Mixing
Axiom of Choice as similar to Parallel Postulate?
7
$begingroup$
I'm looking for resources that have drawn a comparison between the Parallel Postulate and the Axiom of Choice.
That is, if we treat ZFC as an analogue to Euclidean geometry, can we view the development of models of ZF that, for instance, exclude AC and CH, as being similar to geometries that exclude certain axioms of Euclidean geometry?
I haven't been able to locate any, but it's also a really tough idea to frame in academic journal search engines, and this is about the only result I can uncover in general searches: https://thetwomeatmeal.wordpress.com/2010/10/29/the-zermelo-fraenkel-axioms-for-sets/
geometry logic axiom-of-choice
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
$endgroup$
|
show 10 more comments
7
$begingroup$
I'm looking for resources that have drawn a comparison between the Parallel Postulate and the Axiom of Choice.
That is, if we treat ZFC as an analogue to Euclidean geometry, can we view the development of models of ZF that, for instance, exclude AC and CH, as being similar to geometries that exclude certain axioms of Euclidean geometry?
I haven't been able to locate any, but it's also a really tough idea to frame in academic journal search engines, and this is about the only result I can uncover in general searches: https://thetwomeatmeal.wordpress.com/2010/10/29/the-zermelo-fraenkel-axioms-for-sets/
geometry logic axiom-of-choice
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
$endgroup$
3
$begingroup$
There is some fuzzy/soft similaity in that both AC and the parallel postulate are (1) claims that intuitively feel like they should be true, but (2) are somewhat more complex state than the simpler "fundamental" axioms the rest of the theory can be reduced to, (3) therefore people have been searching for a way to reduce them to simpler axiomatic truths, but (4) eventually this search terminated in the discovery that they're independent of the recognized simpler axioms.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:02
3
$begingroup$
Note that this narrative doesn't say that we know there cannot be a property that is equivalent to the axiom in question and is so simple to state that it would have been accepted as an axiom without all the attempts to prove it from other axioms. This seems rather unlikely in either of these cases, but we don't really have a good technical concept of what exactly "simple enough" would mean, so we can't even hope to prove that it is so.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:06
1
$begingroup$
The OP can correct me, but I think the OP is indeed looking for some more abstract parallel like Henning is pointing out, to which we can add that just as there are two separate branches for Euclidian and Non-Euclidian geometry, there are also two branches regarding the AC: one where AC is assumed to be true and one where AC is assumed to be false. Maybe the OP is interested in knowing more about that idea?
$endgroup$
– Bram28
Jan 2 '18 at 17:21
1
$begingroup$
Bram28, Henning, this is indeed what I'm looking for; preferably in a publication of some sort; a discussion of how discarding the parallel postulate enriched geometry--and eventually had real-world application vis a vis Minkowski space-time--alongside a discussion of AC making similar points.
$endgroup$
– Rich Jensen
Jan 2 '18 at 17:33
1
$begingroup$
The axiom of choice is nothing like the axiom of parallels in Euclidean geometry. The parallels postulate was considered an oddity because of its technical and unnatural formulation. The axiom of choice is natural (one could argue about technical, but it's not as technical as Replacement). If any of the ZFC axioms is the odd one out, it would be Foundation (Regularity). That's also the only axiom without much impact on "normal mathematics". It's an odd technical duck, which is very interesting from a set theoretic point and has opposites in Anti Foundation Axioms. But yeah, it's not as sexy.
$endgroup$
– Asaf Karagila♦
Jan 2 '18 at 19:24
|
show 10 more comments
7
7
7
2
$begingroup$
I'm looking for resources that have drawn a comparison between the Parallel Postulate and the Axiom of Choice.
That is, if we treat ZFC as an analogue to Euclidean geometry, can we view the development of models of ZF that, for instance, exclude AC and CH, as being similar to geometries that exclude certain axioms of Euclidean geometry?
