Shepherdson's conditions - a shortcut to the second incompleteness theorem?
Would be great if someone could help me with the following exercise:
(1.5.11. from 'proof theory and logical complexity', Girard, '87)
Let T be a theory in the language $L_0$;
[Remark: That is the language of elementary arithmetic in this case, that is zero, successor, addition, product, prop. connectives and quantifiers, equality and less-than.]
assume that there is a formula $Prov[a,b]$ of $L_0$ such that
[Remark: ...for all natural numbers a, b, n...]
(i) $T vdash A[n] rightarrow T vdash Prov[ulcorner A[x_0] urcorner,n]$
(ii) $T vdash Prov[a,a] rightarrow Prov[ulcorner Prov[x_0,x_0] urcorner,a]$
(iii) $T vdash Prov[a,n] land Prov[langle 19,a,b rangle, n] rightarrow Prov[b,n]$.
Show that, if T is consistent, then $T nvdash neg Prov[ulcorner 0=1 urcorner,x]$.
[Remark: Above every number symbol there should be an overline, showing that we are speaking about numerals of the formal system rather than natural numbers, which is omitted for simplicity.
19 ist the Gödel number of the implication arrow.
The implication arrow in clause (i) isn't really an implication arrow, cause this is a usual (meta-)implication (if-then-statement).]
As I understand, Shepherdson and Bezboruah proved a version or 'variant' of Gödels second incompleteness theorem in some paper, it might have something to do with this, but I guess it would be to hard to work through the paper for me.
Thanks and kind regards,
Ettore
logic foundations proof-theory incompleteness provability
add a comment |
Would be great if someone could help me with the following exercise:
(1.5.11. from 'proof theory and logical complexity', Girard, '87)
Let T be a theory in the language $L_0$;
[Remark: That is the language of elementary arithmetic in this case, that is zero, successor, addition, product, prop. connectives and quantifiers, equality and less-than.]
assume that there is a formula $Prov[a,b]$ of $L_0$ such that
[Remark: ...for all natural numbers a, b, n...]
(i) $T vdash A[n] rightarrow T vdash Prov[ulcorner A[x_0] urcorner,n]$
(ii) $T vdash Prov[a,a] rightarrow Prov[ulcorner Prov[x_0,x_0] urcorner,a]$
(iii) $T vdash Prov[a,n] land Prov[langle 19,a,b rangle, n] rightarrow Prov[b,n]$.
Show that, if T is consistent, then $T nvdash neg Prov[ulcorner 0=1 urcorner,x]$.
[Remark: Above every number symbol there should be an overline, showing that we are speaking about numerals of the formal system rather than natural numbers, which is omitted for simplicity.
19 ist the Gödel number of the implication arrow.
The implication arrow in clause (i) isn't really an implication arrow, cause this is a usual (meta-)implication (if-then-statement).]
As I understand, Shepherdson and Bezboruah proved a version or 'variant' of Gödels second incompleteness theorem in some paper, it might have something to do with this, but I guess it would be to hard to work through the paper for me.
Thanks and kind regards,
Ettore
logic foundations proof-theory incompleteness provability
You haven't asked any question.
– Derek Elkins
Dec 12 '18 at 8:26
Yo Derek, cheers... :) Yap, that's probably true, I guess...well...I wrote I need help with this exercise...so, my question is: How does it work? Why is right what I am supposed to show in this exercise?
– Ettore
Dec 12 '18 at 8:47
add a comment |
Would be great if someone could help me with the following exercise:
(1.5.11. from 'proof theory and logical complexity', Girard, '87)
Let T be a theory in the language $L_0$;
[Remark: That is the language of elementary arithmetic in this case, that is zero, successor, addition, product, prop. connectives and quantifiers, equality and less-than.]
assume that there is a formula $Prov[a,b]$ of $L_0$ such that
[Remark: ...for all natural numbers a, b, n...]
(i) $T vdash A[n] rightarrow T vdash Prov[ulcorner A[x_0] urcorner,n]$
(ii) $T vdash Prov[a,a] rightarrow Prov[ulcorner Prov[x_0,x_0] urcorner,a]$
(iii) $T vdash Prov[a,n] land Prov[langle 19,a,b rangle, n] rightarrow Prov[b,n]$.
