Was there ever an axiom rendered a theorem?
$begingroup$
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 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 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♦
2 days ago
1
$begingroup$
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 days ago
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
2 days ago
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♦
2 days ago
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
2 days ago
|
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 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 math-history axioms
logic math-history axioms
edited 13 hours ago
community wiki
3 revs, 2 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♦
2 days ago
1
$begingroup$
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 days ago
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
2 days ago
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♦
2 days ago
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
2 days ago
|
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♦
2 days ago
1
$begingroup$
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 days ago
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
2 days ago
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♦
2 days ago
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
2 days ago
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♦
2 days 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 days ago
1
1
$begingroup$
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 days 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 days ago
2
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
2 days ago
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
2 days ago
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♦
2 days 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♦
2 days ago
2
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
2 days ago
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
2 days ago
|
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
2 days ago
6
$begingroup$
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
2 days ago
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3179606%2fwas-there-ever-an-axiom-rendered-a-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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 2 days ago
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 2 days ago
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
2 days ago
6
$begingroup$
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
2 days 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.
$endgroup$
$begingroup$
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
2 days ago
6
$begingroup$
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
2 days 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.
$endgroup$
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 2 days ago
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
2 days ago
6
$begingroup$
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
2 days 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
2 days ago
6
$begingroup$
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
2 days 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
2 days 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
2 days ago
6
6
$begingroup$
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
2 days ago
$begingroup$
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
2 days ago
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3179606%2fwas-there-ever-an-axiom-rendered-a-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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♦
2 days ago
1
$begingroup$
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 days ago
2
$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
2 days ago
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♦
2 days ago
2
$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
2 days ago