Equi-consistency of inaccessible cardinals [duplicate]

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  The reason why the existence of inaccessible cardinals cannot be proven in ZFC
  
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  Proving that the existence of strongly inaccessible cardinals is independent from ZFC?
  
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I have trouble understanding this wiki page, concerning inaccessible cardinals.

First "These statements are strong enough to imply the consistency of ZFC". I guess what they mean by that is if $kappa$ is a strongly inaccessible cardinal, then $V_kappa$ is a model of ZFC, where $V$ is the Von Neumann hierarchy.

Then they argue that a proof in ZFC, assuming ZFC is consistent, of the consistency of ZFC+$kappa$ is impossible. Because it would prove the consistency of ZFC, violating Gödel's incompleteness theorem.

I don't understand this, I imagine a relative consistency proof would rather be a model transformation : given a model of ZFC as input, produce a model of ZFC+$kappa$ as output. Then we could continue producing $V_kappa$ to get another model of ZFC. Where is the contradiction in that ? To me it seems we first assumed a model of ZFC, then produced another one. ZFC didn't produce a model of itself from scratch.

set-theory large-cardinals

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asked 2 days ago

V. Semeria

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marked as duplicate by Asaf Karagila set-theory
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This question already has an answer here:

• 
  The reason why the existence of inaccessible cardinals cannot be proven in ZFC
  
  3 answers
  
  


• 
  Proving that the existence of strongly inaccessible cardinals is independent from ZFC?
  
  1 answer
  
  

I have trouble understanding this wiki page, concerning inaccessible cardinals.

First "These statements are strong enough to imply the consistency of ZFC". I guess what they mean by that is if $kappa$ is a strongly inaccessible cardinal, then $V_kappa$ is a model of ZFC, where $V$ is the Von Neumann hierarchy.

Then they argue that a proof in ZFC, assuming ZFC is consistent, of the consistency of ZFC+$kappa$ is impossible. Because it would prove the consistency of ZFC, violating Gödel's incompleteness theorem.

I don't understand this, I imagine a relative consistency proof would rather be a model transformation : given a model of ZFC as input, produce a model of ZFC+$kappa$ as output. Then we could continue producing $V_kappa$ to get another model of ZFC. Where is the contradiction in that ? To me it seems we first assumed a model of ZFC, then produced another one. ZFC didn't produce a model of itself from scratch.

set-theory large-cardinals

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asked 2 days ago

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marked as duplicate by Asaf Karagila set-theory
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up vote
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down vote

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This question already has an answer here:

• 
  The reason why the existence of inaccessible cardinals cannot be proven in ZFC
  
  3 answers
  
  


• 
  Proving that the existence of strongly inaccessible cardinals is independent from ZFC?
  
  1 answer
  
  

I have trouble understanding this wiki page, concerning inaccessible cardinals.

First "These statements are strong enough to imply the consistency of ZFC". I guess what they mean by that is if $kappa$ is a strongly inaccessible cardinal, then $V_kappa$ is a model of ZFC, where $V$ is the Von Neumann hierarchy.

Then they argue that a proof in ZFC, assuming ZFC is consistent, of the consistency of ZFC+$kappa$ is impossible. Because it would prove the consistency of ZFC, violating Gödel's incompleteness theorem.

I don't understand this, I imagine a relative consistency proof would rather be a model transformation : given a model of ZFC as input, produce a model of ZFC+$kappa$ as output. Then we could continue producing $V_kappa$ to get another model of ZFC. Where is the contradiction in that ? To me it seems we first assumed a model of ZFC, then produced another one. ZFC didn't produce a model of itself from scratch.

set-theory large-cardinals

share|cite|improve this question

asked 2 days ago

V. Semeria

1586

This question already has an answer here:

• 
  The reason why the existence of inaccessible cardinals cannot be proven in ZFC
  
  3 answers
  
  


• 
  Proving that the existence of strongly inaccessible cardinals is independent from ZFC?
  
  1 answer
  
  

I have trouble understanding this wiki page, concerning inaccessible cardinals.

First "These statements are strong enough to imply the consistency of ZFC". I guess what they mean by that is if $kappa$ is a strongly inaccessible cardinal, then $V_kappa$ is a model of ZFC, where $V$ is the Von Neumann hierarchy.

Then they argue that a proof in ZFC, assuming ZFC is consistent, of the consistency of ZFC+$kappa$ is impossible. Because it would prove the consistency of ZFC, violating Gödel's incompleteness theorem.

I don't understand this, I imagine a relative consistency proof would rather be a model transformation : given a model of ZFC as input, produce a model of ZFC+$kappa$ as output. Then we could continue producing $V_kappa$ to get another model of ZFC. Where is the contradiction in that ? To me it seems we first assumed a model of ZFC, then produced another one. ZFC didn't produce a model of itself from scratch.

This question already has an answer here:

• 
  The reason why the existence of inaccessible cardinals cannot be proven in ZFC
  
  3 answers
  
  


• 
  Proving that the existence of strongly inaccessible cardinals is independent from ZFC?
  
  1 answer
  
  

set-theory large-cardinals

share|cite|improve this question

asked 2 days ago

V. Semeria

1586

share|cite|improve this question

share|cite|improve this question

asked 2 days ago

V. Semeria

1586

asked 2 days ago

V. Semeria

1586

asked 2 days ago

V. Semeria

1586

1586

marked as duplicate by Asaf Karagila set-theory
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This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

marked as duplicate by Asaf Karagila set-theory
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This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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1 Answer
1

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oldest

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up vote
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down vote



accepted

Apply the incompleteness theorem to $ZFC+exists textan inaccessible $

share|cite|improve this answer

answered 2 days ago

Rene Schipperus

31.9k11957

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So simple... sorry I didn't see that myself
  – V. Semeria
  2 days ago
  

  
  
  
  

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote



accepted

Apply the incompleteness theorem to $ZFC+exists textan inaccessible $

share|cite|improve this answer

answered 2 days ago

Rene Schipperus

31.9k11957

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So simple... sorry I didn't see that myself
  – V. Semeria
  2 days ago
  

  
  
  
  

add a comment | 

up vote
2
down vote



accepted

Apply the incompleteness theorem to $ZFC+exists textan inaccessible $

share|cite|improve this answer

answered 2 days ago

Rene Schipperus

31.9k11957

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So simple... sorry I didn't see that myself
  – V. Semeria
  2 days ago
  

  
  
  
  

add a comment | 

up vote
2
down vote



accepted

up vote
2
down vote



accepted

Apply the incompleteness theorem to $ZFC+exists textan inaccessible $

share|cite|improve this answer

answered 2 days ago

Rene Schipperus

31.9k11957

Apply the incompleteness theorem to $ZFC+exists textan inaccessible $

share|cite|improve this answer

answered 2 days ago

Rene Schipperus

31.9k11957

share|cite|improve this answer

share|cite|improve this answer

answered 2 days ago

Rene Schipperus

31.9k11957

answered 2 days ago

Rene Schipperus

31.9k11957

answered 2 days ago

Rene Schipperus

31.9k11957

31.9k11957

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So simple... sorry I didn't see that myself
  – V. Semeria
  2 days ago
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So simple... sorry I didn't see that myself
  – V. Semeria
  2 days ago
  

  
  
  
  

So simple... sorry I didn't see that myself
– V. Semeria
2 days ago

So simple... sorry I didn't see that myself
– V. Semeria
2 days ago

add a comment |