Why, historically, did Gödel think CH was false?Is Hilbert's second problem about the real numbers or the...
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Why, historically, did Gödel think CH was false?
Is Hilbert's second problem about the real numbers or the natural numbers?Viewing forcing as a result about countable transitive modelsWhy is the Power Set Operation Inherently Vague?Class models of $mathsf{ZFC}$ and consistency resultsIncompleteness theorems in encoding schemes other than Gödel numbering“Representation” of classes by sets in Bernays's set theoryWas Gödel's entire argument actually formalizable when it was written?How did product rule come about historically?Is there actually a universal notion of computability?What is the status of the Axiom of limitation of size? (adrift for almost a century now)
$begingroup$
Gödel was the first to show that ~CH was not provable from ZFC. However, he also thought CH was false in his view of the "Platonic" reality of set theory. It seems this view was also somewhat common among set theorists of a Platonist bent, until Cohen's later forcing result.
Does anyone know what Gödel's reasoning was for CH being false? Did he ever write anything about it, addressing his views on the subject?
soft-question set-theory math-history
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
$endgroup$
add a comment |
$begingroup$
Gödel was the first to show that ~CH was not provable from ZFC. However, he also thought CH was false in his view of the "Platonic" reality of set theory. It seems this view was also somewhat common among set theorists of a Platonist bent, until Cohen's later forcing result.
Does anyone know what Gödel's reasoning was for CH being false? Did he ever write anything about it, addressing his views on the subject?
soft-question set-theory math-history
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
$endgroup$
$begingroup$
Are you asking for a source for the statement that Gödel though CH was false?
$endgroup$
– Lee Mosher
9 hours ago
$begingroup$
One might also look at Godel's collected works volume 2 for history and commentary on the 1947/1964 exposition, and Volume 3 about his unpublished 1970 notes. Also, Kanamori's "Godel and Set theory". There is also discussion of Godel's beliefs on CH in Maddy's "Believing the Axioms I" and Koellner's "On the question of absolute undecidability."
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
I would add that Cohen's result didn't change the fact that set theorists of a Platonist bent tend to regard the CH as false (though it may have convinced a few to not be of a Platonist bent). I don't know much about this, but my understanding is that Godel had some esoteric reasons for believing $mathfrak c =aleph_2,$ whereas the dominant view in the aftermath of Cohen was that it was much larger, perhaps even weakly inaccessible. (Although there have been serious proposals that imply $mathfrak c =aleph_2,$ and even CH, more recently.)
$endgroup$
– spaceisdarkgreen
6 hours ago
add a comment |
$begingroup$
Gödel was the first to show that ~CH was not provable from ZFC. However, he also thought CH was false in his view of the "Platonic" reality of set theory. It seems this view was also somewhat common among set theorists of a Platonist bent, until Cohen's later forcing result.
Does anyone know what Gödel's reasoning was for CH being false? Did he ever write anything about it, addressing his views on the subject?
soft-question set-theory math-history
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
$endgroup$
Gödel was the first to show that ~CH was not provable from ZFC. However, he also thought CH was false in his view of the "Platonic" reality of set theory. It seems this view was also somewhat common among set theorists of a Platonist bent, until Cohen's later forcing result.
Does anyone know what Gödel's reasoning was for CH being false? Did he ever write anything about it, addressing his views on the subject?
soft-question set-theory math-history
soft-question set-theory math-history
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
asked 9 hours ago
Mike BattagliaMike Battaglia
1,5821128
1,5821128
$begingroup$
Are you asking for a source for the statement that Gödel though CH was false?
$endgroup$
– Lee Mosher
9 hours ago
$begingroup$
One might also look at Godel's collected works volume 2 for history and commentary on the 1947/1964 exposition, and Volume 3 about his unpublished 1970 notes. Also, Kanamori's "Godel and Set theory". There is also discussion of Godel's beliefs on CH in Maddy's "Believing the Axioms I" and Koellner's "On the question of absolute undecidability."
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
I would add that Cohen's result didn't change the fact that set theorists of a Platonist bent tend to regard the CH as false (though it may have convinced a few to not be of a Platonist bent). I don't know much about this, but my understanding is that Godel had some esoteric reasons for believing $mathfrak c =aleph_2,$ whereas the dominant view in the aftermath of Cohen was that it was much larger, perhaps even weakly inaccessible. (Although there have been serious proposals that imply $mathfrak c =aleph_2,$ and even CH, more recently.)
$endgroup$
– spaceisdarkgreen
6 hours ago
add a comment |
$begingroup$
Are you asking for a source for the statement that Gödel though CH was false?
