What are the arguments of the mathematicians who objected against the ontological proof Gödel offered?
$begingroup$
Q:
What are the arguments of the mathematicians who objected against the ontological argument/proof Gödel offered?
$$
begin{array}{rl} text{Ax. 1.} & left{P(varphi) wedge Box ; forall x[varphi(x) to psi(x)]right} to P(psi) \ text{Ax. 2.} & P(neg varphi) leftrightarrow neg P(varphi) \ text{Th. 1.} & P(varphi) to Diamond ; exists x[varphi(x)] \ text{Df. 1.} & G(x) iff forall varphi [P(varphi) to varphi(x)] \ text{Ax. 3.} & P(G) \ text{Th. 2.} & Diamond ; exists x ; G(x) \ text{Df. 2.} & varphi text{ ess } x iff varphi(x) wedge forall psi left{psi(x) to Box ; forall y[varphi(y) to psi(y)]right} \ text{Ax. 4.} & P(varphi) to Box ; P(varphi) \ text{Th. 3.} & G(x) to G text{ ess } x \ text{Df. 3.} & E(x) iff forall varphi[varphi text{ ess } x to Box ; exists y ; varphi(y)] \ text{Ax. 5.} & P(E) \ text{Th. 4.} & Box ; exists x ; G(x) end{array} $$
I want to learn. Because, Gödel's proof was checked on the computer. Well, How can mathematicians criticize, if Gödel's proof is controlled by the computer and known to be true?I cannot understand how an object that is external is mathematically proven. What did the computer actually confirm?
I want to know the problems in Gödel's proof.
Any book or link you recommend , or answer will satisfy me.
soft-question proof-explanation book-recommendation math-history philosophy
asked Jan 11 at 13:12
ElementaryElementary
360111
$endgroup$
|
show 2 more comments
$begingroup$
Q:
What are the arguments of the mathematicians who objected against the ontological argument/proof Gödel offered?
$$
begin{array}{rl} text{Ax. 1.} & left{P(varphi) wedge Box ; forall x[varphi(x) to psi(x)]right} to P(psi) \ text{Ax. 2.} & P(neg varphi) leftrightarrow neg P(varphi) \ text{Th. 1.} & P(varphi) to Diamond ; exists x[varphi(x)] \ text{Df. 1.} & G(x) iff forall varphi [P(varphi) to varphi(x)] \ text{Ax. 3.} & P(G) \ text{Th. 2.} & Diamond ; exists x ; G(x) \ text{Df. 2.} & varphi text{ ess } x iff varphi(x) wedge forall psi left{psi(x) to Box ; forall y[varphi(y) to psi(y)]right} \ text{Ax. 4.} & P(varphi) to Box ; P(varphi) \ text{Th. 3.} & G(x) to G text{ ess } x \ text{Df. 3.} & E(x) iff forall varphi[varphi text{ ess } x to Box ; exists y ; varphi(y)] \ text{Ax. 5.} & P(E) \ text{Th. 4.} & Box ; exists x ; G(x) end{array} $$
I want to learn. Because, Gödel's proof was checked on the computer. Well, How can mathematicians criticize, if Gödel's proof is controlled by the computer and known to be true?I cannot understand how an object that is external is mathematically proven. What did the computer actually confirm?
I want to know the problems in Gödel's proof.
Any book or link you recommend , or answer will satisfy me.
soft-question proof-explanation book-recommendation math-history philosophy
asked Jan 11 at 13:12
ElementaryElementary
360111
$endgroup$
2
$begingroup$
You can start with a dedicated book regrding the proof : Melvin Fitting, Types, Tableaus, and Gödel’s God, Kluwer (2002).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:22
4
$begingroup$
Checked means "formally correct", i.e. no logical errors found. True means, formally correct and derived from true axioms: this last point is the crux of the matter, and computers cannot do this (up to now).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:24
3
$begingroup$
I want to know something I don't know. Because I don't know and want to know. Why downvote, I can not understand..
$endgroup$
– Elementary
Jan 11 at 13:29
1
$begingroup$
Personal (and not answering your question): everything we know about God (including existence) is something that is revealed by God Himself.
$endgroup$
– drhab
Jan 11 at 13:36
2
$begingroup$
There is a also a summary of the situation on the Stanford Encyclopedia of Philosophy, at plato.stanford.edu/entries/ontological-arguments/#GodOntArg . The general issue of ontological arguments is philosophical, rather than mathematical.