I haven't been able to locate any, but it's also a really tough idea to frame in academic journal search engines, and this is about the only result I can uncover in general searches: https://thetwomeatmeal.wordpress.com/2010/10/29/the-zermelo-fraenkel-axioms-for-sets/
geometry logic axiom-of-choice
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
$endgroup$
I'm looking for resources that have drawn a comparison between the Parallel Postulate and the Axiom of Choice.
That is, if we treat ZFC as an analogue to Euclidean geometry, can we view the development of models of ZF that, for instance, exclude AC and CH, as being similar to geometries that exclude certain axioms of Euclidean geometry?
I haven't been able to locate any, but it's also a really tough idea to frame in academic journal search engines, and this is about the only result I can uncover in general searches: https://thetwomeatmeal.wordpress.com/2010/10/29/the-zermelo-fraenkel-axioms-for-sets/
geometry logic axiom-of-choice
geometry logic axiom-of-choice
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
edited Jan 28 at 20:02
Maria Mazur
48.3k1260121
48.3k1260121
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
asked Jan 2 '18 at 16:46
Rich JensenRich Jensen
1476
1476
3
$begingroup$
There is some fuzzy/soft similaity in that both AC and the parallel postulate are (1) claims that intuitively feel like they should be true, but (2) are somewhat more complex state than the simpler "fundamental" axioms the rest of the theory can be reduced to, (3) therefore people have been searching for a way to reduce them to simpler axiomatic truths, but (4) eventually this search terminated in the discovery that they're independent of the recognized simpler axioms.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:02
3
$begingroup$
Note that this narrative doesn't say that we know there cannot be a property that is equivalent to the axiom in question and is so simple to state that it would have been accepted as an axiom without all the attempts to prove it from other axioms. This seems rather unlikely in either of these cases, but we don't really have a good technical concept of what exactly "simple enough" would mean, so we can't even hope to prove that it is so.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:06
1
$begingroup$
The OP can correct me, but I think the OP is indeed looking for some more abstract parallel like Henning is pointing out, to which we can add that just as there are two separate branches for Euclidian and Non-Euclidian geometry, there are also two branches regarding the AC: one where AC is assumed to be true and one where AC is assumed to be false. Maybe the OP is interested in knowing more about that idea?
$endgroup$
– Bram28
Jan 2 '18 at 17:21
1
$begingroup$
Bram28, Henning, this is indeed what I'm looking for; preferably in a publication of some sort; a discussion of how discarding the parallel postulate enriched geometry--and eventually had real-world application vis a vis Minkowski space-time--alongside a discussion of AC making similar points.
$endgroup$
– Rich Jensen
Jan 2 '18 at 17:33
1
$begingroup$
The axiom of choice is nothing like the axiom of parallels in Euclidean geometry. The parallels postulate was considered an oddity because of its technical and unnatural formulation. The axiom of choice is natural (one could argue about technical, but it's not as technical as Replacement). If any of the ZFC axioms is the odd one out, it would be Foundation (Regularity). That's also the only axiom without much impact on "normal mathematics". It's an odd technical duck, which is very interesting from a set theoretic point and has opposites in Anti Foundation Axioms. But yeah, it's not as sexy.
$endgroup$
– Asaf Karagila♦
Jan 2 '18 at 19:24
|
show 10 more comments
3
$begingroup$
There is some fuzzy/soft similaity in that both AC and the parallel postulate are (1) claims that intuitively feel like they should be true, but (2) are somewhat more complex state than the simpler "fundamental" axioms the rest of the theory can be reduced to, (3) therefore people have been searching for a way to reduce them to simpler axiomatic truths, but (4) eventually this search terminated in the discovery that they're independent of the recognized simpler axioms.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:02
3
$begingroup$
Note that this narrative doesn't say that we know there cannot be a property that is equivalent to the axiom in question and is so simple to state that it would have been accepted as an axiom without all the attempts to prove it from other axioms. This seems rather unlikely in either of these cases, but we don't really have a good technical concept of what exactly "simple enough" would mean, so we can't even hope to prove that it is so.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:06
1
$begingroup$
The OP can correct me, but I think the OP is indeed looking for some more abstract parallel like Henning is pointing out, to which we can add that just as there are two separate branches for Euclidian and Non-Euclidian geometry, there are also two branches regarding the AC: one where AC is assumed to be true and one where AC is assumed to be false. Maybe the OP is interested in knowing more about that idea?