$endgroup$
– Lee Mosher
9 hours ago
$begingroup$
One might also look at Godel's collected works volume 2 for history and commentary on the 1947/1964 exposition, and Volume 3 about his unpublished 1970 notes. Also, Kanamori's "Godel and Set theory". There is also discussion of Godel's beliefs on CH in Maddy's "Believing the Axioms I" and Koellner's "On the question of absolute undecidability."
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
I would add that Cohen's result didn't change the fact that set theorists of a Platonist bent tend to regard the CH as false (though it may have convinced a few to not be of a Platonist bent). I don't know much about this, but my understanding is that Godel had some esoteric reasons for believing $mathfrak c =aleph_2,$ whereas the dominant view in the aftermath of Cohen was that it was much larger, perhaps even weakly inaccessible. (Although there have been serious proposals that imply $mathfrak c =aleph_2,$ and even CH, more recently.)
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
Are you asking for a source for the statement that Gödel though CH was false?
$endgroup$
– Lee Mosher
9 hours ago
$begingroup$
Are you asking for a source for the statement that Gödel though CH was false?
$endgroup$
– Lee Mosher
9 hours ago
$begingroup$
One might also look at Godel's collected works volume 2 for history and commentary on the 1947/1964 exposition, and Volume 3 about his unpublished 1970 notes. Also, Kanamori's "Godel and Set theory". There is also discussion of Godel's beliefs on CH in Maddy's "Believing the Axioms I" and Koellner's "On the question of absolute undecidability."
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
One might also look at Godel's collected works volume 2 for history and commentary on the 1947/1964 exposition, and Volume 3 about his unpublished 1970 notes. Also, Kanamori's "Godel and Set theory". There is also discussion of Godel's beliefs on CH in Maddy's "Believing the Axioms I" and Koellner's "On the question of absolute undecidability."
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
I would add that Cohen's result didn't change the fact that set theorists of a Platonist bent tend to regard the CH as false (though it may have convinced a few to not be of a Platonist bent). I don't know much about this, but my understanding is that Godel had some esoteric reasons for believing $mathfrak c =aleph_2,$ whereas the dominant view in the aftermath of Cohen was that it was much larger, perhaps even weakly inaccessible. (Although there have been serious proposals that imply $mathfrak c =aleph_2,$ and even CH, more recently.)
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
I would add that Cohen's result didn't change the fact that set theorists of a Platonist bent tend to regard the CH as false (though it may have convinced a few to not be of a Platonist bent). I don't know much about this, but my understanding is that Godel had some esoteric reasons for believing $mathfrak c =aleph_2,$ whereas the dominant view in the aftermath of Cohen was that it was much larger, perhaps even weakly inaccessible. (Although there have been serious proposals that imply $mathfrak c =aleph_2,$ and even CH, more recently.)
$endgroup$
– spaceisdarkgreen
6 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
There is a classical survey of Gödel about the continuum hypothesis:
"What is Cantor's Continuum Problem", K. Gödel, The American Mathematical Monthly, Vol. 54, No. 9 (Nov., 1947), pp. 515-525
In section 4, he discusses "in what sense and in which direction a solution of the continuum problem may be expected". While this is of course just a survey, it still represents some of Gödel's individual thoughts about the subject at the time.
A barrier free link is right now e.g. this.
edited 2 hours ago
spaceisdarkgreen
33.8k21753
answered 8 hours ago
blubblub
3,169829
$endgroup$
7
$begingroup$
Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful.
$endgroup$
– David Richerby
5 hours ago
$begingroup$
@DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $mathfrak c=aleph_2,$ but that too was considered a failure.
$endgroup$
– spaceisdarkgreen
3 hours ago
$begingroup$
@spaceisdarkgreen Thanks -- I edited the one-sentence summary that it implies results Goedel found implausible into the answer. blub, I hope that's OK; please do edit if not.
$endgroup$
– David Richerby
2 hours ago
$begingroup$
@DavidRicherby I'm not 100% on whether it's appropriate to make this kind of an edit to a non CW post (blub may well disagree with me). So I reverted it for now. I agree with you and the 7 others that a summary of Godel's thoughts would be good to have in the answer. (But even if they opt not to, my comment will be visible.)
$endgroup$
– spaceisdarkgreen
2 hours ago
add a comment |
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1 Answer
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1 Answer
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votes
$begingroup$
There is a classical survey of Gödel about the continuum hypothesis:
"What is Cantor's Continuum Problem", K. Gödel, The American Mathematical Monthly, Vol. 54, No. 9 (Nov., 1947), pp. 515-525
In section 4, he discusses "in what sense and in which direction a solution of the continuum problem may be expected". While this is of course just a survey, it still represents some of Gödel's individual thoughts about the subject at the time.
