Mixing
Godel's theorem incompleteness, truth vs.provability
$begingroup$
I know this question has been investigated in other threads, but I would like to pose yet another question on Gödel's theorem incompleteness, and truth in 'the standard model' compared with provability in the 'formal model'.
I will use the language of Gensler's little book "Gödel's Theorem Simplified" (see here for a summary).
For what Gensler calls System C, he gives an outline for how to construct a wff in system C. The system C the symbols are /, (, ), - , n (for a variable), nn (another variable),...etc.).
The reference numbers for system C formulas (like Godel numbers) are defined too.
For example the ref # for / is 1, the reference # for the variable n is 8, etc.
Then Gensler gives instructions for how to create a formula called F which states
(F) "There is no system C proof for the son of the system C formula with reference # n ".
Here n is a variable. So the system C formula with reference # n exists, call that system C formula XX, and F is the statement that the system C son of XX, SonXX, is not a theorem in system C).
F is a string of symbols of system C.The son of F is obtained by taking the reference number (an actual whole number like 112278 etc) for F, writing that whole number as the system C string //.../ (with reference # of F many /'s appearing) and replacing each occurrence the variable n in (F) with that many ///.../.
The result is a new system C formula, the son of F, called G.
The 'standard model' interpretation of G is then
(G) "There is no system C proof for the son of the system C formula with reference # equal to the reference # of F."
In other words, there is no system C proof of G itself.
Then Gensler and other sources indicate this standard interpretation of G is true, so it is a true 'ordinary arithmetic' statement, for which the corresponding system C formula has no proof.
It seems to me that what you can really say is this:
If you assume that the standard interpretation of G is a true statement in the ordinary standard model, then the corresponding system C formula is an unprovable system C formula. So you have a 'true' ordinary arith statement which cannot be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is true.
If you assume that the standard interpretation of G is a false statement in the ordinary standard model, then the corresponding system C formula is an provable system C formula.So you have a 'false' ordinary arith statement which can be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is false.
But how do you know that one of the two statements " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" must be valid?
Isn't the best you can say that IF you could determine that G were a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" THEN you can draw conclusions about non provability or provability of G as a system C formula?
logic incompleteness provability
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
asked Jan 23 at 2:51
grell6954grell6954
143
$endgroup$
|
show 1 more comment
$begingroup$
I know this question has been investigated in other threads, but I would like to pose yet another question on Gödel's theorem incompleteness, and truth in 'the standard model' compared with provability in the 'formal model'.
I will use the language of Gensler's little book "Gödel's Theorem Simplified" (see here for a summary).
For what Gensler calls System C, he gives an outline for how to construct a wff in system C. The system C the symbols are /, (, ), - , n (for a variable), nn (another variable),...etc.).
The reference numbers for system C formulas (like Godel numbers) are defined too.
For example the ref # for / is 1, the reference # for the variable n is 8, etc.
Then Gensler gives instructions for how to create a formula called F which states
(F) "There is no system C proof for the son of the system C formula with reference # n ".
Here n is a variable. So the system C formula with reference # n exists, call that system C formula XX, and F is the statement that the system C son of XX, SonXX, is not a theorem in system C).
F is a string of symbols of system C.The son of F is obtained by taking the reference number (an actual whole number like 112278 etc) for F, writing that whole number as the system C string //.../ (with reference # of F many /'s appearing) and replacing each occurrence the variable n in (F) with that many ///.../.
The result is a new system C formula, the son of F, called G.
The 'standard model' interpretation of G is then
(G) "There is no system C proof for the son of the system C formula with reference # equal to the reference # of F."
In other words, there is no system C proof of G itself.
Then Gensler and other sources indicate this standard interpretation of G is true, so it is a true 'ordinary arithmetic' statement, for which the corresponding system C formula has no proof.
It seems to me that what you can really say is this:
If you assume that the standard interpretation of G is a true statement in the ordinary standard model, then the corresponding system C formula is an unprovable system C formula. So you have a 'true' ordinary arith statement which cannot be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is true.