$endgroup$
– Bram28
Jan 2 '18 at 17:21
1
$begingroup$
Bram28, Henning, this is indeed what I'm looking for; preferably in a publication of some sort; a discussion of how discarding the parallel postulate enriched geometry--and eventually had real-world application vis a vis Minkowski space-time--alongside a discussion of AC making similar points.
$endgroup$
– Rich Jensen
Jan 2 '18 at 17:33
1
$begingroup$
The axiom of choice is nothing like the axiom of parallels in Euclidean geometry. The parallels postulate was considered an oddity because of its technical and unnatural formulation. The axiom of choice is natural (one could argue about technical, but it's not as technical as Replacement). If any of the ZFC axioms is the odd one out, it would be Foundation (Regularity). That's also the only axiom without much impact on "normal mathematics". It's an odd technical duck, which is very interesting from a set theoretic point and has opposites in Anti Foundation Axioms. But yeah, it's not as sexy.
$endgroup$
– Asaf Karagila♦
Jan 2 '18 at 19:24
3
3
$begingroup$
There is some fuzzy/soft similaity in that both AC and the parallel postulate are (1) claims that intuitively feel like they should be true, but (2) are somewhat more complex state than the simpler "fundamental" axioms the rest of the theory can be reduced to, (3) therefore people have been searching for a way to reduce them to simpler axiomatic truths, but (4) eventually this search terminated in the discovery that they're independent of the recognized simpler axioms.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:02
$begingroup$
There is some fuzzy/soft similaity in that both AC and the parallel postulate are (1) claims that intuitively feel like they should be true, but (2) are somewhat more complex state than the simpler "fundamental" axioms the rest of the theory can be reduced to, (3) therefore people have been searching for a way to reduce them to simpler axiomatic truths, but (4) eventually this search terminated in the discovery that they're independent of the recognized simpler axioms.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:02
3
3
$begingroup$
Note that this narrative doesn't say that we know there cannot be a property that is equivalent to the axiom in question and is so simple to state that it would have been accepted as an axiom without all the attempts to prove it from other axioms. This seems rather unlikely in either of these cases, but we don't really have a good technical concept of what exactly "simple enough" would mean, so we can't even hope to prove that it is so.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:06
$begingroup$
Note that this narrative doesn't say that we know there cannot be a property that is equivalent to the axiom in question and is so simple to state that it would have been accepted as an axiom without all the attempts to prove it from other axioms. This seems rather unlikely in either of these cases, but we don't really have a good technical concept of what exactly "simple enough" would mean, so we can't even hope to prove that it is so.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:06
1
1
$begingroup$
The OP can correct me, but I think the OP is indeed looking for some more abstract parallel like Henning is pointing out, to which we can add that just as there are two separate branches for Euclidian and Non-Euclidian geometry, there are also two branches regarding the AC: one where AC is assumed to be true and one where AC is assumed to be false. Maybe the OP is interested in knowing more about that idea?
$endgroup$
– Bram28
Jan 2 '18 at 17:21
$begingroup$
The OP can correct me, but I think the OP is indeed looking for some more abstract parallel like Henning is pointing out, to which we can add that just as there are two separate branches for Euclidian and Non-Euclidian geometry, there are also two branches regarding the AC: one where AC is assumed to be true and one where AC is assumed to be false. Maybe the OP is interested in knowing more about that idea?
$endgroup$
– Bram28
Jan 2 '18 at 17:21
1
1
$begingroup$
Bram28, Henning, this is indeed what I'm looking for; preferably in a publication of some sort; a discussion of how discarding the parallel postulate enriched geometry--and eventually had real-world application vis a vis Minkowski space-time--alongside a discussion of AC making similar points.
