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?Why is Zorn's Lemma called a lemma?Can a sequence whose final term is an axiom, be considered a formal proof?Axiom Systems and Formal SystemsWhen the mathematical community consider the inclusion of a new axiom?.A Concept Which Has Been 'Specialized' In the Course of HistoryWhy 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?Why is Zorn's Lemma called a lemma?Can a sequence whose final term is an axiom, be considered a formal proof?Axiom Systems and Formal SystemsWhen the mathematical community consider the inclusion of a new axiom?.A Concept Which Has Been 'Specialized' In the Course of HistoryWhy 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 have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic math-history axioms
asked 2 hours ago
Eyal RothEyal Roth
1293
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
|
show 3 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 have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic math-history axioms
asked 2 hours ago
Eyal RothEyal Roth
1293
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
There are theorems that we take to be axioms for the sake of convenience. For example, in ZF(C), the empty set axiom follows from the axiom of infinity (which, in particular, implies that there exists a set $x$) and the axiom schema of separation (which implies that $ y in x mid y ne y $ is a set), and yet we still (usually) state the empty set axiom as a separate axiom.
$endgroup$
– Clive Newstead
2 hours ago
$begingroup$
For a while, Cantor thought he had proved what we now call the Continuum Hypothesis.
$endgroup$
– Gerry Myerson
2 hours ago
1
$begingroup$
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♦
2 hours ago
$begingroup$
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
2 hours ago
1
$begingroup$
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♦
1 hour ago
|
show 3 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 have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic math-history axioms
asked 2 hours ago
Eyal RothEyal Roth
1293
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
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 have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?
logic math-history axioms
logic math-history axioms
asked 2 hours ago
Eyal RothEyal Roth
1293
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Eyal RothEyal Roth
1293
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 1 hour ago
Eyal Roth
asked 2 hours ago
Eyal RothEyal Roth
1293
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Eyal RothEyal Roth
1293
asked 2 hours ago
Eyal RothEyal Roth
1293
1293
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
There are theorems that we take to be axioms for the sake of convenience. For example, in ZF(C), the empty set axiom follows from the axiom of infinity (which, in particular, implies that there exists a set $x$) and the axiom schema of separation (which implies that $ y in x mid y ne y $ is a set), and yet we still (usually) state the empty set axiom as a separate axiom.
$endgroup$
– Clive Newstead
2 hours ago
$begingroup$
For a while, Cantor thought he had proved what we now call the Continuum Hypothesis.
$endgroup$
– Gerry Myerson
2 hours ago
1
$begingroup$
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♦
2 hours ago
$begingroup$
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
2 hours ago
1
$begingroup$
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♦
1 hour ago
|
show 3 more comments
$begingroup$
There are theorems that we take to be axioms for the sake of convenience. For example, in ZF(C), the empty set axiom follows from the axiom of infinity (which, in particular, implies that there exists a set $x$) and the axiom schema of separation (which implies that $ y in x mid y ne y $ is a set), and yet we still (usually) state the empty set axiom as a separate axiom.
$endgroup$
– Clive Newstead
2 hours ago
$begingroup$
For a while, Cantor thought he had proved what we now call the Continuum Hypothesis.
$endgroup$
– Gerry Myerson
2 hours ago
1
$begingroup$
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♦
2 hours ago
$begingroup$
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
2 hours ago
1
$begingroup$
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♦
1 hour ago
$begingroup$
There are theorems that we take to be axioms for the sake of convenience. For example, in ZF(C), the empty set axiom follows from the axiom of infinity (which, in particular, implies that there exists a set $x$) and the axiom schema of separation (which implies that $ y in x mid y ne y $ is a set), and yet we still (usually) state the empty set axiom as a separate axiom.
$endgroup$
– Clive Newstead
2 hours ago
$begingroup$
There are theorems that we take to be axioms for the sake of convenience. For example, in ZF(C), the empty set axiom follows from the axiom of infinity (which, in particular, implies that there exists a set $x$) and the axiom schema of separation (which implies that $ y in x mid y ne y $ is a set), and yet we still (usually) state the empty set axiom as a separate axiom.
$endgroup$
– Clive Newstead
2 hours ago
$begingroup$
For a while, Cantor thought he had proved what we now call the Continuum Hypothesis.
$endgroup$
– Gerry Myerson
2 hours ago
$begingroup$
For a while, Cantor thought he had proved what we now call the Continuum Hypothesis.
$endgroup$
– Gerry Myerson
2 hours ago
1
1
$begingroup$
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♦
2 hours ago
$begingroup$
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♦
2 hours ago
$begingroup$
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
2 hours ago
$begingroup$
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
2 hours ago
1
1
$begingroup$
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♦
1 hour ago
$begingroup$
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♦
1 hour ago
|
show 3 more comments
2 Answers
2
active
oldest
votes
$begingroup$
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.
answered 1 hour ago
J.G.J.G.
33k23251
$endgroup$
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.
answered 2 hours ago
ShaunShaun
10.4k113686
$endgroup$
$begingroup$
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
1 hour ago
add a comment |
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2 Answers
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2 Answers
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$begingroup$
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.
answered 1 hour ago
J.G.J.G.
33k23251
$endgroup$
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.
answered 1 hour ago
J.G.J.G.
33k23251
$endgroup$
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.
answered 1 hour ago
J.G.J.G.
33k23251
$endgroup$
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 1 hour ago
J.G.J.G.
33k23251
answered 1 hour ago
J.G.J.G.
33k23251
answered 1 hour ago
J.G.J.G.
33k23251
answered 1 hour ago
J.G.J.G.
33k23251
33k23251
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.
answered 2 hours ago
ShaunShaun
10.4k113686
$endgroup$
$begingroup$
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
1 hour ago
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.
answered 2 hours ago
ShaunShaun
10.4k113686
$endgroup$
$begingroup$
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
1 hour ago
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.
answered 2 hours ago
ShaunShaun
10.4k113686
$endgroup$
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.
answered 2 hours ago
ShaunShaun
10.4k113686
edited 2 hours ago
answered 2 hours ago
ShaunShaun
10.4k113686
answered 2 hours ago
ShaunShaun
10.4k113686
answered 2 hours ago
ShaunShaun
10.4k113686
10.4k113686
$begingroup$
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
1 hour ago
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
1 hour ago
$begingroup$
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
1 hour ago
$begingroup$
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
1 hour ago
add a comment |
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
There are theorems that we take to be axioms for the sake of convenience. For example, in ZF(C), the empty set axiom follows from the axiom of infinity (which, in particular, implies that there exists a set $x$) and the axiom schema of separation (which implies that $ y in x mid y ne y $ is a set), and yet we still (usually) state the empty set axiom as a separate axiom.
$endgroup$
– Clive Newstead
2 hours ago
$begingroup$
For a while, Cantor thought he had proved what we now call the Continuum Hypothesis.
$endgroup$
– Gerry Myerson
2 hours ago
1
$begingroup$
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♦
2 hours ago
$begingroup$
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
2 hours ago
1
$begingroup$
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.
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– Asaf Karagila♦
1 hour ago