If you assume that the standard interpretation of G is a false statement in the ordinary standard model, then the corresponding system C formula is an provable system C formula.So you have a 'false' ordinary arith statement which can be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is false.
But how do you know that one of the two statements " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" must be valid?
Isn't the best you can say that IF you could determine that G were a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" THEN you can draw conclusions about non provability or provability of G as a system C formula?
logic incompleteness provability
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
asked Jan 23 at 2:51
grell6954grell6954
143
$endgroup$
1
$begingroup$
The assumption is a part of the theorem itself: Gödel's First Incompleteness Theorem states that, for a model with sufficient "power" (can represent arithmetic), that its consistency implies its incompleteness. It stands to reason, then, that one would assume G to be true. If you wish to prove the opposite (that completeness implies inconsistency), then you would assume that a proof of G exists, thus rendering it false, and your model inconsistent. Now, the theorem does not actually require consistent as an assumption- weaker assumptions work- but that is the one your mentioned book uses.
$endgroup$
– Ispil
Jan 23 at 6:32
$begingroup$
You can see the lenghty post Is PA consistent? do we know it?
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 7:32
$begingroup$
Gödel's Incompleteness Theorems is : "if [system] $C$ is consistent, then [formula] $G$ is unprovable". But also $lnot G$ is unprovable. The two are sentences of $C$ and thus - for every model $mathcal M$ of $C$ - exactly one of them (it does not matter what one) - must be TRUE in $mathcal M$ .
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 8:00
$begingroup$
Thanks for these two replies, which I do find helpful.
$endgroup$
– grell6954
Jan 23 at 15:43
$begingroup$
Here is the way they have helped me understand the theorem. THE KEY IS THAT GODEL’S IMCOMPLETENESS THM SAYS IF A MODEL OF ARITH (LIKE THE STANDARD ONE) IS CONSISTENT THEN IT MUST BE INCOMPLETE. IN THE GODEL/ GENSLER MODEL: ASSUME STANDARD ARITH IS CONSISTENT. IF IT IS ALSO COMPLETE THEN ONE OF THESE MUST BE TRUE: " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" (and consistency implies exactly 1 must be true). So consistency makes completeness impossible and completeness means 1 of these MUST be true.
$endgroup$
– grell6954
Jan 23 at 15:52
|
show 1 more comment
$begingroup$
I know this question has been investigated in other threads, but I would like to pose yet another question on Gödel's theorem incompleteness, and truth in 'the standard model' compared with provability in the 'formal model'.
I will use the language of Gensler's little book "Gödel's Theorem Simplified" (see here for a summary).
For what Gensler calls System C, he gives an outline for how to construct a wff in system C. The system C the symbols are /, (, ), - , n (for a variable), nn (another variable),...etc.).
The reference numbers for system C formulas (like Godel numbers) are defined too.
For example the ref # for / is 1, the reference # for the variable n is 8, etc.
Then Gensler gives instructions for how to create a formula called F which states
(F) "There is no system C proof for the son of the system C formula with reference # n ".
Here n is a variable. So the system C formula with reference # n exists, call that system C formula XX, and F is the statement that the system C son of XX, SonXX, is not a theorem in system C).
F is a string of symbols of system C.The son of F is obtained by taking the reference number (an actual whole number like 112278 etc) for F, writing that whole number as the system C string //.../ (with reference # of F many /'s appearing) and replacing each occurrence the variable n in (F) with that many ///.../.
The result is a new system C formula, the son of F, called G.
The 'standard model' interpretation of G is then
(G) "There is no system C proof for the son of the system C formula with reference # equal to the reference # of F."
In other words, there is no system C proof of G itself.
Then Gensler and other sources indicate this standard interpretation of G is true, so it is a true 'ordinary arithmetic' statement, for which the corresponding system C formula has no proof.
It seems to me that what you can really say is this:
If you assume that the standard interpretation of G is a true statement in the ordinary standard model, then the corresponding system C formula is an unprovable system C formula. So you have a 'true' ordinary arith statement which cannot be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is true.
