provability of a formula in a given theory
Suppose $T$ is an arithmetically sound theory. If there exists a polynomial function $p$ such that $T$ proves for every $n$
$negvarphi(overline{p(n)})rightarrowvarphi(overline{n})$
is it true that for every $n$, $T$ proves $varphi(overline{n})$
logic
asked Nov 29 at 18:01
Daidalos
303
add a comment |
Suppose $T$ is an arithmetically sound theory. If there exists a polynomial function $p$ such that $T$ proves for every $n$
$negvarphi(overline{p(n)})rightarrowvarphi(overline{n})$
is it true that for every $n$, $T$ proves $varphi(overline{n})$
logic
asked Nov 29 at 18:01
Daidalos
303
add a comment |
Suppose $T$ is an arithmetically sound theory. If there exists a polynomial function $p$ such that $T$ proves for every $n$
$negvarphi(overline{p(n)})rightarrowvarphi(overline{n})$
is it true that for every $n$, $T$ proves $varphi(overline{n})$
logic
asked Nov 29 at 18:01
Daidalos
303
Suppose $T$ is an arithmetically sound theory. If there exists a polynomial function $p$ such that $T$ proves for every $n$
$negvarphi(overline{p(n)})rightarrowvarphi(overline{n})$
is it true that for every $n$, $T$ proves $varphi(overline{n})$
logic
logic
asked Nov 29 at 18:01
Daidalos
303
asked Nov 29 at 18:01
Daidalos
303
asked Nov 29 at 18:01
Daidalos
303
asked Nov 29 at 18:01
Daidalos
303
asked Nov 29 at 18:01
Daidalos
303
303
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
No, and this has nothing to do with logic: take $varphi(x)$ to be "$x$ is even" and $p(x)=x+1$. The statement $$forall x(negvarphi(p(x))rightarrowvarphi(x))wedge negvarphi(overline{1})$$ is - to nuke a mosquito - provable in the arithmetically sound theory PA.
answered Nov 29 at 18:21
Noah Schweber
120k10146279
add a comment |
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%2f3018959%2fprovability-of-a-formula-in-a-given-theory%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
No, and this has nothing to do with logic: take $varphi(x)$ to be "$x$ is even" and $p(x)=x+1$. The statement $$forall x(negvarphi(p(x))rightarrowvarphi(x))wedge negvarphi(overline{1})$$ is - to nuke a mosquito - provable in the arithmetically sound theory PA.
answered Nov 29 at 18:21
Noah Schweber
120k10146279
add a comment |
No, and this has nothing to do with logic: take $varphi(x)$ to be "$x$ is even" and $p(x)=x+1$. The statement $$forall x(negvarphi(p(x))rightarrowvarphi(x))wedge negvarphi(overline{1})$$ is - to nuke a mosquito - provable in the arithmetically sound theory PA.
answered Nov 29 at 18:21
Noah Schweber
120k10146279
add a comment |
No, and this has nothing to do with logic: take $varphi(x)$ to be "$x$ is even" and $p(x)=x+1$. The statement $$forall x(negvarphi(p(x))rightarrowvarphi(x))wedge negvarphi(overline{1})$$ is - to nuke a mosquito - provable in the arithmetically sound theory PA.
answered Nov 29 at 18:21
Noah Schweber
120k10146279
No, and this has nothing to do with logic: take $varphi(x)$ to be "$x$ is even" and $p(x)=x+1$. The statement $$forall x(negvarphi(p(x))rightarrowvarphi(x))wedge negvarphi(overline{1})$$ is - to nuke a mosquito - provable in the arithmetically sound theory PA.
answered Nov 29 at 18:21
Noah Schweber
120k10146279
answered Nov 29 at 18:21
Noah Schweber
120k10146279
answered Nov 29 at 18:21
Noah Schweber
120k10146279
answered Nov 29 at 18:21
Noah Schweber
120k10146279
120k10146279
add a comment |
add a comment |
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%2f3018959%2fprovability-of-a-formula-in-a-given-theory%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
