Unprovability.

[Antonio Speltzu]

Is unprovability expressible in P, representable in P or what is it?

[Rupert]

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Consider the property of natural numbers which holds of a given natural number n if and only if n is the Gödel number of a formula which is unprovable in P.

This property is *expressible in the language of P* in the following sense. I can write down a formula with one free variable of type 0 in the language of P, and I can say, relative to the *standard semantics for P* (which can only be formally treated in full generality in a slightly stronger metatheory, Zermelo set theory for example) that for all natural numbers n, this formula is true (in the standard model) under the interpretation which assigns n to the free variable, if and only if n indeed has the property in question. Actually writing down this formula is feasible but labour-intensive, and Gödel gives detailed instructions for how to do in his paper (see Definitions 1-46 together with outline of proof of Proposition V).

I remarked that the full semantics for the language of P can only adequately be treated in Zermelo set theory, but the semantics for the first-order language of arithmetic would be adequate to our purposes here, and that can be formally treated within P itself, or indeed in second-order arithmetic. The details of how to do this were later elaborated on by Tarski. The details of how to formally treat semantics in higher-order languages are not outlined by Gödel in his 1931 paper, but a footnote makes clear that he was aware of how to do it at the time of writing the paper, prior to the appearance of Tarski's publications on the topic. In any case the details are well-known now.

Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.

So, yes the property is *expressible in the language of P* in the above sense. But it is not true that this property is *strongly representable in P*. It is not true that there exists a formula s(x) in the language of P, with one free variable x of type 0, with the property that, for all natural numbers n, if the property holds of n then s(Z(n)) is provable in P, and if the property does not holds of n then ~s(Z(n)) is provable in P. Because only properties whose extension is a recursive set can possibly be strongly representable in P. And this property has an extension which is the complement of a recursively enumerable set which is not recursive. So for that reason the property has an extension which is not a recursive set. Therefore it is not strongly representable in P.

So, the property is expressible in the language of P, in the sense given above, but not strongly representable in P according the standard meaning of "strongly representable", for reasons explained just above.

So there you go. Glad to be of help.

[Antonio Speltzu]

Post by Rupert

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Consider the property of natural numbers which holds of a given natural number n if and only if n is the Gödel number of a formula which is unprovable in P.
This property is *expressible in the language of P* in the following sense. I can write down a formula with one free variable of type 0 in the language of P, and I can say, relative to the *standard semantics for P* (which can only be formally treated in full generality in a slightly stronger metatheory, Zermelo set theory for example) that for all natural numbers n, this formula is true (in the standard model) under the interpretation which assigns n to the free variable, if and only if n indeed has the property in question. Actually writing down this formula is feasible but labour-intensive, and Gödel gives detailed instructions for how to do in his paper (see Definitions 1-46 together with outline of proof of Proposition V).
I remarked that the full semantics for the language of P can only adequately be treated in Zermelo set theory, but the semantics for the first-order language of arithmetic would be adequate to our purposes here, and that can be formally treated within P itself, or indeed in second-order arithmetic. The details of how to do this were later elaborated on by Tarski. The details of how to formally treat semantics in higher-order languages are not outlined by Gödel in his 1931 paper, but a footnote makes clear that he was aware of how to do it at the time of writing the paper, prior to the appearance of Tarski's publications on the topic. In any case the details are well-known now.
Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.
So, yes the property is *expressible in the language of P* in the above sense. But it is not true that this property is *strongly representable in P*. It is not true that there exists a formula s(x) in the language of P, with one free variable x of type 0, with the property that, for all natural numbers n, if the property holds of n then s(Z(n)) is provable in P, and if the property does not holds of n then ~s(Z(n)) is provable in P. Because only properties whose extension is a recursive set can possibly be strongly representable in P. And this property has an extension which is the complement of a recursively enumerable set which is not recursive. So for that reason the property has an extension which is not a recursive set. Therefore it is not strongly representable in P.
So, the property is expressible in the language of P, in the sense given above, but not strongly representable in P according the standard meaning of "strongly representable", for reasons explained just above.
So there you go. Glad to be of help.

Thanks Rupert, your help is priceless.

[Rupert]

Post by Antonio Speltzu
Thanks Rupert, your help is priceless.

Well, whatever else might be said, there can certainly be no doubt that it's unpaid. So "priceless" in that sense, yes.

Am I to understand that you don't really value my efforts to communicate to you what I take to be the truth all that much? Because, after all, at the end of the day I'm just engaging in recreational activity to while away an idle moment. I definitely *honestly believe* that I'm offering you important truths which you would do well to carefully consider and try to understand, but if you're not convinced of that and have no intention of taking me seriously, then hey, I can quite easily find other stuff to do while I'm having a bit of a break from trying to get useful stuff done. You just think I don't really know what I'm talking about? Well fine. I can always shut up and go away. You don't care one way or the other?

[Antonio Speltzu]

Post by Rupert

Post by Antonio Speltzu
Thanks Rupert, your help is priceless.

Well, whatever else might be said, there can certainly be no doubt that it's unpaid. So "priceless" in that sense, yes.
Am I to understand that you don't really value my efforts to communicate to you what I take to be the truth all that much? Because, after all, at the end of the day I'm just engaging in recreational activity to while away an idle moment. I definitely *honestly believe* that I'm offering you important truths which you would do well to carefully consider and try to understand, but if you're not convinced of that and have no intention of taking me seriously, then hey, I can quite easily find other stuff to do while I'm having a bit of a break from trying to get useful stuff done. You just think I don't really know what I'm talking about? Well fine. I can always shut up and go away. You don't care one way or the other?

You misunderstand me, in my land that is a compliment, it means that something is so valuable that it is difficult to put a price on it.

[Rupert]

Post by Antonio Speltzu

Post by Rupert

Post by Antonio Speltzu
Thanks Rupert, your help is priceless.