If you assume that the standard interpretation of G is a false statement in the ordinary standard model, then the corresponding system C formula is an provable system C formula.So you have a 'false' ordinary arith statement which can be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is false.
But how do you know that one of the two statements " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" must be valid?
Isn't the best you can say that IF you could determine that G were a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" THEN you can draw conclusions about non provability or provability of G as a system C formula?
logic incompleteness provability
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
asked Jan 23 at 2:51
grell6954grell6954
143
$endgroup$
I know this question has been investigated in other threads, but I would like to pose yet another question on Gödel's theorem incompleteness, and truth in 'the standard model' compared with provability in the 'formal model'.
I will use the language of Gensler's little book "Gödel's Theorem Simplified" (see here for a summary).
For what Gensler calls System C, he gives an outline for how to construct a wff in system C. The system C the symbols are /, (, ), - , n (for a variable), nn (another variable),...etc.).
The reference numbers for system C formulas (like Godel numbers) are defined too.
For example the ref # for / is 1, the reference # for the variable n is 8, etc.
Then Gensler gives instructions for how to create a formula called F which states
(F) "There is no system C proof for the son of the system C formula with reference # n ".
Here n is a variable. So the system C formula with reference # n exists, call that system C formula XX, and F is the statement that the system C son of XX, SonXX, is not a theorem in system C).
F is a string of symbols of system C.The son of F is obtained by taking the reference number (an actual whole number like 112278 etc) for F, writing that whole number as the system C string //.../ (with reference # of F many /'s appearing) and replacing each occurrence the variable n in (F) with that many ///.../.
The result is a new system C formula, the son of F, called G.
The 'standard model' interpretation of G is then
(G) "There is no system C proof for the son of the system C formula with reference # equal to the reference # of F."
In other words, there is no system C proof of G itself.
Then Gensler and other sources indicate this standard interpretation of G is true, so it is a true 'ordinary arithmetic' statement, for which the corresponding system C formula has no proof.
It seems to me that what you can really say is this:
If you assume that the standard interpretation of G is a true statement in the ordinary standard model, then the corresponding system C formula is an unprovable system C formula. So you have a 'true' ordinary arith statement which cannot be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is true.
If you assume that the standard interpretation of G is a false statement in the ordinary standard model, then the corresponding system C formula is an provable system C formula.So you have a 'false' ordinary arith statement which can be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is false.
But how do you know that one of the two statements " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" must be valid?
Isn't the best you can say that IF you could determine that G were a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" THEN you can draw conclusions about non provability or provability of G as a system C formula?
logic incompleteness provability
logic incompleteness provability
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
asked Jan 23 at 2:51
grell6954grell6954
143
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
asked Jan 23 at 2:51
grell6954grell6954
143
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
edited Jan 23 at 5:50
Ruggiero Rilievi
179112
179112
asked Jan 23 at 2:51
grell6954grell6954
143
asked Jan 23 at 2:51
grell6954grell6954
143
asked Jan 23 at 2:51
grell6954grell6954
143
143
1
$begingroup$
The assumption is a part of the theorem itself: Gödel's First Incompleteness Theorem states that, for a model with sufficient "power" (can represent arithmetic), that its consistency implies its incompleteness. It stands to reason, then, that one would assume G to be true. If you wish to prove the opposite (that completeness implies inconsistency), then you would assume that a proof of G exists, thus rendering it false, and your model inconsistent. Now, the theorem does not actually require consistent as an assumption- weaker assumptions work- but that is the one your mentioned book uses.
$endgroup$
– Ispil
Jan 23 at 6:32
$begingroup$
You can see the lenghty post Is PA consistent? do we know it?
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 7:32
$begingroup$
Gödel's Incompleteness Theorems is : "if [system] $C$ is consistent, then [formula] $G$ is unprovable". But also $lnot G$ is unprovable. The two are sentences of $C$ and thus - for every model $mathcal M$ of $C$ - exactly one of them (it does not matter what one) - must be TRUE in $mathcal M$ .
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 8:00
$begingroup$
Thanks for these two replies, which I do find helpful.
