:36
The infinite version of Ramsey's Theorem follows from the finite version via compactness.
– vhspdfg
Aug 6 '16 at 23:36
3
3
What about the simple consequence of the elimination of quantifiers for the fields $mathbb{C}$ and $mathbb{R}$: the projection of a semi-algebraic set is again semi-algebraic. (In the case of $mathbb{R}$, this is called the Tarski-Seidenberg theorem.)
– russoo
Aug 9 '16 at 16:54
What about the simple consequence of the elimination of quantifiers for the fields $mathbb{C}$ and $mathbb{R}$: the projection of a semi-algebraic set is again semi-algebraic. (In the case of $mathbb{R}$, this is called the Tarski-Seidenberg theorem.)
– russoo
Aug 9 '16 at 16:54
2
2
Moreover, there is an easy generalization of the four color theorem to infinite graphs which uses the theorem for the finite case and compactness.
– russoo
Aug 9 '16 at 16:59
Moreover, there is an easy generalization of the four color theorem to infinite graphs which uses the theorem for the finite case and compactness.
– russoo
Aug 9 '16 at 16:59
1
1
@dav11 Exercise. Consider an infinite planar graph and an appropriate language and theory such that, if that graph did not admit a four-coloring, then by compactness there would be some finite subset of the theory (which we arrange to correspond to a finite subset of the graph) which does not admit a four-coloring.
– vhspdfg
Aug 14 '16 at 16:45
@dav11 Exercise. Consider an infinite planar graph and an appropriate language and theory such that, if that graph did not admit a four-coloring, then by compactness there would be some finite subset of the theory (which we arrange to correspond to a finite subset of the graph) which does not admit a four-coloring.
– vhspdfg
Aug 14 '16 at 16:45
1
1
It occurs that we can show the uniqueness of a countable atomless Boolean algebra by $omega$-categoricity if we wished to avoid back-and-forth. We can also show the existence of an algebraic closure of a field by compactness.
– vhspdfg
Aug 14 '16 at 16:46
It occurs that we can show the uniqueness of a countable atomless Boolean algebra by $omega$-categoricity if we wished to avoid back-and-forth. We can also show the existence of an algebraic closure of a field by compactness.
– vhspdfg
Aug 14 '16 at 16:46
|
show 6 more comments
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5
The infinite version of Ramsey's Theorem follows from the finite version via compactness.
– vhspdfg
Aug 6 '16 at 23:36
3
What about the simple consequence of the elimination of quantifiers for the fields $mathbb{C}$ and $mathbb{R}$: the projection of a semi-algebraic set is again semi-algebraic. (In the case of $mathbb{R}$, this is called the Tarski-Seidenberg theorem.)
– russoo
Aug 9 '16 at 16:54
2
Moreover, there is an easy generalization of the four color theorem to infinite graphs which uses the theorem for the finite case and compactness.
– russoo
Aug 9 '16 at 16:59
1
@dav11 Exercise. Consider an infinite planar graph and an appropriate language and theory such that, if that graph did not admit a four-coloring, then by compactness there would be some finite subset of the theory (which we arrange to correspond to a finite subset of the graph) which does not admit a four-coloring.
– vhspdfg
Aug 14 '16 at 16:45
1
It occurs that we can show the uniqueness of a countable atomless Boolean algebra by $omega$-categoricity if we wished to avoid back-and-forth. We can also show the existence of an algebraic closure of a field by compactness.
– vhspdfg
Aug 14 '16 at 16:46