Well, whatever else might be said, there can certainly be no doubt that it's unpaid. So "priceless" in that sense, yes.
Am I to understand that you don't really value my efforts to communicate to you what I take to be the truth all that much? Because, after all, at the end of the day I'm just engaging in recreational activity to while away an idle moment. I definitely *honestly believe* that I'm offering you important truths which you would do well to carefully consider and try to understand, but if you're not convinced of that and have no intention of taking me seriously, then hey, I can quite easily find other stuff to do while I'm having a bit of a break from trying to get useful stuff done. You just think I don't really know what I'm talking about? Well fine. I can always shut up and go away. You don't care one way or the other?

You misunderstand me, in my land that is a compliment, it means that something is so valuable that it is difficult to put a price on it.

Oh okay. Sorry to get annoyed then, didn't mean to misunderstand.

"Priceless" sometimes means that in English but it also can mean "funny".

Sorry about the difficulties in communicating. And yeah sure, happy to help.

[Khong Dong]

Post by Rupert

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.

So what's your "reasonable hourly rate" to write down one of the two statements (which each should take about 30 seconds to write)

- TRUE(N |= (G(PA) ⋀ cGC))
- FALSE(N |= (G(PA) ⋀ cGC))

where N is the so called "standard" arithmetic structure?

[Khong Dong]

Post by Khong Dong

Post by Rupert

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.

So what's your "reasonable hourly rate" to write down one of the two statements (which each should take about 30 seconds to write)
- TRUE(N |= (G(PA) ⋀ cGC))
- FALSE(N |= (G(PA) ⋀ cGC))
where N is the so called "standard" arithmetic structure?

It's all about your being "pretty happy" with "full truth-determinateness for first-order language of arithmetic" of course.
(So far sci.logic hasn't seen any "full truth-determinateness" on any of the two statements to evaluate what would a "reasonable hourly rate" be yet.)

[Khong Dong]

Post by Khong Dong

Post by Khong Dong

Post by Rupert

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.

So what's your "reasonable hourly rate" to write down one of the two statements (which each should take about 30 seconds to write)
- TRUE(N |= (G(PA) ⋀ cGC))
- FALSE(N |= (G(PA) ⋀ cGC))
where N is the so called "standard" arithmetic structure?

It's all about your being "pretty happy" with "full truth-determinateness for first-order language of arithmetic" of course.
(So far sci.logic hasn't seen any "full truth-determinateness" on any of the two statements to evaluate what would a "reasonable hourly rate" be yet.)

Once, you advise us on your "reasonable hourly rate", Rupert, I'll will reflect on your "full truth-determinateness" and Professor Hamkins' apparent indefiniteness of arithmetic truth below (to see if I should spend some money for you writing down one of the two 30-second statements above):

<quote>

2. Indefinite arithmetic truth
Let us begin with what may seem naively to be a surprising case,
where we have two models of set theory with the same structure of
arithmetic <N,+, ·, 0, 1, <>, but different theories of arithmetic truth.

<quote>

(Note his seemingly epistemological "different theories of arithmetic truth".)

[Rupert]

Post by Khong Dong

Post by Khong Dong

Post by Khong Dong

Post by Rupert

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.

So what's your "reasonable hourly rate" to write down one of the two statements (which each should take about 30 seconds to write)
- TRUE(N |= (G(PA) ⋀ cGC))
- FALSE(N |= (G(PA) ⋀ cGC))
where N is the so called "standard" arithmetic structure?

It's all about your being "pretty happy" with "full truth-determinateness for first-order language of arithmetic" of course.
(So far sci.logic hasn't seen any "full truth-determinateness" on any of the two statements to evaluate what would a "reasonable hourly rate" be yet.)

<quote>
2. Indefinite arithmetic truth
Let us begin with what may seem naively to be a surprising case,
where we have two models of set theory with the same structure of
arithmetic <N,+, ·, 0, 1, <>, but different theories of arithmetic truth.
<quote>
(Note his seemingly epistemological "different theories of arithmetic truth".)

Yeah sure. Which task? The task of writing out the formula in P which strongly represents the property of unprovability? 100 euros per hour is fine for that one, and I could undertake to get the job done in no more than 20 hours.

Yeah sure, I know Hamkins' work. His concept of a multiverse where well-foundedness is not absolute is interesting indeed, and I attended a talk about mathematical topics related to that, given by Victoria Gitman, at a recent conference. It's all very good stuff, and I don't know what his take would be on full truth-determinateness for first-order language of arithmetic, I did actually already agree that that was a philosophically contested matter and that an expert could reasonably disagree, I stated that it was indeed my view and that I was open to having a debate about it. I didn't say that no well-informed person could possibly disagree. And I don't know what Hamkins' take is on this particular question, but I do know that he favours a multiverse view with regard to set theory, and also the view that truth-value of CH is indeterminate. He discussed that with me in verbal face-to-face conversation over lunch, while he was visiting Amsterdam to give a talk, in quite recent memory.

So yeah cool. Of course it's totally fine to advocate the view that first-order language of arithmetic is not fully truth-determinate. I'm just not fully convinced right now that you yourself have made a strong case for that view in your previous posts on this newsgroup, that's all. I'm not in any way ruling out that Hamkins might have had something interesting to say about it in his published work. I simply stated what my current view is, and said I was happy to debate it.

Now then. How else can I help?

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Khong Dong

Post by Khong Dong

Post by Rupert

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.

So what's your "reasonable hourly rate" to write down one of the two statements (which each should take about 30 seconds to write)
- TRUE(N |= (G(PA) ⋀ cGC))
- FALSE(N |= (G(PA) ⋀ cGC))
where N is the so called "standard" arithmetic structure?