$endgroup$
– grell6954
Jan 23 at 15:43
$begingroup$
Here is the way they have helped me understand the theorem. THE KEY IS THAT GODEL’S IMCOMPLETENESS THM SAYS IF A MODEL OF ARITH (LIKE THE STANDARD ONE) IS CONSISTENT THEN IT MUST BE INCOMPLETE. IN THE GODEL/ GENSLER MODEL: ASSUME STANDARD ARITH IS CONSISTENT. IF IT IS ALSO COMPLETE THEN ONE OF THESE MUST BE TRUE: " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" (and consistency implies exactly 1 must be true). So consistency makes completeness impossible and completeness means 1 of these MUST be true.
$endgroup$
– grell6954
Jan 23 at 15:52
|
show 1 more comment
1
$begingroup$
The assumption is a part of the theorem itself: Gödel's First Incompleteness Theorem states that, for a model with sufficient "power" (can represent arithmetic), that its consistency implies its incompleteness. It stands to reason, then, that one would assume G to be true. If you wish to prove the opposite (that completeness implies inconsistency), then you would assume that a proof of G exists, thus rendering it false, and your model inconsistent. Now, the theorem does not actually require consistent as an assumption- weaker assumptions work- but that is the one your mentioned book uses.
$endgroup$
– Ispil
Jan 23 at 6:32
$begingroup$
You can see the lenghty post Is PA consistent? do we know it?
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 7:32
$begingroup$
Gödel's Incompleteness Theorems is : "if [system] $C$ is consistent, then [formula] $G$ is unprovable". But also $lnot G$ is unprovable. The two are sentences of $C$ and thus - for every model $mathcal M$ of $C$ - exactly one of them (it does not matter what one) - must be TRUE in $mathcal M$ .
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 8:00
$begingroup$
Thanks for these two replies, which I do find helpful.
$endgroup$
– grell6954
Jan 23 at 15:43
$begingroup$
Here is the way they have helped me understand the theorem. THE KEY IS THAT GODEL’S IMCOMPLETENESS THM SAYS IF A MODEL OF ARITH (LIKE THE STANDARD ONE) IS CONSISTENT THEN IT MUST BE INCOMPLETE. IN THE GODEL/ GENSLER MODEL: ASSUME STANDARD ARITH IS CONSISTENT. IF IT IS ALSO COMPLETE THEN ONE OF THESE MUST BE TRUE: " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" (and consistency implies exactly 1 must be true). So consistency makes completeness impossible and completeness means 1 of these MUST be true.
$endgroup$
– grell6954
Jan 23 at 15:52
1
1
$begingroup$
The assumption is a part of the theorem itself: Gödel's First Incompleteness Theorem states that, for a model with sufficient "power" (can represent arithmetic), that its consistency implies its incompleteness. It stands to reason, then, that one would assume G to be true. If you wish to prove the opposite (that completeness implies inconsistency), then you would assume that a proof of G exists, thus rendering it false, and your model inconsistent. Now, the theorem does not actually require consistent as an assumption- weaker assumptions work- but that is the one your mentioned book uses.
$endgroup$
– Ispil
Jan 23 at 6:32
$begingroup$
The assumption is a part of the theorem itself: Gödel's First Incompleteness Theorem states that, for a model with sufficient "power" (can represent arithmetic), that its consistency implies its incompleteness. It stands to reason, then, that one would assume G to be true. If you wish to prove the opposite (that completeness implies inconsistency), then you would assume that a proof of G exists, thus rendering it false, and your model inconsistent. Now, the theorem does not actually require consistent as an assumption- weaker assumptions work- but that is the one your mentioned book uses.
$endgroup$
– Ispil
Jan 23 at 6:32
$begingroup$
You can see the lenghty post Is PA consistent? do we know it?
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 7:32
$begingroup$
You can see the lenghty post Is PA consistent? do we know it?
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 7:32
$begingroup$
Gödel's Incompleteness Theorems is : "if [system] $C$ is consistent, then [formula] $G$ is unprovable". But also $lnot G$ is unprovable. The two are sentences of $C$ and thus - for every model $mathcal M$ of $C$ - exactly one of them (it does not matter what one) - must be TRUE in $mathcal M$ .
