Are the HBL derivability conditions necessary for Gödel's incompleteness theorems? (For Löb's theorem?)
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I am currently working with 'proof theory and logical complexity', a monograph on proof theory.
In the exercises of the first chapter one was asked to prove Löb's theorem (https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem) and two related/similar statements.
(The first chapter essentially gives the setup to prove and proves both incompleteness theorems and some results around them.)
Searching the internet I came across the Hilbert-Bernays-Löb 'derivability conditions' (https://en.wikipedia.org/wiki/Hilbert%E2%80%93Bernays_provability_conditions), that are - apparently - necessary to prove Löb's theorem.
The author never mentions them explicitly, and while the first is very plausible and the second not unplausible, I read that the third's proof is tedious and difficult.
The author never gives a proof for them, and I don't recall that one would have been needed, so I ask myself whether it is possible to prove the incompleteness theorems without them, or if they follow easily from some result, especially since Löb's theorem is in this case an exercise, and would be quite hard if the author intends the student to have the idea and the proof of the derivability conditions all by himself. Or maybe Löb's theorem is itself provable without them?
Thanks,
Ettore
PS: I added the question also on mathoverflow:
https://mathoverflow.net/questions/319402/are-the-hbl-derivability-conditions-necessary-for-g%C3%B6dels-incompleteness-theorem
logic proof-theory incompleteness meta-math provability
asked Dec 6 '18 at 10:39
EttoreEttore
989
$endgroup$
add a comment |
$begingroup$
I am currently working with 'proof theory and logical complexity', a monograph on proof theory.
In the exercises of the first chapter one was asked to prove Löb's theorem (https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem) and two related/similar statements.
(The first chapter essentially gives the setup to prove and proves both incompleteness theorems and some results around them.)
Searching the internet I came across the Hilbert-Bernays-Löb 'derivability conditions' (https://en.wikipedia.org/wiki/Hilbert%E2%80%93Bernays_provability_conditions), that are - apparently - necessary to prove Löb's theorem.
The author never mentions them explicitly, and while the first is very plausible and the second not unplausible, I read that the third's proof is tedious and difficult.
The author never gives a proof for them, and I don't recall that one would have been needed, so I ask myself whether it is possible to prove the incompleteness theorems without them, or if they follow easily from some result, especially since Löb's theorem is in this case an exercise, and would be quite hard if the author intends the student to have the idea and the proof of the derivability conditions all by himself. Or maybe Löb's theorem is itself provable without them?
Thanks,
Ettore
PS: I added the question also on mathoverflow:
https://mathoverflow.net/questions/319402/are-the-hbl-derivability-conditions-necessary-for-g%C3%B6dels-incompleteness-theorem
logic proof-theory incompleteness meta-math provability
asked Dec 6 '18 at 10:39
EttoreEttore
989
$endgroup$
1
$begingroup$
Not strictly needed; G's original proof does not use them. They are used for a "abstract" (i.e. generalized) version of the theorem. See e.g. B.Buldt, The Scope of Gödel’s First Incompleteness Theorem (2014).
$endgroup$
– Mauro ALLEGRANZA
Dec 6 '18 at 10:58
add a comment |
$begingroup$
I am currently working with 'proof theory and logical complexity', a monograph on proof theory.
In the exercises of the first chapter one was asked to prove Löb's theorem (https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem) and two related/similar statements.
(The first chapter essentially gives the setup to prove and proves both incompleteness theorems and some results around them.)
Searching the internet I came across the Hilbert-Bernays-Löb 'derivability conditions' (https://en.wikipedia.org/wiki/Hilbert%E2%80%93Bernays_provability_conditions), that are - apparently - necessary to prove Löb's theorem.
The author never mentions them explicitly, and while the first is very plausible and the second not unplausible, I read that the third's proof is tedious and difficult.