It's all about your being "pretty happy" with "full truth-determinateness for first-order language of arithmetic" of course.
(So far sci.logic hasn't seen any "full truth-determinateness" on any of the two statements to evaluate what would a "reasonable hourly rate" be yet.)

<quote>
2. Indefinite arithmetic truth
Let us begin with what may seem naively to be a surprising case,
where we have two models of set theory with the same structure of
arithmetic <N,+, ·, 0, 1, <>, but different theories of arithmetic truth.
<quote>
(Note his seemingly epistemological "different theories of arithmetic truth".)

Yeah sure. Which task? The task of writing out the formula in P which strongly represents the property of unprovability?

No. Just write _one_ of the two (30-second-writting) sentences I wrote above, providing that (a) you have in your mind a proof for it and (b) allowing me to mention your name to MO saying that you can prove that sentence is true.

Post by Rupert
100 euros per hour is fine for that one, and I could undertake to get the job done in no more than 20 hours.

Sure. As stipulated, I don't mind paying you for your 30-second writing - with that rate. In fact I'd double that rate as a friend.

Post by Rupert
Yeah sure, I know Hamkins' work. His concept of a multiverse where well-foundedness is not absolute is interesting indeed, and I attended a talk about mathematical topics related to that, given by Victoria Gitman, at a recent conference. It's all very good stuff, and I don't know what his take would be on full truth-determinateness for first-order language of arithmetic,

You'd know - if you read his Theorem 1. in his paper immediately below the part I've quoted above.

Post by Rupert
I did actually already agree that that was a philosophically contested matter and that an expert could reasonably disagree, I stated that it was indeed my view and that I was open to having a debate about it. I didn't say that no well-informed person could possibly disagree. And I don't know what Hamkins' take is on this particular question, but I do know that he favours a multiverse view with regard to set theory, and also the view that truth-value of CH is indeterminate. He discussed that with me in verbal face-to-face conversation over lunch, while he was visiting Amsterdam to give a talk, in quite recent memory.

Last time I check, Hamkins' Theorem 1. is a _theorem_ with _proof_ and hence there isn't much room for one to wiggle "philosophy", theology "view" or the like.
In any rate, Hamkins' theorem on indefiniteness of arithmetic truth would directly contradict with your subjective "view" of "full truth-determinateness", although
his indefiniteness of arithmetic truth would be virtually the same as as my main thesis of general uncertainty of arithmetic truths in my "quantum_Mathematics.pdf":

https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing

Post by Rupert
So yeah cool. Of course it's totally fine to advocate the view that first-order language of arithmetic is not fully truth-determinate. I'm just not fully convinced right now that you yourself have made a strong case for that view in your previous posts on this newsgroup, that's all.

I did, in "quantum_Mathematics.pdf" mentioned above. Like many other papers (yours, Hamkins', ... included), it has certain amount of both English and technical formulae (formalization), but if you don't understand it and/or are unable to _correctly_ point out any error then that would be a different matter.

Post by Rupert
I'm not in any way ruling out that Hamkins might have had something interesting to say about it in his published work. I simply stated what my current view is, and said I was happy to debate it.

You can debate Hamkins' indefiniteness of arithmetic truth or my uncertainty of arithmetic truth: face value, they're genuinely different wording renditions of the same fact in mathematical logic (reasoning).

Post by Rupert
Now then. How else can I help?

Again, honestly admitting that as we speak you don't have "the standard structure" due to the truth-value uncertainty/indefiniteness of at least one specific formula sentence (say cGC). Can you help admitting that?

[Rupert]

Post by Khong Dong

Post by Rupert
Yeah sure. Which task? The task of writing out the formula in P which strongly represents the property of unprovability?

No. Just write _one_ of the two (30-second-writting) sentences I wrote above, providing that (a) you have in your mind a proof for it and (b) allowing me to mention your name to MO saying that you can prove that sentence is true.

First of all, those strings of symbols that you wrote above are not, in fact, unabbreviated sentences in any first-order language, and if what you have in mind is that they could be disabbreviated into sentences in a first-order language, I couldn't really say that I know right now which first-order language you have in mind. I think in both cases you probably mean to say something which I could express by writing down a sentence in the first-order language of arithmetic. With regard to *those* two sentences - and obviously I'd only be guessing that you would agree those are reasonable paraphrases of what you want to say, it's really very hard ever to know with you - the situation would be: I don't currently know how to prove either of those two sentences from any set of axioms which are widely regarded as justified. As of course you know perfectly well. Rather bizarre that for some reason I have to explain it to you all over again. But yeah.

Post by Khong Dong

Post by Rupert
100 euros per hour is fine for that one, and I could undertake to get the job done in no more than 20 hours.

Sure. As stipulated, I don't mind paying you for your 30-second writing - with that rate. In fact I'd double that rate as a friend.

But now that we have clarified what task you are talking about here, of course you know perfectly well that I never said I was able to do that task, and you know that I am not in fact able to do it. This isn't any kind of surprise to you or any other reader of this post, and you haven't given any clear reason why there is anything noteworthy about that. So this is a rather stupid conversation.

Post by Khong Dong

Post by Rupert
Yeah sure, I know Hamkins' work. His concept of a multiverse where well-foundedness is not absolute is interesting indeed, and I attended a talk about mathematical topics related to that, given by Victoria Gitman, at a recent conference. It's all very good stuff, and I don't know what his take would be on full truth-determinateness for first-order language of arithmetic,

You'd know - if you read his Theorem 1. in his paper immediately below the part I've quoted above.

Possibly.

Post by Khong Dong

Post by Rupert
I did actually already agree that that was a philosophically contested matter and that an expert could reasonably disagree, I stated that it was indeed my view and that I was open to having a debate about it. I didn't say that no well-informed person could possibly disagree. And I don't know what Hamkins' take is on this particular question, but I do know that he favours a multiverse view with regard to set theory, and also the view that truth-value of CH is indeterminate. He discussed that with me in verbal face-to-face conversation over lunch, while he was visiting Amsterdam to give a talk, in quite recent memory.