A barrier free link is right now e.g. this.
edited 2 hours ago
spaceisdarkgreen
33.8k21753
answered 8 hours ago
blubblub
3,169829
$endgroup$
7
$begingroup$
Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful.
$endgroup$
– David Richerby
5 hours ago
$begingroup$
@DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $mathfrak c=aleph_2,$ but that too was considered a failure.
$endgroup$
– spaceisdarkgreen
3 hours ago
$begingroup$
@spaceisdarkgreen Thanks -- I edited the one-sentence summary that it implies results Goedel found implausible into the answer. blub, I hope that's OK; please do edit if not.
$endgroup$
– David Richerby
2 hours ago
$begingroup$
@DavidRicherby I'm not 100% on whether it's appropriate to make this kind of an edit to a non CW post (blub may well disagree with me). So I reverted it for now. I agree with you and the 7 others that a summary of Godel's thoughts would be good to have in the answer. (But even if they opt not to, my comment will be visible.)
$endgroup$
– spaceisdarkgreen
2 hours ago
add a comment |
$begingroup$
There is a classical survey of Gödel about the continuum hypothesis:
"What is Cantor's Continuum Problem", K. Gödel, The American Mathematical Monthly, Vol. 54, No. 9 (Nov., 1947), pp. 515-525
In section 4, he discusses "in what sense and in which direction a solution of the continuum problem may be expected". While this is of course just a survey, it still represents some of Gödel's individual thoughts about the subject at the time.
A barrier free link is right now e.g. this.
edited 2 hours ago
spaceisdarkgreen
33.8k21753
answered 8 hours ago
blubblub
3,169829
$endgroup$
7
$begingroup$
Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful.
$endgroup$
– David Richerby
5 hours ago
$begingroup$
@DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $mathfrak c=aleph_2,$ but that too was considered a failure.
$endgroup$
– spaceisdarkgreen
3 hours ago
$begingroup$
@spaceisdarkgreen Thanks -- I edited the one-sentence summary that it implies results Goedel found implausible into the answer. blub, I hope that's OK; please do edit if not.
$endgroup$
– David Richerby
2 hours ago
$begingroup$
@DavidRicherby I'm not 100% on whether it's appropriate to make this kind of an edit to a non CW post (blub may well disagree with me). So I reverted it for now. I agree with you and the 7 others that a summary of Godel's thoughts would be good to have in the answer. (But even if they opt not to, my comment will be visible.)
$endgroup$
– spaceisdarkgreen
2 hours ago
add a comment |
$begingroup$
There is a classical survey of Gödel about the continuum hypothesis:
"What is Cantor's Continuum Problem", K. Gödel, The American Mathematical Monthly, Vol. 54, No. 9 (Nov., 1947), pp. 515-525
In section 4, he discusses "in what sense and in which direction a solution of the continuum problem may be expected". While this is of course just a survey, it still represents some of Gödel's individual thoughts about the subject at the time.
A barrier free link is right now e.g. this.
edited 2 hours ago
spaceisdarkgreen
33.8k21753
answered 8 hours ago
blubblub
3,169829
$endgroup$
There is a classical survey of Gödel about the continuum hypothesis:
"What is Cantor's Continuum Problem", K. Gödel, The American Mathematical Monthly, Vol. 54, No. 9 (Nov., 1947), pp. 515-525
In section 4, he discusses "in what sense and in which direction a solution of the continuum problem may be expected". While this is of course just a survey, it still represents some of Gödel's individual thoughts about the subject at the time.
A barrier free link is right now e.g. this.
edited 2 hours ago
spaceisdarkgreen
33.8k21753
answered 8 hours ago
blubblub
3,169829
edited 2 hours ago
spaceisdarkgreen
33.8k21753
edited 2 hours ago
spaceisdarkgreen
33.8k21753
edited 2 hours ago
spaceisdarkgreen
33.8k21753
33.8k21753
answered 8 hours ago
blubblub
3,169829
answered 8 hours ago
blubblub
3,169829
answered 8 hours ago
blubblub
3,169829
3,169829
7
$begingroup$
Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful.
$endgroup$
– David Richerby
5 hours ago
$begingroup$
@DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $mathfrak c=aleph_2,$ but that too was considered a failure.
$endgroup$
– spaceisdarkgreen
3 hours ago
$begingroup$
@spaceisdarkgreen Thanks -- I edited the one-sentence summary that it implies results Goedel found implausible into the answer. blub, I hope that's OK; please do edit if not.