$endgroup$
– Carl Mummert
Jan 11 at 13:36
|
show 2 more comments
$begingroup$
Q:
What are the arguments of the mathematicians who objected against the ontological argument/proof Gödel offered?
$$
begin{array}{rl} text{Ax. 1.} & left{P(varphi) wedge Box ; forall x[varphi(x) to psi(x)]right} to P(psi) \ text{Ax. 2.} & P(neg varphi) leftrightarrow neg P(varphi) \ text{Th. 1.} & P(varphi) to Diamond ; exists x[varphi(x)] \ text{Df. 1.} & G(x) iff forall varphi [P(varphi) to varphi(x)] \ text{Ax. 3.} & P(G) \ text{Th. 2.} & Diamond ; exists x ; G(x) \ text{Df. 2.} & varphi text{ ess } x iff varphi(x) wedge forall psi left{psi(x) to Box ; forall y[varphi(y) to psi(y)]right} \ text{Ax. 4.} & P(varphi) to Box ; P(varphi) \ text{Th. 3.} & G(x) to G text{ ess } x \ text{Df. 3.} & E(x) iff forall varphi[varphi text{ ess } x to Box ; exists y ; varphi(y)] \ text{Ax. 5.} & P(E) \ text{Th. 4.} & Box ; exists x ; G(x) end{array} $$
I want to learn. Because, Gödel's proof was checked on the computer. Well, How can mathematicians criticize, if Gödel's proof is controlled by the computer and known to be true?I cannot understand how an object that is external is mathematically proven. What did the computer actually confirm?
I want to know the problems in Gödel's proof.
Any book or link you recommend , or answer will satisfy me.
soft-question proof-explanation book-recommendation math-history philosophy
asked Jan 11 at 13:12
ElementaryElementary
360111
$endgroup$
Q:
What are the arguments of the mathematicians who objected against the ontological argument/proof Gödel offered?
$$
begin{array}{rl} text{Ax. 1.} & left{P(varphi) wedge Box ; forall x[varphi(x) to psi(x)]right} to P(psi) \ text{Ax. 2.} & P(neg varphi) leftrightarrow neg P(varphi) \ text{Th. 1.} & P(varphi) to Diamond ; exists x[varphi(x)] \ text{Df. 1.} & G(x) iff forall varphi [P(varphi) to varphi(x)] \ text{Ax. 3.} & P(G) \ text{Th. 2.} & Diamond ; exists x ; G(x) \ text{Df. 2.} & varphi text{ ess } x iff varphi(x) wedge forall psi left{psi(x) to Box ; forall y[varphi(y) to psi(y)]right} \ text{Ax. 4.} & P(varphi) to Box ; P(varphi) \ text{Th. 3.} & G(x) to G text{ ess } x \ text{Df. 3.} & E(x) iff forall varphi[varphi text{ ess } x to Box ; exists y ; varphi(y)] \ text{Ax. 5.} & P(E) \ text{Th. 4.} & Box ; exists x ; G(x) end{array} $$
I want to learn. Because, Gödel's proof was checked on the computer. Well, How can mathematicians criticize, if Gödel's proof is controlled by the computer and known to be true?I cannot understand how an object that is external is mathematically proven. What did the computer actually confirm?
I want to know the problems in Gödel's proof.
Any book or link you recommend , or answer will satisfy me.
soft-question proof-explanation book-recommendation math-history philosophy
soft-question proof-explanation book-recommendation math-history philosophy
asked Jan 11 at 13:12
ElementaryElementary
360111
asked Jan 11 at 13:12
ElementaryElementary
360111
edited Jan 11 at 14:57
Elementary
asked Jan 11 at 13:12
ElementaryElementary
360111
asked Jan 11 at 13:12
ElementaryElementary
360111
asked Jan 11 at 13:12
ElementaryElementary
360111
360111
2
$begingroup$
You can start with a dedicated book regrding the proof : Melvin Fitting, Types, Tableaus, and Gödel’s God, Kluwer (2002).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:22
4
$begingroup$
Checked means "formally correct", i.e. no logical errors found. True means, formally correct and derived from true axioms: this last point is the crux of the matter, and computers cannot do this (up to now).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:24
3
$begingroup$
I want to know something I don't know. Because I don't know and want to know. Why downvote, I can not understand..
