Did S. Lang prove Kuratowski–Zorn lemma without Axiom of choice or Well-ordering theorem?How do we know we need the axiom of choice for some theorem?Proving that Zorn's Lemma implies the axiom of choiceBernstein sets, Well-Ordering theorem vs Axiom of ChoiceExistence of infinite set and axiom schema of replacement imply axiom of infinityProving the Axiom of Choice for countable setsUsing an induction argument to show that $forall ninBbbN$, $(n,n+1)capBbbNneqemptyset$Axiom of Choice implies Well-Ordering Principle
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Did S. Lang prove Kuratowski–Zorn lemma without Axiom of choice or Well-ordering theorem?
How do we know we need the axiom of choice for some theorem?Proving that Zorn's Lemma implies the axiom of choiceBernstein sets, Well-Ordering theorem vs Axiom of ChoiceExistence of infinite set and axiom schema of replacement imply axiom of infinityProving the Axiom of Choice for countable setsUsing an induction argument to show that $forall ninBbbN$, $(n,n+1)capBbbNneqemptyset$Axiom of Choice implies Well-Ordering Principle
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In S. Lang' Algebra, Appendix 2, the author proved Zorn's Lemma. After a carefully reading of the proof, I failed to see either Axiom of choice or Well-ordering theorem were assumed in his proof. So did he use any equivalent forms of Zorn's Lemma to prove it at all? If not, can Zorn's Lemma be proven (as Lang did) using other axioms of set theory?
proof-verification elementary-set-theory
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ZurielZuriel
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In S. Lang' Algebra, Appendix 2, the author proved Zorn's Lemma. After a carefully reading of the proof, I failed to see either Axiom of choice or Well-ordering theorem were assumed in his proof. So did he use any equivalent forms of Zorn's Lemma to prove it at all? If not, can Zorn's Lemma be proven (as Lang did) using other axioms of set theory?
proof-verification elementary-set-theory
asked 8 hours ago
ZurielZuriel
2,02612 silver badges28 bronze badges
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In S. Lang' Algebra, Appendix 2, the author proved Zorn's Lemma. After a carefully reading of the proof, I failed to see either Axiom of choice or Well-ordering theorem were assumed in his proof. So did he use any equivalent forms of Zorn's Lemma to prove it at all? If not, can Zorn's Lemma be proven (as Lang did) using other axioms of set theory?
proof-verification elementary-set-theory
asked 8 hours ago
ZurielZuriel
2,02612 silver badges28 bronze badges
$endgroup$
In S. Lang' Algebra, Appendix 2, the author proved Zorn's Lemma. After a carefully reading of the proof, I failed to see either Axiom of choice or Well-ordering theorem were assumed in his proof. So did he use any equivalent forms of Zorn's Lemma to prove it at all? If not, can Zorn's Lemma be proven (as Lang did) using other axioms of set theory?
proof-verification elementary-set-theory
proof-verification elementary-set-theory
asked 8 hours ago
ZurielZuriel
2,02612 silver badges28 bronze badges
asked 8 hours ago
ZurielZuriel
2,02612 silver badges28 bronze badges
asked 8 hours ago
ZurielZuriel
2,02612 silver badges28 bronze badges
asked 8 hours ago
ZurielZuriel
2,02612 silver badges28 bronze badges
asked 8 hours ago
ZurielZuriel
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Zorn's Lemma is equivalent to the Axiom of Choice and to the Well-Ordering Theorem. You cannot prove one of these statements without assuming another (Lang alludes to this on page 881). At a brief glance, it seems that Lang uses the Axiom of Choice on page 884, in the proof of Corollary 2.4. There, he constructs a function $f : A to A$ by choosing, for each $x in A$, an element $y_x in A$ such that $y_x > x$. This is precisely an application of the axiom of choice! To be more explicit, he is making use of the existence of a choice function $A to coprod_x in A y in A : y > x$.
answered 8 hours ago
diracdeltafunkdiracdeltafunk
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Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
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– Zuriel
1 hour ago
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Zorn's Lemma is equivalent to the Axiom of Choice and to the Well-Ordering Theorem. You cannot prove one of these statements without assuming another (Lang alludes to this on page 881). At a brief glance, it seems that Lang uses the Axiom of Choice on page 884, in the proof of Corollary 2.4. There, he constructs a function $f : A to A$ by choosing, for each $x in A$, an element $y_x in A$ such that $y_x > x$. This is precisely an application of the axiom of choice! To be more explicit, he is making use of the existence of a choice function $A to coprod_x in A y in A : y > x$.
answered 8 hours ago
diracdeltafunkdiracdeltafunk
8544 silver badges14 bronze badges
$endgroup$
$begingroup$
Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
$endgroup$
– Zuriel
1 hour ago
add a comment
|
$begingroup$
Zorn's Lemma is equivalent to the Axiom of Choice and to the Well-Ordering Theorem. You cannot prove one of these statements without assuming another (Lang alludes to this on page 881). At a brief glance, it seems that Lang uses the Axiom of Choice on page 884, in the proof of Corollary 2.4. There, he constructs a function $f : A to A$ by choosing, for each $x in A$, an element $y_x in A$ such that $y_x > x$. This is precisely an application of the axiom of choice! To be more explicit, he is making use of the existence of a choice function $A to coprod_x in A y in A : y > x$.
answered 8 hours ago
diracdeltafunkdiracdeltafunk
8544 silver badges14 bronze badges
$endgroup$
$begingroup$
Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
$endgroup$
– Zuriel
1 hour ago
add a comment
|
$begingroup$
Zorn's Lemma is equivalent to the Axiom of Choice and to the Well-Ordering Theorem. You cannot prove one of these statements without assuming another (Lang alludes to this on page 881). At a brief glance, it seems that Lang uses the Axiom of Choice on page 884, in the proof of Corollary 2.4. There, he constructs a function $f : A to A$ by choosing, for each $x in A$, an element $y_x in A$ such that $y_x > x$. This is precisely an application of the axiom of choice! To be more explicit, he is making use of the existence of a choice function $A to coprod_x in A y in A : y > x$.
answered 8 hours ago
diracdeltafunkdiracdeltafunk
8544 silver badges14 bronze badges
$endgroup$
Zorn's Lemma is equivalent to the Axiom of Choice and to the Well-Ordering Theorem. You cannot prove one of these statements without assuming another (Lang alludes to this on page 881). At a brief glance, it seems that Lang uses the Axiom of Choice on page 884, in the proof of Corollary 2.4. There, he constructs a function $f : A to A$ by choosing, for each $x in A$, an element $y_x in A$ such that $y_x > x$. This is precisely an application of the axiom of choice! To be more explicit, he is making use of the existence of a choice function $A to coprod_x in A y in A : y > x$.
answered 8 hours ago
diracdeltafunkdiracdeltafunk
8544 silver badges14 bronze badges
answered 8 hours ago
diracdeltafunkdiracdeltafunk
8544 silver badges14 bronze badges
answered 8 hours ago
diracdeltafunkdiracdeltafunk
8544 silver badges14 bronze badges
answered 8 hours ago
diracdeltafunkdiracdeltafunk
8544 silver badges14 bronze badges
8544 silver badges14 bronze badges
$begingroup$
Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
$endgroup$
– Zuriel
1 hour ago
add a comment
|
$begingroup$
Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
$endgroup$
– Zuriel
1 hour ago
$begingroup$
Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
$endgroup$
– Zuriel
1 hour ago
$begingroup$
Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
$endgroup$
– Zuriel
1 hour ago
add a comment
|
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