$endgroup$
– David Richerby
2 hours ago
$begingroup$
@DavidRicherby I'm not 100% on whether it's appropriate to make this kind of an edit to a non CW post (blub may well disagree with me). So I reverted it for now. I agree with you and the 7 others that a summary of Godel's thoughts would be good to have in the answer. (But even if they opt not to, my comment will be visible.)
$endgroup$
– spaceisdarkgreen
2 hours ago
add a comment |
7
$begingroup$
Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful.
$endgroup$
– David Richerby
5 hours ago
$begingroup$
@DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $mathfrak c=aleph_2,$ but that too was considered a failure.
$endgroup$
– spaceisdarkgreen
3 hours ago
$begingroup$
@spaceisdarkgreen Thanks -- I edited the one-sentence summary that it implies results Goedel found implausible into the answer. blub, I hope that's OK; please do edit if not.
$endgroup$
– David Richerby
2 hours ago
$begingroup$
@DavidRicherby I'm not 100% on whether it's appropriate to make this kind of an edit to a non CW post (blub may well disagree with me). So I reverted it for now. I agree with you and the 7 others that a summary of Godel's thoughts would be good to have in the answer. (But even if they opt not to, my comment will be visible.)
$endgroup$
– spaceisdarkgreen
2 hours ago
7
7
$begingroup$
Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful.
$endgroup$
– David Richerby
5 hours ago
$begingroup$
Could you at least give a short summary of the argument? Even if it's just at the level of "He was worried that CH implies that unicorns cannot exist", that would be helpful.
$endgroup$
– David Richerby
5 hours ago
$begingroup$
@DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $mathfrak c=aleph_2,$ but that too was considered a failure.
$endgroup$
– spaceisdarkgreen
3 hours ago
$begingroup$
@DavidRicherby He does not really give a (strong) argument in this reference. He only says he feels that several results in descriptive set theory that the Polish school had shown follow from CH are implausible (see p 523). I think it is safe to say that there was never any wide agreement with Godel that these were so implausible to be worth singling out. In later work, he attempted to give a detailed argument that $mathfrak c=aleph_2,$ but that too was considered a failure.
$endgroup$
– spaceisdarkgreen
3 hours ago
$begingroup$
@spaceisdarkgreen Thanks -- I edited the one-sentence summary that it implies results Goedel found implausible into the answer. blub, I hope that's OK; please do edit if not.
$endgroup$
– David Richerby
2 hours ago
$begingroup$
@spaceisdarkgreen Thanks -- I edited the one-sentence summary that it implies results Goedel found implausible into the answer. blub, I hope that's OK; please do edit if not.
$endgroup$
– David Richerby
2 hours ago
$begingroup$
@DavidRicherby I'm not 100% on whether it's appropriate to make this kind of an edit to a non CW post (blub may well disagree with me). So I reverted it for now. I agree with you and the 7 others that a summary of Godel's thoughts would be good to have in the answer. (But even if they opt not to, my comment will be visible.)
$endgroup$
– spaceisdarkgreen
2 hours ago
$begingroup$
@DavidRicherby I'm not 100% on whether it's appropriate to make this kind of an edit to a non CW post (blub may well disagree with me). So I reverted it for now. I agree with you and the 7 others that a summary of Godel's thoughts would be good to have in the answer. (But even if they opt not to, my comment will be visible.)
$endgroup$
– spaceisdarkgreen
2 hours ago
add a comment |
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$begingroup$
Are you asking for a source for the statement that Gödel though CH was false?
$endgroup$
– Lee Mosher
9 hours ago
$begingroup$
One might also look at Godel's collected works volume 2 for history and commentary on the 1947/1964 exposition, and Volume 3 about his unpublished 1970 notes. Also, Kanamori's "Godel and Set theory". There is also discussion of Godel's beliefs on CH in Maddy's "Believing the Axioms I" and Koellner's "On the question of absolute undecidability."
$endgroup$
– spaceisdarkgreen
6 hours ago
$begingroup$
I would add that Cohen's result didn't change the fact that set theorists of a Platonist bent tend to regard the CH as false (though it may have convinced a few to not be of a Platonist bent). I don't know much about this, but my understanding is that Godel had some esoteric reasons for believing $mathfrak c =aleph_2,$ whereas the dominant view in the aftermath of Cohen was that it was much larger, perhaps even weakly inaccessible. (Although there have been serious proposals that imply $mathfrak c =aleph_2,$ and even CH, more recently.)
$endgroup$
– spaceisdarkgreen
6 hours ago