$endgroup$
– Elementary
Jan 11 at 13:29
1
$begingroup$
Personal (and not answering your question): everything we know about God (including existence) is something that is revealed by God Himself.
$endgroup$
– drhab
Jan 11 at 13:36
2
$begingroup$
There is a also a summary of the situation on the Stanford Encyclopedia of Philosophy, at plato.stanford.edu/entries/ontological-arguments/#GodOntArg . The general issue of ontological arguments is philosophical, rather than mathematical.
$endgroup$
– Carl Mummert
Jan 11 at 13:36
|
show 2 more comments
2
$begingroup$
You can start with a dedicated book regrding the proof : Melvin Fitting, Types, Tableaus, and Gödel’s God, Kluwer (2002).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:22
4
$begingroup$
Checked means "formally correct", i.e. no logical errors found. True means, formally correct and derived from true axioms: this last point is the crux of the matter, and computers cannot do this (up to now).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:24
3
$begingroup$
I want to know something I don't know. Because I don't know and want to know. Why downvote, I can not understand..
$endgroup$
– Elementary
Jan 11 at 13:29
1
$begingroup$
Personal (and not answering your question): everything we know about God (including existence) is something that is revealed by God Himself.
$endgroup$
– drhab
Jan 11 at 13:36
2
$begingroup$
There is a also a summary of the situation on the Stanford Encyclopedia of Philosophy, at plato.stanford.edu/entries/ontological-arguments/#GodOntArg . The general issue of ontological arguments is philosophical, rather than mathematical.
$endgroup$
– Carl Mummert
Jan 11 at 13:36
2
2
$begingroup$
You can start with a dedicated book regrding the proof : Melvin Fitting, Types, Tableaus, and Gödel’s God, Kluwer (2002).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:22
$begingroup$
You can start with a dedicated book regrding the proof : Melvin Fitting, Types, Tableaus, and Gödel’s God, Kluwer (2002).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:22
4
4
$begingroup$
Checked means "formally correct", i.e. no logical errors found. True means, formally correct and derived from true axioms: this last point is the crux of the matter, and computers cannot do this (up to now).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:24
$begingroup$
Checked means "formally correct", i.e. no logical errors found. True means, formally correct and derived from true axioms: this last point is the crux of the matter, and computers cannot do this (up to now).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:24
3
3
$begingroup$
I want to know something I don't know. Because I don't know and want to know. Why downvote, I can not understand..
$endgroup$
– Elementary
Jan 11 at 13:29
$begingroup$
I want to know something I don't know. Because I don't know and want to know. Why downvote, I can not understand..
$endgroup$
– Elementary
Jan 11 at 13:29
1
1
$begingroup$
Personal (and not answering your question): everything we know about God (including existence) is something that is revealed by God Himself.
$endgroup$
– drhab
Jan 11 at 13:36
$begingroup$
Personal (and not answering your question): everything we know about God (including existence) is something that is revealed by God Himself.
$endgroup$
– drhab
Jan 11 at 13:36
2
2
$begingroup$
There is a also a summary of the situation on the Stanford Encyclopedia of Philosophy, at plato.stanford.edu/entries/ontological-arguments/#GodOntArg . The general issue of ontological arguments is philosophical, rather than mathematical.
$endgroup$
– Carl Mummert
Jan 11 at 13:36
$begingroup$
There is a also a summary of the situation on the Stanford Encyclopedia of Philosophy, at plato.stanford.edu/entries/ontological-arguments/#GodOntArg . The general issue of ontological arguments is philosophical, rather than mathematical.
$endgroup$
– Carl Mummert
Jan 11 at 13:36
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The proof is of course, sound. But to claim that this argument proves that God as we understand exists, you also need to argue that the axioms really hold in the real world, which I believe would be extremely hard.
Edit : Another criticism comes from the fact that the same axioms can be used to prove that there must be infinitely many semi-gods.
answered Jan 11 at 13:15
LeventLevent
2,729925
$endgroup$
$begingroup$
"Semi-god" - nice! (+1 for that)
$endgroup$
– Peter
Jan 11 at 14:18
2
$begingroup$
If you happen to know the details of the proof : Can we prove this way also that Santa Claus exists ? I only ask because I do not understand how Goedel wanted to link God with mathematics.
$endgroup$
– Peter
Jan 11 at 14:26
1
$begingroup$
I do not know every detail of the proof, but basically it says that there are beings missing almost none of the "good" properties. I also don't think Gödel was really trying to prove that the God exists and as far as I remember, he only trying to formalize the ontological proof.