Show that, if T is consistent, then $T nvdash neg Prov[ulcorner 0=1 urcorner,x]$.
[Remark: Above every number symbol there should be an overline, showing that we are speaking about numerals of the formal system rather than natural numbers, which is omitted for simplicity.
19 ist the Gödel number of the implication arrow.
The implication arrow in clause (i) isn't really an implication arrow, cause this is a usual (meta-)implication (if-then-statement).]
As I understand, Shepherdson and Bezboruah proved a version or 'variant' of Gödels second incompleteness theorem in some paper, it might have something to do with this, but I guess it would be to hard to work through the paper for me.
Thanks and kind regards,
Ettore
logic foundations proof-theory incompleteness provability
Would be great if someone could help me with the following exercise:
(1.5.11. from 'proof theory and logical complexity', Girard, '87)
Let T be a theory in the language $L_0$;
[Remark: That is the language of elementary arithmetic in this case, that is zero, successor, addition, product, prop. connectives and quantifiers, equality and less-than.]
assume that there is a formula $Prov[a,b]$ of $L_0$ such that
[Remark: ...for all natural numbers a, b, n...]
(i) $T vdash A[n] rightarrow T vdash Prov[ulcorner A[x_0] urcorner,n]$
(ii) $T vdash Prov[a,a] rightarrow Prov[ulcorner Prov[x_0,x_0] urcorner,a]$
(iii) $T vdash Prov[a,n] land Prov[langle 19,a,b rangle, n] rightarrow Prov[b,n]$.
Show that, if T is consistent, then $T nvdash neg Prov[ulcorner 0=1 urcorner,x]$.
[Remark: Above every number symbol there should be an overline, showing that we are speaking about numerals of the formal system rather than natural numbers, which is omitted for simplicity.
19 ist the Gödel number of the implication arrow.
The implication arrow in clause (i) isn't really an implication arrow, cause this is a usual (meta-)implication (if-then-statement).]
As I understand, Shepherdson and Bezboruah proved a version or 'variant' of Gödels second incompleteness theorem in some paper, it might have something to do with this, but I guess it would be to hard to work through the paper for me.
Thanks and kind regards,
Ettore
logic foundations proof-theory incompleteness provability
logic foundations proof-theory incompleteness provability
asked Dec 11 '18 at 14:31
Ettore
969
969
You haven't asked any question.
– Derek Elkins
Dec 12 '18 at 8:26
Yo Derek, cheers... :) Yap, that's probably true, I guess...well...I wrote I need help with this exercise...so, my question is: How does it work? Why is right what I am supposed to show in this exercise?
– Ettore
Dec 12 '18 at 8:47
add a comment |
You haven't asked any question.
– Derek Elkins
Dec 12 '18 at 8:26
Yo Derek, cheers... :) Yap, that's probably true, I guess...well...I wrote I need help with this exercise...so, my question is: How does it work? Why is right what I am supposed to show in this exercise?
– Ettore
Dec 12 '18 at 8:47
You haven't asked any question.
– Derek Elkins
Dec 12 '18 at 8:26
You haven't asked any question.
– Derek Elkins
Dec 12 '18 at 8:26
Yo Derek, cheers... :) Yap, that's probably true, I guess...well...I wrote I need help with this exercise...so, my question is: How does it work? Why is right what I am supposed to show in this exercise?
– Ettore
Dec 12 '18 at 8:47
Yo Derek, cheers... :) Yap, that's probably true, I guess...well...I wrote I need help with this exercise...so, my question is: How does it work? Why is right what I am supposed to show in this exercise?
– Ettore
Dec 12 '18 at 8:47
add a comment |
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You haven't asked any question.
– Derek Elkins
Dec 12 '18 at 8:26
Yo Derek, cheers... :) Yap, that's probably true, I guess...well...I wrote I need help with this exercise...so, my question is: How does it work? Why is right what I am supposed to show in this exercise?
– Ettore
Dec 12 '18 at 8:47