The author never gives a proof for them, and I don't recall that one would have been needed, so I ask myself whether it is possible to prove the incompleteness theorems without them, or if they follow easily from some result, especially since Löb's theorem is in this case an exercise, and would be quite hard if the author intends the student to have the idea and the proof of the derivability conditions all by himself. Or maybe Löb's theorem is itself provable without them?
Thanks,
Ettore
PS: I added the question also on mathoverflow:
https://mathoverflow.net/questions/319402/are-the-hbl-derivability-conditions-necessary-for-g%C3%B6dels-incompleteness-theorem
logic proof-theory incompleteness meta-math provability
asked Dec 6 '18 at 10:39
EttoreEttore
989
$endgroup$
I am currently working with 'proof theory and logical complexity', a monograph on proof theory.
In the exercises of the first chapter one was asked to prove Löb's theorem (https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem) and two related/similar statements.
(The first chapter essentially gives the setup to prove and proves both incompleteness theorems and some results around them.)
Searching the internet I came across the Hilbert-Bernays-Löb 'derivability conditions' (https://en.wikipedia.org/wiki/Hilbert%E2%80%93Bernays_provability_conditions), that are - apparently - necessary to prove Löb's theorem.
The author never mentions them explicitly, and while the first is very plausible and the second not unplausible, I read that the third's proof is tedious and difficult.
The author never gives a proof for them, and I don't recall that one would have been needed, so I ask myself whether it is possible to prove the incompleteness theorems without them, or if they follow easily from some result, especially since Löb's theorem is in this case an exercise, and would be quite hard if the author intends the student to have the idea and the proof of the derivability conditions all by himself. Or maybe Löb's theorem is itself provable without them?
Thanks,
Ettore
PS: I added the question also on mathoverflow:
https://mathoverflow.net/questions/319402/are-the-hbl-derivability-conditions-necessary-for-g%C3%B6dels-incompleteness-theorem
logic proof-theory incompleteness meta-math provability
logic proof-theory incompleteness meta-math provability
asked Dec 6 '18 at 10:39
EttoreEttore
989
asked Dec 6 '18 at 10:39
EttoreEttore
989
edited Dec 24 '18 at 10:33
Ettore
asked Dec 6 '18 at 10:39
EttoreEttore
989
asked Dec 6 '18 at 10:39
EttoreEttore
989
asked Dec 6 '18 at 10:39
EttoreEttore
989
989
1
$begingroup$
Not strictly needed; G's original proof does not use them. They are used for a "abstract" (i.e. generalized) version of the theorem. See e.g. B.Buldt, The Scope of Gödel’s First Incompleteness Theorem (2014).
$endgroup$
– Mauro ALLEGRANZA
Dec 6 '18 at 10:58
add a comment |
1
$begingroup$
Not strictly needed; G's original proof does not use them. They are used for a "abstract" (i.e. generalized) version of the theorem. See e.g. B.Buldt, The Scope of Gödel’s First Incompleteness Theorem (2014).
$endgroup$
– Mauro ALLEGRANZA
Dec 6 '18 at 10:58
1
1
$begingroup$
Not strictly needed; G's original proof does not use them. They are used for a "abstract" (i.e. generalized) version of the theorem. See e.g. B.Buldt, The Scope of Gödel’s First Incompleteness Theorem (2014).
$endgroup$
– Mauro ALLEGRANZA
Dec 6 '18 at 10:58
$begingroup$
Not strictly needed; G's original proof does not use them. They are used for a "abstract" (i.e. generalized) version of the theorem. See e.g. B.Buldt, The Scope of Gödel’s First Incompleteness Theorem (2014).
$endgroup$
– Mauro ALLEGRANZA
Dec 6 '18 at 10:58
add a comment |
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$begingroup$
Not strictly needed; G's original proof does not use them. They are used for a "abstract" (i.e. generalized) version of the theorem. See e.g. B.Buldt, The Scope of Gödel’s First Incompleteness Theorem (2014).
$endgroup$
– Mauro ALLEGRANZA
Dec 6 '18 at 10:58