$endgroup$
– Levent
Jan 11 at 14:30
$begingroup$
OK, thanks for clarifying. So he did not claim that he actually proved God's existence mathematically ?
$endgroup$
– Peter
Jan 11 at 14:31
2
$begingroup$
No, he definitely did not.
$endgroup$
– Levent
Jan 11 at 14:32
add a comment |
Your Answer
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1 Answer
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1 Answer
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$begingroup$
The proof is of course, sound. But to claim that this argument proves that God as we understand exists, you also need to argue that the axioms really hold in the real world, which I believe would be extremely hard.
Edit : Another criticism comes from the fact that the same axioms can be used to prove that there must be infinitely many semi-gods.
answered Jan 11 at 13:15
LeventLevent
2,729925
$endgroup$
$begingroup$
"Semi-god" - nice! (+1 for that)
$endgroup$
– Peter
Jan 11 at 14:18
2
$begingroup$
If you happen to know the details of the proof : Can we prove this way also that Santa Claus exists ? I only ask because I do not understand how Goedel wanted to link God with mathematics.
$endgroup$
– Peter
Jan 11 at 14:26
1
$begingroup$
I do not know every detail of the proof, but basically it says that there are beings missing almost none of the "good" properties. I also don't think Gödel was really trying to prove that the God exists and as far as I remember, he only trying to formalize the ontological proof.
$endgroup$
– Levent
Jan 11 at 14:30
$begingroup$
OK, thanks for clarifying. So he did not claim that he actually proved God's existence mathematically ?
$endgroup$
– Peter
Jan 11 at 14:31
2
$begingroup$
No, he definitely did not.
$endgroup$
– Levent
Jan 11 at 14:32
add a comment |
$begingroup$
The proof is of course, sound. But to claim that this argument proves that God as we understand exists, you also need to argue that the axioms really hold in the real world, which I believe would be extremely hard.
Edit : Another criticism comes from the fact that the same axioms can be used to prove that there must be infinitely many semi-gods.
answered Jan 11 at 13:15
LeventLevent
2,729925
$endgroup$
$begingroup$
"Semi-god" - nice! (+1 for that)
$endgroup$
– Peter
Jan 11 at 14:18
2
$begingroup$
If you happen to know the details of the proof : Can we prove this way also that Santa Claus exists ? I only ask because I do not understand how Goedel wanted to link God with mathematics.
$endgroup$
– Peter
Jan 11 at 14:26
1
$begingroup$
I do not know every detail of the proof, but basically it says that there are beings missing almost none of the "good" properties. I also don't think Gödel was really trying to prove that the God exists and as far as I remember, he only trying to formalize the ontological proof.
$endgroup$
– Levent
Jan 11 at 14:30
$begingroup$
OK, thanks for clarifying. So he did not claim that he actually proved God's existence mathematically ?
$endgroup$
– Peter
Jan 11 at 14:31
2
$begingroup$
No, he definitely did not.
$endgroup$
– Levent
Jan 11 at 14:32
add a comment |
$begingroup$
The proof is of course, sound. But to claim that this argument proves that God as we understand exists, you also need to argue that the axioms really hold in the real world, which I believe would be extremely hard.
Edit : Another criticism comes from the fact that the same axioms can be used to prove that there must be infinitely many semi-gods.
answered Jan 11 at 13:15
LeventLevent
2,729925
$endgroup$
The proof is of course, sound. But to claim that this argument proves that God as we understand exists, you also need to argue that the axioms really hold in the real world, which I believe would be extremely hard.
Edit : Another criticism comes from the fact that the same axioms can be used to prove that there must be infinitely many semi-gods.
answered Jan 11 at 13:15
LeventLevent
2,729925
answered Jan 11 at 13:15
LeventLevent
2,729925
answered Jan 11 at 13:15
LeventLevent
2,729925
answered Jan 11 at 13:15
LeventLevent
2,729925
2,729925
$begingroup$
"Semi-god" - nice! (+1 for that)
$endgroup$
– Peter
Jan 11 at 14:18
2
$begingroup$
If you happen to know the details of the proof : Can we prove this way also that Santa Claus exists ? I only ask because I do not understand how Goedel wanted to link God with mathematics.
