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
asked Dec 11 '18 at 14:31
Ettore
969
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
asked Dec 11 '18 at 14:31
Ettore
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 |
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
asked Dec 11 '18 at 14:31
Ettore
969
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
asked Dec 11 '18 at 14:31
Ettore
969
asked Dec 11 '18 at 14:31
Ettore
969
asked Dec 11 '18 at 14:31
Ettore
969
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3035338%2fshepherdsons-conditions-a-shortcut-to-the-second-incompleteness-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3035338%2fshepherdsons-conditions-a-shortcut-to-the-second-incompleteness-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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