Last time I check, Hamkins' Theorem 1. is a _theorem_ with _proof_ and hence there isn't much room for one to wiggle "philosophy", theology "view" or the like.

Can you give the citation, please? I'll take a look at this Theorem 1 of which you speak. If it called a theorem in a published work of Hamkins, then yes, in that case I have little doubt that it can indeed be proved. Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

Post by Khong Dong
In any rate, Hamkins' theorem on indefiniteness of arithmetic truth would directly contradict with your subjective "view" of "full truth-determinateness", although

Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic. I also confidently predict that it does not in any reasonable sense have the slightest degree of resemblance or commonality of content with anything you have ever written.

Post by Khong Dong
https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing

Post by Rupert
So yeah cool. Of course it's totally fine to advocate the view that first-order language of arithmetic is not fully truth-determinate. I'm just not fully convinced right now that you yourself have made a strong case for that view in your previous posts on this newsgroup, that's all.

I did, in "quantum_Mathematics.pdf" mentioned above. Like many other papers (yours, Hamkins', ... included), it has certain amount of both English and technical formulae (formalization), but if you don't understand it and/or are unable to _correctly_ point out any error then that would be a different matter.

I'll write a peer-reviewer's report of it for you for 5000 euros if you're interested in that.

Post by Khong Dong

Post by Rupert
I'm not in any way ruling out that Hamkins might have had something interesting to say about it in his published work. I simply stated what my current view is, and said I was happy to debate it.

You can debate Hamkins' indefiniteness of arithmetic truth or my uncertainty of arithmetic truth: face value, they're genuinely different wording renditions of the same fact in mathematical logic (reasoning).

I can do anything I want to, can't I. So yeah, what's the title of this paper of Hamkins?

Post by Khong Dong

Post by Rupert
Now then. How else can I help?

Again, honestly admitting that as we speak you don't have "the standard structure" due to the truth-value uncertainty/indefiniteness of at least one specific formula sentence (say cGC). Can you help admitting that?

My previous comments on that matter have been adequate.

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Rupert
Yeah sure. Which task? The task of writing out the formula in P which strongly represents the property of unprovability?

No. Just write _one_ of the two (30-second-writting) sentences I wrote above, providing that (a) you have in your mind a proof for it and (b) allowing me to mention your name to MO saying that you can prove that sentence is true.

First of all, those strings of symbols that you wrote above are not, in fact, unabbreviated sentences in any first-order language, and if what you have in mind is that they could be disabbreviated into sentences in a first-order language, I couldn't really say that I know right now which first-order language you have in mind. I think in both cases you probably mean to say something which I could express by writing down a sentence in the first-order language of arithmetic. With regard to *those* two sentences - and obviously I'd only be guessing that you would agree those are reasonable paraphrases of what you want to say, it's really very hard ever to know with you - the situation would be: I don't currently know how to prove either of those two sentences from any set of axioms which are widely regarded as justified. As of course you know perfectly well. Rather bizarre that for some reason I have to explain it to you all over again. But yeah.

Post by Khong Dong

Post by Rupert
100 euros per hour is fine for that one, and I could undertake to get the job done in no more than 20 hours.

Sure. As stipulated, I don't mind paying you for your 30-second writing - with that rate. In fact I'd double that rate as a friend.

But now that we have clarified what task you are talking about here, of course you know perfectly well that I never said I was able to do that task, and you know that I am not in fact able to do it. This isn't any kind of surprise to you or any other reader of this post, and you haven't given any clear reason why there is anything noteworthy about that.

I did. (You've ignored it).

Post by Rupert
So this is a rather stupid conversation.

The voice of one who was on drug or possessed by an inquisition-spirit.

Post by Rupert

Post by Khong Dong

Post by Rupert
Yeah sure, I know Hamkins' work. His concept of a multiverse where well-foundedness is not absolute is interesting indeed, and I attended a talk about mathematical topics related to that, given by Victoria Gitman, at a recent conference. It's all very good stuff, and I don't know what his take would be on full truth-determinateness for first-order language of arithmetic,

You'd know - if you read his Theorem 1. in his paper immediately below the part I've quoted above.

Possibly.

Post by Khong Dong

Post by Rupert
I did actually already agree that that was a philosophically contested matter and that an expert could reasonably disagree, I stated that it was indeed my view and that I was open to having a debate about it. I didn't say that no well-informed person could possibly disagree. And I don't know what Hamkins' take is on this particular question, but I do know that he favours a multiverse view with regard to set theory, and also the view that truth-value of CH is indeterminate. He discussed that with me in verbal face-to-face conversation over lunch, while he was visiting Amsterdam to give a talk, in quite recent memory.

Last time I check, Hamkins' Theorem 1. is a _theorem_ with _proof_ and hence there isn't much room for one to wiggle "philosophy", theology "view" or the like.

Can you give the citation, please? I'll take a look at this Theorem 1 of which you speak. If it called a theorem in a published work of Hamkins, then yes, in that case I have little doubt that it can indeed be proved.

Alluding to Theorem 1, his "Abstract" -- in https://arxiv.org/pdf/1312.0670.pdf -- starts with:

"We prove that the satisfaction relation N |= ϕ[a ⃗ ] of
first-order logic is not absolute between models of set theory having
the structure N and the formulas ϕ all in common."

Note his "We prove".

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Post by Rupert

Post by Khong Dong
In any rate, Hamkins' theorem on indefiniteness of arithmetic truth would directly contradict with your subjective "view" of "full truth-determinateness", although

Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

Post by Rupert
I also confidently predict that it does not in any reasonable sense have the slightest degree of resemblance or commonality of content with anything you have ever written.

