Was there ever an axiom rendered a theorem?How can the axiom of choice be called “axiom” if it is false in Cohen's model?What is the difference between an axiom and a postulate?Sufficient Conditions for proving $V$ is a vector spaceCan a sequence whose final term is an axiom, be considered a formal proof?Why is Zorn's Lemma called a lemma?Axiom Systems and Formal SystemsWhen the mathematical community consider the inclusion of a new axiom?.Why is the axiom of choice not taught from the start to mathematics undergraduates?Why is the Generalization Axiom considered a Pure Axiom?Euclid's Elements missing axiom of M. Pasch examplesZermelo-Fraenkel set theory and Hilbert's axioms for geometryWhich is the first theorem in Euclid's Elements which uses Pasch's Axiom?Axiom of Choice — Why is it an axiom and not a theorem?Is consistency an axiom of mathematics?Redunduncy of Pasch's Axiom of Hilbert's Foundations of Geometry
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Was there ever an axiom rendered a theorem?
How can the axiom of choice be called “axiom” if it is false in Cohen's model?What is the difference between an axiom and a postulate?Sufficient Conditions for proving $V$ is a vector spaceCan a sequence whose final term is an axiom, be considered a formal proof?Why is Zorn's Lemma called a lemma?Axiom Systems and Formal SystemsWhen the mathematical community consider the inclusion of a new axiom?.Why is the axiom of choice not taught from the start to mathematics undergraduates?Why is the Generalization Axiom considered a Pure Axiom?Euclid's Elements missing axiom of M. Pasch examplesZermelo-Fraenkel set theory and Hilbert's axioms for geometryWhich is the first theorem in Euclid's Elements which uses Pasch's Axiom?Axiom of Choice — Why is it an axiom and not a theorem?Is consistency an axiom of mathematics?Redunduncy of Pasch's Axiom of Hilbert's Foundations of Geometry
$begingroup$
In the history of mathematics, are there notable examples of theorems which have been first considered axioms?
Alternatively, was there any statement first considered an axiom that later has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic soft-question math-history axioms
$endgroup$
|
show 8 more comments
$begingroup$
In the history of mathematics, are there notable examples of theorems which have been first considered axioms?
Alternatively, was there any statement first considered an axiom that later has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic soft-question math-history axioms
$endgroup$
2
$begingroup$
All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
$endgroup$
– Asaf Karagila♦
Apr 8 at 13:16
2
$begingroup$
I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
$endgroup$
– quarague
Apr 8 at 13:26
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:38
4
$begingroup$
And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:41
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
Apr 8 at 16:05
|
show 8 more comments
$begingroup$
In the history of mathematics, are there notable examples of theorems which have been first considered axioms?
Alternatively, was there any statement first considered an axiom that later has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic soft-question math-history axioms
$endgroup$
In the history of mathematics, are there notable examples of theorems which have been first considered axioms?
Alternatively, was there any statement first considered an axiom that later has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic soft-question math-history axioms
logic soft-question math-history axioms
edited Apr 13 at 19:09
community wiki
4 revs, 3 users 57%
Eyal Roth
2
$begingroup$
All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
$endgroup$
– Asaf Karagila♦
Apr 8 at 13:16
2
$begingroup$
I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
$endgroup$
– quarague
Apr 8 at 13:26
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:38
4
$begingroup$
And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:41
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
Apr 8 at 16:05
|
show 8 more comments
2
$begingroup$
All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
$endgroup$
– Asaf Karagila♦
Apr 8 at 13:16
2
$begingroup$
I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
$endgroup$
– quarague
Apr 8 at 13:26
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:38
4
$begingroup$
And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:41
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
Apr 8 at 16:05
2
2
$begingroup$
All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
$endgroup$
– Asaf Karagila♦
Apr 8 at 13:16
$begingroup$
All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
$endgroup$
– Asaf Karagila♦
Apr 8 at 13:16
2
2
$begingroup$
I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
$endgroup$
– quarague
Apr 8 at 13:26
$begingroup$
I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
$endgroup$
– quarague
Apr 8 at 13:26
2
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:38
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:38
4
4
$begingroup$
And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:41
$begingroup$
And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
$endgroup$
– Asaf Karagila♦
Apr 8 at 14:41
2
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
Apr 8 at 16:05
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
Apr 8 at 16:05
|
show 8 more comments
3 Answers
3
active
oldest
votes
$begingroup$
The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms
In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.
http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf
$endgroup$
add a comment |
$begingroup$
Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.
$endgroup$
add a comment |
$begingroup$
Yes, everywhere. What is an axiom from one theory can be a theorem in another.
Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.
Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.
Also, watch this Feynman clip.
$endgroup$
$begingroup$
That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
$endgroup$
– Eyal Roth
Apr 8 at 13:57
8
$begingroup$
These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
$endgroup$
– Ross Millikan
Apr 8 at 20:03
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms
In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.
http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf
$endgroup$
add a comment |
$begingroup$
The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms
In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.
http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf
$endgroup$
add a comment |
$begingroup$
The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms
In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.
http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf
$endgroup$
The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms
In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.
http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf
answered Apr 8 at 18:18
community wiki
roy smith
add a comment |
add a comment |
$begingroup$
Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.
$endgroup$
add a comment |
$begingroup$
Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.
$endgroup$
add a comment |
$begingroup$
Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.
$endgroup$
Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.
answered Apr 8 at 14:26
community wiki
J.G.
add a comment |
add a comment |
$begingroup$
Yes, everywhere. What is an axiom from one theory can be a theorem in another.
Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.
Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.
Also, watch this Feynman clip.
$endgroup$
$begingroup$
That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
$endgroup$
– Eyal Roth
Apr 8 at 13:57
8
$begingroup$
These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
$endgroup$
– Ross Millikan
Apr 8 at 20:03
add a comment |
$begingroup$
Yes, everywhere. What is an axiom from one theory can be a theorem in another.
Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.
Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.
Also, watch this Feynman clip.
$endgroup$
$begingroup$
That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
$endgroup$
– Eyal Roth
Apr 8 at 13:57
8
$begingroup$
These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
$endgroup$
– Ross Millikan
Apr 8 at 20:03
add a comment |
$begingroup$
Yes, everywhere. What is an axiom from one theory can be a theorem in another.
Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.
Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.
Also, watch this Feynman clip.
$endgroup$
Yes, everywhere. What is an axiom from one theory can be a theorem in another.
Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.
Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.
Also, watch this Feynman clip.
edited Apr 8 at 13:07
community wiki
2 revs
Shaun
$begingroup$
That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
$endgroup$
– Eyal Roth
Apr 8 at 13:57
8
$begingroup$
These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
$endgroup$
– Ross Millikan
Apr 8 at 20:03
add a comment |
$begingroup$
That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
$endgroup$
– Eyal Roth
Apr 8 at 13:57
8
$begingroup$
These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
$endgroup$
– Ross Millikan
Apr 8 at 20:03
$begingroup$
That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
$endgroup$
– Eyal Roth
Apr 8 at 13:57
$begingroup$
That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
$endgroup$
– Eyal Roth
Apr 8 at 13:57
8
8
$begingroup$
These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
$endgroup$
– Ross Millikan
Apr 8 at 20:03
$begingroup$
These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
$endgroup$
– Ross Millikan
Apr 8 at 20:03
add a comment |
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All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
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– Asaf Karagila♦
Apr 8 at 13:16
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I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
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– quarague
Apr 8 at 13:26
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Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
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– Asaf Karagila♦
Apr 8 at 14:38
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And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
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– Asaf Karagila♦
Apr 8 at 14:41
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I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
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– Bram28
Apr 8 at 16:05