$endgroup$
– Peter
Jan 11 at 14:26
1
$begingroup$
I do not know every detail of the proof, but basically it says that there are beings missing almost none of the "good" properties. I also don't think Gödel was really trying to prove that the God exists and as far as I remember, he only trying to formalize the ontological proof.
$endgroup$
– Levent
Jan 11 at 14:30
$begingroup$
OK, thanks for clarifying. So he did not claim that he actually proved God's existence mathematically ?
$endgroup$
– Peter
Jan 11 at 14:31
2
$begingroup$
No, he definitely did not.
$endgroup$
– Levent
Jan 11 at 14:32
add a comment |
$begingroup$
"Semi-god" - nice! (+1 for that)
$endgroup$
– Peter
Jan 11 at 14:18
2
$begingroup$
If you happen to know the details of the proof : Can we prove this way also that Santa Claus exists ? I only ask because I do not understand how Goedel wanted to link God with mathematics.
$endgroup$
– Peter
Jan 11 at 14:26
1
$begingroup$
I do not know every detail of the proof, but basically it says that there are beings missing almost none of the "good" properties. I also don't think Gödel was really trying to prove that the God exists and as far as I remember, he only trying to formalize the ontological proof.
$endgroup$
– Levent
Jan 11 at 14:30
$begingroup$
OK, thanks for clarifying. So he did not claim that he actually proved God's existence mathematically ?
$endgroup$
– Peter
Jan 11 at 14:31
2
$begingroup$
No, he definitely did not.
$endgroup$
– Levent
Jan 11 at 14:32
$begingroup$
"Semi-god" - nice! (+1 for that)
$endgroup$
– Peter
Jan 11 at 14:18
$begingroup$
"Semi-god" - nice! (+1 for that)
$endgroup$
– Peter
Jan 11 at 14:18
2
2
$begingroup$
If you happen to know the details of the proof : Can we prove this way also that Santa Claus exists ? I only ask because I do not understand how Goedel wanted to link God with mathematics.
$endgroup$
– Peter
Jan 11 at 14:26
$begingroup$
If you happen to know the details of the proof : Can we prove this way also that Santa Claus exists ? I only ask because I do not understand how Goedel wanted to link God with mathematics.
$endgroup$
– Peter
Jan 11 at 14:26
1
1
$begingroup$
I do not know every detail of the proof, but basically it says that there are beings missing almost none of the "good" properties. I also don't think Gödel was really trying to prove that the God exists and as far as I remember, he only trying to formalize the ontological proof.
$endgroup$
– Levent
Jan 11 at 14:30
$begingroup$
I do not know every detail of the proof, but basically it says that there are beings missing almost none of the "good" properties. I also don't think Gödel was really trying to prove that the God exists and as far as I remember, he only trying to formalize the ontological proof.
$endgroup$
– Levent
Jan 11 at 14:30
$begingroup$
OK, thanks for clarifying. So he did not claim that he actually proved God's existence mathematically ?
$endgroup$
– Peter
Jan 11 at 14:31
$begingroup$
OK, thanks for clarifying. So he did not claim that he actually proved God's existence mathematically ?
$endgroup$
– Peter
Jan 11 at 14:31
2
2
$begingroup$
No, he definitely did not.
$endgroup$
– Levent
Jan 11 at 14:32
$begingroup$
No, he definitely did not.
$endgroup$
– Levent
Jan 11 at 14:32
add a comment |
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$begingroup$
You can start with a dedicated book regrding the proof : Melvin Fitting, Types, Tableaus, and Gödel’s God, Kluwer (2002).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:22
4
$begingroup$
Checked means "formally correct", i.e. no logical errors found. True means, formally correct and derived from true axioms: this last point is the crux of the matter, and computers cannot do this (up to now).
$endgroup$
– Mauro ALLEGRANZA
Jan 11 at 13:24
3
$begingroup$
I want to know something I don't know. Because I don't know and want to know. Why downvote, I can not understand..
$endgroup$
– Elementary
Jan 11 at 13:29
1
$begingroup$
Personal (and not answering your question): everything we know about God (including existence) is something that is revealed by God Himself.
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– drhab
Jan 11 at 13:36
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There is a also a summary of the situation on the Stanford Encyclopedia of Philosophy, at plato.stanford.edu/entries/ontological-arguments/#GodOntArg . The general issue of ontological arguments is philosophical, rather than mathematical.
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– Carl Mummert
Jan 11 at 13:36