Except that "Trust but verify" and so far you've had zero verification of your utterance. Don't just "confidently predict" - post after post: show precisely one problem of my paper that would lead to contradiction to, e.g., Hamkins' Theorem 1. Can you show that? Or would it just be "garbage"/"stupid" (your words) ramblings you're going to continue?

Post by Rupert

Post by Khong Dong
https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing

Post by Rupert
So yeah cool. Of course it's totally fine to advocate the view that first-order language of arithmetic is not fully truth-determinate. I'm just not fully convinced right now that you yourself have made a strong case for that view in your previous posts on this newsgroup, that's all.

Again, Hamkins' paper begins with "We prove", not "We view".

(And again my paper is in https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing).

[Khong Dong]

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Yeah sure. Which task? The task of writing out the formula in P which strongly represents the property of unprovability?

No. Just write _one_ of the two (30-second-writting) sentences I wrote above, providing that (a) you have in your mind a proof for it and (b) allowing me to mention your name to MO saying that you can prove that sentence is true.

First of all, those strings of symbols that you wrote above are not, in fact, unabbreviated sentences in any first-order language, and if what you have in mind is that they could be disabbreviated into sentences in a first-order language, I couldn't really say that I know right now which first-order language you have in mind. I think in both cases you probably mean to say something which I could express by writing down a sentence in the first-order language of arithmetic. With regard to *those* two sentences - and obviously I'd only be guessing that you would agree those are reasonable paraphrases of what you want to say, it's really very hard ever to know with you - the situation would be: I don't currently know how to prove either of those two sentences from any set of axioms which are widely regarded as justified. As of course you know perfectly well. Rather bizarre that for some reason I have to explain it to you all over again. But yeah.

Post by Khong Dong

Post by Rupert
100 euros per hour is fine for that one, and I could undertake to get the job done in no more than 20 hours.

Sure. As stipulated, I don't mind paying you for your 30-second writing - with that rate. In fact I'd double that rate as a friend.

But now that we have clarified what task you are talking about here, of course you know perfectly well that I never said I was able to do that task, and you know that I am not in fact able to do it. This isn't any kind of surprise to you or any other reader of this post, and you haven't given any clear reason why there is anything noteworthy about that.

I did. (You've ignored it).

Post by Rupert
So this is a rather stupid conversation.

The voice of one who was on drug or possessed by an inquisition-spirit.

Post by Rupert

Post by Khong Dong

Post by Rupert
Yeah sure, I know Hamkins' work. His concept of a multiverse where well-foundedness is not absolute is interesting indeed, and I attended a talk about mathematical topics related to that, given by Victoria Gitman, at a recent conference. It's all very good stuff, and I don't know what his take would be on full truth-determinateness for first-order language of arithmetic,

You'd know - if you read his Theorem 1. in his paper immediately below the part I've quoted above.

Possibly.

Post by Khong Dong

Post by Rupert
I did actually already agree that that was a philosophically contested matter and that an expert could reasonably disagree, I stated that it was indeed my view and that I was open to having a debate about it. I didn't say that no well-informed person could possibly disagree. And I don't know what Hamkins' take is on this particular question, but I do know that he favours a multiverse view with regard to set theory, and also the view that truth-value of CH is indeterminate. He discussed that with me in verbal face-to-face conversation over lunch, while he was visiting Amsterdam to give a talk, in quite recent memory.

Last time I check, Hamkins' Theorem 1. is a _theorem_ with _proof_ and hence there isn't much room for one to wiggle "philosophy", theology "view" or the like.

Can you give the citation, please? I'll take a look at this Theorem 1 of which you speak. If it called a theorem in a published work of Hamkins, then yes, in that case I have little doubt that it can indeed be proved.

"We prove that the satisfaction relation N |= ϕ[a ⃗ ] of
first-order logic is not absolute between models of set theory having
the structure N and the formulas ϕ all in common."
Note his "We prove".

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Post by Rupert

Post by Khong Dong
In any rate, Hamkins' theorem on indefiniteness of arithmetic truth would directly contradict with your subjective "view" of "full truth-determinateness", although

Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

Post by Rupert
I also confidently predict that it does not in any reasonable sense have the slightest degree of resemblance or commonality of content with anything you have ever written.

Except that "Trust but verify" and so far you've had zero verification of your utterance. Don't just "confidently predict" - post after post: show precisely one problem of my paper that would lead to contradiction to, e.g., Hamkins' Theorem 1. Can you show that? Or would it just be "garbage"/"stupid" (your words) ramblings you're going to continue?

Post by Rupert

Post by Khong Dong
https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing

Post by Rupert
So yeah cool. Of course it's totally fine to advocate the view that first-order language of arithmetic is not fully truth-determinate. I'm just not fully convinced right now that you yourself have made a strong case for that view in your previous posts on this newsgroup, that's all.

Again, Hamkins' paper begins with "We prove", not "We view".
(And again my paper is in https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing).

Although you were shown multiple times Gödel's encoding and definition of primes are invalid, Rupert, for a die-hard fan of Gödel's incompleteness "theorems", I thought you might be interested in how Hamkins' paper would distance itself from Incompleteness on the matters of indefiniteness/uncertainty of arithmetic truths in general:

<quote>

This is a stronger kind of non-absoluteness phenomenon than
the usual observation, via the incompleteness theorem, that models of
set theory can disagree on arithmetic truth, for here we have models
of set theory, which disagree about arithmetic truth, yet agree on the
structure in which that truth resides.

</quote>

So, pick your choice and go ahead and show that your "full truth-determinateness" -- antonym of Theorem 1 -- is provable, or the proof of Theorem 1 is invalid.

[Rupert]

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

Post by Khong Dong

Post by Rupert
Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

My statement does not contradict his Theorem 1, as explained above.

