Mdthbs

Can an NPC use the teleport spell to affect an object they can see with the scry spell?

Found a minor bug, affecting 1% of users. What should QA do?

Find all matrices satisfy

What's the correct way to determine turn order in this situation?

Driving test in New Zealand?

Using 4K Skyrim Textures when running 1920 x 1080 display resolution?

Why is the time of useful consciousness only seconds at high altitudes?

Airport Security - advanced check, 4th amendment breach

Was the whistleblower in the Ukraine scandal legally required to make his report?

Was "чёрствый" ever a synonym for fresh in Russian?

Manager told a colleague of mine I was getting fired soon

Tikz background color of node multilayer

How dangerous are my worn rims?

Non-electric Laser

Did the Humans find out about Gaius Baltar's role in the sabotage of the fleet?

Is right click on tables bad UX

How to prove (A v B), (A → C), (B → D) therefore (C v D)

Is "weekend warrior" derogatory?

What are some ways to season that don't rely on garlic and onions?

Does the US Armed Forces refuse to recruit anyone with an IQ less than 83?

How is the speed of nucleons in the nucleus measured?

As an interviewer, how to conduct interviews with candidates you already know will be rejected?

Was Smaug sealed inside the Lonely Mountain?

Did S. Lang prove Kuratowski–Zorn lemma without Axiom of choice or Well-ordering theorem?

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 $forall ninBbb{N}$, $(n,n+1)capBbb{N}neqemptyset$Axiom of Choice implies Well-Ordering Principle

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{
margin-bottom:0;
}
.everyonelovesstackoverflow{position:absolute;height:1px;width:1px;opacity:0;top:0;left:0;pointer-events:none;}

2

$begingroup$

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

share|cite|improve this question

asked 9 hours ago

ZurielZuriel

2,0261212 silver badges2828 bronze badges

$endgroup$

add a comment
 | 

2

$begingroup$

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

share|cite|improve this question

asked 9 hours ago

ZurielZuriel

2,0261212 silver badges2828 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

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

share|cite|improve this question

asked 9 hours ago

ZurielZuriel

2,0261212 silver badges2828 bronze badges

$endgroup$

share|cite|improve this question

asked 9 hours ago

ZurielZuriel

2,0261212 silver badges2828 bronze badges

share|cite|improve this question

asked 9 hours ago

ZurielZuriel

2,0261212 silver badges2828 bronze badges

asked 9 hours ago

ZurielZuriel

2,0261212 silver badges2828 bronze badges

asked 9 hours ago

ZurielZuriel

2,0261212 silver badges2828 bronze badges

1 Answer
1

active

oldest

votes

9

$begingroup$

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 $A to coprod_{x in A} {y in A : y > x}$.

share|cite|improve this answer

answered 8 hours ago

diracdeltafunkdiracdeltafunk

86444 silver badges1414 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
  $endgroup$
  – Zuriel
  2 hours ago
  

  
  
  
  

add a comment
 | 

Your Answer

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/4.0/"u003ecc by-sa 4.0 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
});


}
});

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3372390%2fdid-s-lang-prove-kuratowski-zorn-lemma-without-axiom-of-choice-or-well-ordering%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

9

$begingroup$

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 $A to coprod_{x in A} {y in A : y > x}$.

share|cite|improve this answer

answered 8 hours ago

diracdeltafunkdiracdeltafunk

86444 silver badges1414 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
  $endgroup$
  – Zuriel
  2 hours ago
  

  
  
  
  

add a comment
 | 

9

$begingroup$

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 $A to coprod_{x in A} {y in A : y > x}$.

share|cite|improve this answer

answered 8 hours ago

diracdeltafunkdiracdeltafunk

86444 silver badges1414 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you! I wish Lang had mentioned the usage of Axiom of Choice more explicitly.
  $endgroup$
  – Zuriel
  2 hours 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 $A to coprod_{x in A} {y in A : y > x}$.

share|cite|improve this answer

answered 8 hours ago

diracdeltafunkdiracdeltafunk

86444 silver badges1414 bronze badges

$endgroup$

share|cite|improve this answer

answered 8 hours ago

diracdeltafunkdiracdeltafunk

86444 silver badges1414 bronze badges

answered 8 hours ago

diracdeltafunkdiracdeltafunk

86444 silver badges1414 bronze badges

answered 8 hours ago

diracdeltafunkdiracdeltafunk

86444 silver badges1414 bronze badges