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 8:00
$begingroup$
Gödel's Incompleteness Theorems is : "if [system] $C$ is consistent, then [formula] $G$ is unprovable". But also $lnot G$ is unprovable. The two are sentences of $C$ and thus - for every model $mathcal M$ of $C$ - exactly one of them (it does not matter what one) - must be TRUE in $mathcal M$ .
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 8:00
$begingroup$
Thanks for these two replies, which I do find helpful.
$endgroup$
– grell6954
Jan 23 at 15:43
$begingroup$
Thanks for these two replies, which I do find helpful.
$endgroup$
– grell6954
Jan 23 at 15:43
$begingroup$
Here is the way they have helped me understand the theorem. THE KEY IS THAT GODEL’S IMCOMPLETENESS THM SAYS IF A MODEL OF ARITH (LIKE THE STANDARD ONE) IS CONSISTENT THEN IT MUST BE INCOMPLETE. IN THE GODEL/ GENSLER MODEL: ASSUME STANDARD ARITH IS CONSISTENT. IF IT IS ALSO COMPLETE THEN ONE OF THESE MUST BE TRUE: " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" (and consistency implies exactly 1 must be true). So consistency makes completeness impossible and completeness means 1 of these MUST be true.
$endgroup$
– grell6954
Jan 23 at 15:52
$begingroup$
Here is the way they have helped me understand the theorem. THE KEY IS THAT GODEL’S IMCOMPLETENESS THM SAYS IF A MODEL OF ARITH (LIKE THE STANDARD ONE) IS CONSISTENT THEN IT MUST BE INCOMPLETE. IN THE GODEL/ GENSLER MODEL: ASSUME STANDARD ARITH IS CONSISTENT. IF IT IS ALSO COMPLETE THEN ONE OF THESE MUST BE TRUE: " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" (and consistency implies exactly 1 must be true). So consistency makes completeness impossible and completeness means 1 of these MUST be true.
$endgroup$
– grell6954
Jan 23 at 15:52
|
show 1 more 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%2f3084013%2fgodels-theorem-incompleteness-truth-vs-provability%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.
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%2f3084013%2fgodels-theorem-incompleteness-truth-vs-provability%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
1
$begingroup$
The assumption is a part of the theorem itself: Gödel's First Incompleteness Theorem states that, for a model with sufficient "power" (can represent arithmetic), that its consistency implies its incompleteness. It stands to reason, then, that one would assume G to be true. If you wish to prove the opposite (that completeness implies inconsistency), then you would assume that a proof of G exists, thus rendering it false, and your model inconsistent. Now, the theorem does not actually require consistent as an assumption- weaker assumptions work- but that is the one your mentioned book uses.
$endgroup$
– Ispil
Jan 23 at 6:32
$begingroup$
You can see the lenghty post Is PA consistent? do we know it?
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 7:32
$begingroup$
Gödel's Incompleteness Theorems is : "if [system] $C$ is consistent, then [formula] $G$ is unprovable". But also $lnot G$ is unprovable. The two are sentences of $C$ and thus - for every model $mathcal M$ of $C$ - exactly one of them (it does not matter what one) - must be TRUE in $mathcal M$ .
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 8:00
$begingroup$
Thanks for these two replies, which I do find helpful.
$endgroup$
– grell6954
Jan 23 at 15:43
$begingroup$
Here is the way they have helped me understand the theorem. THE KEY IS THAT GODEL’S IMCOMPLETENESS THM SAYS IF A MODEL OF ARITH (LIKE THE STANDARD ONE) IS CONSISTENT THEN IT MUST BE INCOMPLETE. IN THE GODEL/ GENSLER MODEL: ASSUME STANDARD ARITH IS CONSISTENT. IF IT IS ALSO COMPLETE THEN ONE OF THESE MUST BE TRUE: " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" (and consistency implies exactly 1 must be true). So consistency makes completeness impossible and completeness means 1 of these MUST be true.
$endgroup$
– grell6954
Jan 23 at 15:52