Post by Khong Dong

Post by Rupert
I also confidently predict that it does not in any reasonable sense have the slightest degree of resemblance or commonality of content with anything you have ever written.

Except that "Trust but verify" and so far you've had zero verification of your utterance. Don't just "confidently predict" - post after post: show precisely one problem of my paper that would lead to contradiction to, e.g., Hamkins' Theorem 1. Can you show that? Or would it just be "garbage"/"stupid" (your words) ramblings you're going to continue?

Your paper? I think we said 5000 euros for me peer-reviewing your paper. I just looked at Hamkins' paper, and explained why his Theorem 1 does not contradict my statement.

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

So "a sentence in the language of arithmetic" is different from "a sentence in the first-order language of arithmetic"? Really?

It also says in there:

<quote>

in the sense that there is in M1 and M2 an
arithmetic sentence σ, such that M1 thinks σ is true, but M2 thinks it
is false.

M1 believes N |= σ
M2 believes N |= ¬σ

</quote>

So, is σ above "a sentence in the language of arithmetic" or "a sentence in the first-order language of arithmetic"?

Post by Rupert

Post by Khong Dong

Post by Rupert
Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

My statement does not contradict his Theorem 1, as explained above.

Wrong.

[Rupert]

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

So "a sentence in the language of arithmetic" is different from "a sentence in the first-order language of arithmetic"? Really?

It says "arithmetic sentence", doesn't it? And yes, different from "sentence in first-order language of arithmetic". Yes, really.

Post by Khong Dong
<quote>
in the sense that there is in M1 and M2 an
arithmetic sentence σ, such that M1 thinks σ is true, but M2 thinks it
is false.
M1 believes N |= σ
M2 believes N |= ¬σ
</quote>

Yes, that's right. An arithmetic sentence being a sentence in the first order language of arithmetic *with parameters from the structure under consideration*. Which is different to what I meant.

Post by Khong Dong
So, is σ above "a sentence in the language of arithmetic" or "a sentence in the first-order language of arithmetic"?

Well, as noted Hamkins says "arithmetic sentence", and I believe I have now typed out what he means by that enough times.

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

My statement does not contradict his Theorem 1, as explained above.

Wrong.

You sure?

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

So "a sentence in the language of arithmetic" is different from "a sentence in the first-order language of arithmetic"? Really?

It says "arithmetic sentence", doesn't it? And yes, different from "sentence in first-order language of arithmetic". Yes, really.

Post by Khong Dong
<quote>
in the sense that there is in M1 and M2 an
arithmetic sentence σ, such that M1 thinks σ is true, but M2 thinks it
is false.
M1 believes N |= σ
M2 believes N |= ¬σ
</quote>

Yes, that's right. An arithmetic sentence being a sentence in the first order language of arithmetic *with parameters from the structure under consideration*. Which is different to what I meant.

Post by Khong Dong
So, is σ above "a sentence in the language of arithmetic" or "a sentence in the first-order language of arithmetic"?

Well, as noted Hamkins says "arithmetic sentence", and I believe I have now typed out what he means by that enough times.

So you will understand that in the context of (Theorem 1) notations "N |= σ", "N |= ¬σ", where N would stand for a language structure for the first-order language of arithmetic, σ is a formula in the first-order language of arithmetic. Great.

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

My statement does not contradict his Theorem 1, as explained above.

Wrong.

You sure?

Positive.

[Rupert]

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

So "a sentence in the language of arithmetic" is different from "a sentence in the first-order language of arithmetic"? Really?

It says "arithmetic sentence", doesn't it? And yes, different from "sentence in first-order language of arithmetic". Yes, really.

Post by Khong Dong
<quote>
in the sense that there is in M1 and M2 an
arithmetic sentence σ, such that M1 thinks σ is true, but M2 thinks it
is false.
M1 believes N |= σ
M2 believes N |= ¬σ
</quote>

Yes, that's right. An arithmetic sentence being a sentence in the first order language of arithmetic *with parameters from the structure under consideration*. Which is different to what I meant.

Post by Khong Dong
So, is σ above "a sentence in the language of arithmetic" or "a sentence in the first-order language of arithmetic"?

Well, as noted Hamkins says "arithmetic sentence", and I believe I have now typed out what he means by that enough times.

So you will understand that in the context of (Theorem 1) notations "N |= σ", "N |= ¬σ", where N would stand for a language structure for the first-order language of arithmetic, σ is a formula in the first-order language of arithmetic. Great.

With parameters from the structure N, yes. Not a parameter-free sentence in that language.

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

My statement does not contradict his Theorem 1, as explained above.

Wrong.

You sure?

Positive.

Well, I believe that I have given a satisfactory explanation of why you're mistaken about that, if only you would take the trouble to understand it, but if you're *absolutely sure* that I've got that wrong, but you don't want to argue the point, then there it is; we shall agree to disagree.

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

So "a sentence in the language of arithmetic" is different from "a sentence in the first-order language of arithmetic"? Really?

It says "arithmetic sentence", doesn't it? And yes, different from "sentence in first-order language of arithmetic". Yes, really.

Post by Khong Dong
<quote>
in the sense that there is in M1 and M2 an
arithmetic sentence σ, such that M1 thinks σ is true, but M2 thinks it
is false.
M1 believes N |= σ
M2 believes N |= ¬σ
</quote>

Yes, that's right. An arithmetic sentence being a sentence in the first order language of arithmetic *with parameters from the structure under consideration*. Which is different to what I meant.

Post by Khong Dong
So, is σ above "a sentence in the language of arithmetic" or "a sentence in the first-order language of arithmetic"?

Well, as noted Hamkins says "arithmetic sentence", and I believe I have now typed out what he means by that enough times.

So you will understand that in the context of (Theorem 1) notations "N |= σ", "N |= ¬σ", where N would stand for a language structure for the first-order language of arithmetic, σ is a formula in the first-order language of arithmetic. Great.