$endgroup$
– Rich Jensen
Jan 2 '18 at 17:33
$begingroup$
Bram28, Henning, this is indeed what I'm looking for; preferably in a publication of some sort; a discussion of how discarding the parallel postulate enriched geometry--and eventually had real-world application vis a vis Minkowski space-time--alongside a discussion of AC making similar points.
$endgroup$
– Rich Jensen
Jan 2 '18 at 17:33
1
1
$begingroup$
The axiom of choice is nothing like the axiom of parallels in Euclidean geometry. The parallels postulate was considered an oddity because of its technical and unnatural formulation. The axiom of choice is natural (one could argue about technical, but it's not as technical as Replacement). If any of the ZFC axioms is the odd one out, it would be Foundation (Regularity). That's also the only axiom without much impact on "normal mathematics". It's an odd technical duck, which is very interesting from a set theoretic point and has opposites in Anti Foundation Axioms. But yeah, it's not as sexy.
$endgroup$
– Asaf Karagila♦
Jan 2 '18 at 19:24
$begingroup$
The axiom of choice is nothing like the axiom of parallels in Euclidean geometry. The parallels postulate was considered an oddity because of its technical and unnatural formulation. The axiom of choice is natural (one could argue about technical, but it's not as technical as Replacement). If any of the ZFC axioms is the odd one out, it would be Foundation (Regularity). That's also the only axiom without much impact on "normal mathematics". It's an odd technical duck, which is very interesting from a set theoretic point and has opposites in Anti Foundation Axioms. But yeah, it's not as sexy.
$endgroup$
– Asaf Karagila♦
Jan 2 '18 at 19:24
|
show 10 more comments
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3
$begingroup$
There is some fuzzy/soft similaity in that both AC and the parallel postulate are (1) claims that intuitively feel like they should be true, but (2) are somewhat more complex state than the simpler "fundamental" axioms the rest of the theory can be reduced to, (3) therefore people have been searching for a way to reduce them to simpler axiomatic truths, but (4) eventually this search terminated in the discovery that they're independent of the recognized simpler axioms.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:02
3
$begingroup$
Note that this narrative doesn't say that we know there cannot be a property that is equivalent to the axiom in question and is so simple to state that it would have been accepted as an axiom without all the attempts to prove it from other axioms. This seems rather unlikely in either of these cases, but we don't really have a good technical concept of what exactly "simple enough" would mean, so we can't even hope to prove that it is so.
$endgroup$
– Henning Makholm
Jan 2 '18 at 17:06
1
$begingroup$
The OP can correct me, but I think the OP is indeed looking for some more abstract parallel like Henning is pointing out, to which we can add that just as there are two separate branches for Euclidian and Non-Euclidian geometry, there are also two branches regarding the AC: one where AC is assumed to be true and one where AC is assumed to be false. Maybe the OP is interested in knowing more about that idea?
$endgroup$
– Bram28
Jan 2 '18 at 17:21
1
$begingroup$
Bram28, Henning, this is indeed what I'm looking for; preferably in a publication of some sort; a discussion of how discarding the parallel postulate enriched geometry--and eventually had real-world application vis a vis Minkowski space-time--alongside a discussion of AC making similar points.
$endgroup$
– Rich Jensen
Jan 2 '18 at 17:33
1
$begingroup$
The axiom of choice is nothing like the axiom of parallels in Euclidean geometry. The parallels postulate was considered an oddity because of its technical and unnatural formulation. The axiom of choice is natural (one could argue about technical, but it's not as technical as Replacement). If any of the ZFC axioms is the odd one out, it would be Foundation (Regularity). That's also the only axiom without much impact on "normal mathematics". It's an odd technical duck, which is very interesting from a set theoretic point and has opposites in Anti Foundation Axioms. But yeah, it's not as sexy.
$endgroup$
– Asaf Karagila♦
Jan 2 '18 at 19:24