With parameters from the structure N, yes. Not a parameter-free sentence in that language.

You're confused between Theorem 1 as a statement in which σ is stated as just a formula of the first-order language of arithmetic, and ONE certain specific formula in ONE proof Hamkins' paper comes up with. Others might come up with a much simpler σ in ANOTHER proof -- of Theorem 1.

By my quantum Mathematics paper, for example, cGC does witnesses the truth of Theorem 1, hence (σ ↔ cGC) there -- no "parameters from the structure N" is required.

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Give the citation. I confidently predict that I will be able to explain why it is that this theorem does not give me compelling reason why I have to abandon my stance on the full truth-determinateness of the first-order language of arithmetic.

Go right ahead and show that your antonym of Theorem 1 is provable, now that you have Hamkins' paper (citation).

My statement does not contradict his Theorem 1, as explained above.

Wrong.

You sure?

Positive.

Well, I believe that I have given a satisfactory explanation of why you're mistaken about that, if only you would take the trouble to understand it, but if you're *absolutely sure* that I've got that wrong, but you don't want to argue the point, then there it is; we shall agree to disagree.

You 're mistaken: technical matters are very cut-and-dried. Your arithmetic full truth-determinateness in N is a _negation_ of the statement of Hamkins' Theorem 1 -- and you're wrong.

[Rupert]

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

So "a sentence in the language of arithmetic" is different from "a sentence in the first-order language of arithmetic"? Really?

It says "arithmetic sentence", doesn't it? And yes, different from "sentence in first-order language of arithmetic". Yes, really.

Post by Khong Dong
<quote>
in the sense that there is in M1 and M2 an
arithmetic sentence σ, such that M1 thinks σ is true, but M2 thinks it
is false.
M1 believes N |= σ
M2 believes N |= ¬σ
</quote>

Yes, that's right. An arithmetic sentence being a sentence in the first order language of arithmetic *with parameters from the structure under consideration*. Which is different to what I meant.

Post by Khong Dong
So, is σ above "a sentence in the language of arithmetic" or "a sentence in the first-order language of arithmetic"?

Well, as noted Hamkins says "arithmetic sentence", and I believe I have now typed out what he means by that enough times.

So you will understand that in the context of (Theorem 1) notations "N |= σ", "N |= ¬σ", where N would stand for a language structure for the first-order language of arithmetic, σ is a formula in the first-order language of arithmetic. Great.

With parameters from the structure N, yes. Not a parameter-free sentence in that language.

You're confused between Theorem 1 as a statement in which σ is stated as just a formula of the first-order language of arithmetic, and ONE certain specific formula in ONE proof Hamkins' paper comes up with. Others might come up with a much simpler σ in ANOTHER proof -- of Theorem 1.

I'm not confused. The phenomenon Hamkins is talking about is only possible when σ has parameters from the structure.

Post by Khong Dong
By my quantum Mathematics paper, for example, cGC does witnesses the truth of Theorem 1, hence (σ ↔ cGC) there -- no "parameters from the structure N" is required.

Since cGC is a parameter-free sentence, the truth-value of "N |= σ" would have to be the same between M1 and M2 for all structures N. This is trivial. I will write out a detailed proof for you for 100 euros, and help you understand it by offering you tutoring for 100 euros per hour.

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
Whether you correctly understand what it says is a moot point of course. If you gave the citation, I could offer thoughts on that topic.

It's more like whether you correctly understand what it (Hamkins' indefiniteness of arithmetic truth) says is a moot point naturally, since "indefiniteness of arithmetic truth" is an antonym of your "full truth-determinateness", while a synonym of my "uncertainty of arithmetic truth".

Yes, got it now, thank you for the citation. What Hamkins means by "arithmetic sentence" in Theorem 1 is a sentence in the language of arithmetic including parameters from the structure under consideration, which may be a non-standard model of arithmetic. That's not the same as a sentence in the first-order language of arithmetic.

So "a sentence in the language of arithmetic" is different from "a sentence in the first-order language of arithmetic"? Really?

It says "arithmetic sentence", doesn't it? And yes, different from "sentence in first-order language of arithmetic". Yes, really.

Post by Khong Dong
<quote>
in the sense that there is in M1 and M2 an
arithmetic sentence σ, such that M1 thinks σ is true, but M2 thinks it
is false.
M1 believes N |= σ
M2 believes N |= ¬σ
</quote>

Yes, that's right. An arithmetic sentence being a sentence in the first order language of arithmetic *with parameters from the structure under consideration*. Which is different to what I meant.

Post by Khong Dong
So, is σ above "a sentence in the language of arithmetic" or "a sentence in the first-order language of arithmetic"?

Well, as noted Hamkins says "arithmetic sentence", and I believe I have now typed out what he means by that enough times.

So you will understand that in the context of (Theorem 1) notations "N |= σ", "N |= ¬σ", where N would stand for a language structure for the first-order language of arithmetic, σ is a formula in the first-order language of arithmetic. Great.

With parameters from the structure N, yes. Not a parameter-free sentence in that language.

You're confused between Theorem 1 as a statement in which σ is stated as just a formula of the first-order language of arithmetic, and ONE certain specific formula in ONE proof Hamkins' paper comes up with. Others might come up with a much simpler σ in ANOTHER proof -- of Theorem 1.

I'm not confused. The phenomenon Hamkins is talking about is only possible when σ has parameters from the structure.

You're confused between a theorem's statement and its proof, and also confused between the meaning of "phenomenon" and that of "theorem": we're arguing about Hamkins' "Theorem 1", not "Phenomenon 1". Not to mention that you're also mistaken: the conclusion of Hamkins' Theorem 1 does NOT insist on the form (parameterized or otherwise) of the formula σ -- and neither does the hypothesis or the whole statement of Theorem 1.

Post by Rupert

Post by Khong Dong
By my quantum Mathematics paper, for example, cGC does witnesses the truth of Theorem 1, hence (σ ↔ cGC) there -- no "parameters from the structure N" is required.

Since cGC is a parameter-free sentence, the truth-value of "N |= σ" would have to be the same between M1 and M2 for all structures N. This is trivial. I will write out a detailed proof for you for 100 euros, and help you understand it by offering you tutoring for 100 euros per hour.

You're wrong, bluffing. Undecide(True(N |= σ)) is proven in some way in "4.6.1 GC (Goldbach Conjecture) related theorems" of my paper, where (σ ↔ cGC), and undecide(True(N |= σ)) is a different rendition of Hamkins' Theorem 1's statement. So far you're unable to cite any specific error on either of the two papers.

[Rupert]

I will write a peer-reviewer's report of your paper for 200 euros, and I will offer you further tutoring about understanding Hamkins' paper properly for 100 euros per hour. I have said enough at this stage. If my comments don't convince you, that's fine. You are allowed to have your own opinion.

[Khong Dong]

Post by Rupert
I will write a peer-reviewer's report of your paper for 200 euros, and I will offer you further tutoring about understanding Hamkins' paper properly for 100 euros per hour. I have said enough at this stage. If my comments don't convince you, that's fine. You are allowed to have your own opinion.

Which is a humorous, comical way you've admitted you actually couldn't find any error on undecide(True(N |= σ)) in either Hamkins' or my rendition of the fact.

[Rupert]

Post by Khong Dong

Post by Rupert
I will write a peer-reviewer's report of your paper for 200 euros, and I will offer you further tutoring about understanding Hamkins' paper properly for 100 euros per hour. I have said enough at this stage. If my comments don't convince you, that's fine. You are allowed to have your own opinion.

Which is a humorous, comical way you've admitted you actually couldn't find any error on undecide(True(N |= σ)) in either Hamkins' or my rendition of the fact.

Well, if you're saying you're having a chuckle over it, then you know, that's joyful. There needs to be more laughter in the world.

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Rupert
I will write a peer-reviewer's report of your paper for 200 euros, and I will offer you further tutoring about understanding Hamkins' paper properly for 100 euros per hour. I have said enough at this stage. If my comments don't convince you, that's fine. You are allowed to have your own opinion.

Which is a humorous, comical way you've admitted you actually couldn't find any error on undecide(True(N |= σ)) in either Hamkins' or my rendition of the fact.

Well, if you're saying you're having a chuckle over it, then you know, that's joyful. There needs to be more laughter in the world.

I forgot a couple of words. The fuller statement is:

"Which is a humorous, comical -- but wimpy and disingenuous -- way you've admitted you actually couldn't find any error on undecide(True(N |= σ)) in either Hamkins' or my rendition of the fact."

[Rupert]

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Rupert
I will write a peer-reviewer's report of your paper for 200 euros, and I will offer you further tutoring about understanding Hamkins' paper properly for 100 euros per hour. I have said enough at this stage. If my comments don't convince you, that's fine. You are allowed to have your own opinion.

Which is a humorous, comical way you've admitted you actually couldn't find any error on undecide(True(N |= σ)) in either Hamkins' or my rendition of the fact.

Well, if you're saying you're having a chuckle over it, then you know, that's joyful. There needs to be more laughter in the world.

"Which is a humorous, comical -- but wimpy and disingenuous -- way you've admitted you actually couldn't find any error on undecide(True(N |= σ)) in either Hamkins' or my rendition of the fact."

Ah okay. You thought you might throw in a bit of putting aspersions on my masculinity for good measure, just to drive the point home. And accusations of dishonesty as well. Well, you know, keep up the good work. :)

So, reading through the remarks I've made in this thread, you really are totally convinced that deep down I know you're right? That's what you're saying to me here? Well, as a report of your mental state, I admit I don't know that you're deliberately lying. It's curious to reflect on how exactly your mind works. And you seem pretty keen to have me put further effort into bothering to argue the point with you without you paying me any money. That would be a fair surmise about what you would like to see happen?

Well then. I'm drinking a cup of tea.

[Rupert]

Post by Khong Dong

Post by Rupert

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Khong Dong sometimes raises concerns about whether we have a clear conception of the standard model based on there being sentences whose truth-value we do not currently know, but none of this is relevant to the fact that we do know how to give a formal mathematical treatment of semantics, and everything that I have said above can be translated into a formal language and given a machine-checkable formally correct proof in the above stated metatheories. I would be willing to actually perform this task for a reasonable hourly rate. So the question of formal mathematical correctness is settled, the finer philosophical points can be debated I suppose. I personally am pretty happy with full truth-determinateness for first-order language of arithmetic along with many other professional philosophers, but certainly we can talk.

So what's your "reasonable hourly rate" to write down one of the two statements (which each should take about 30 seconds to write)
- TRUE(N |= (G(PA) ⋀ cGC))
- FALSE(N |= (G(PA) ⋀ cGC))
where N is the so called "standard" arithmetic structure?

You're asking how much I would charge you to do that, as a friend?

Hmmm... 50 euros would probably be enough.

[Khong Dong]

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Your "what is it?" is the key for a _valid_ answer.

"We shall write |-F... as an abbreviation for ... is a theorem if F. When no confusion results, we omit the subscript F."
(Shoenfield, Mathematical Logic, pg. 6.)

If instead of "|-F..." we have "F |- ...", then, given a formal system P and a formula F, we'd have the following definitions:

- provable(P,F) ⇔ (P |- F)
- unprovable(P,F) ⇔ neg(P |- F)
- undecidable(P,F) ⇔ unprovable(P |- F) and unprovable(P |- ~F)
- CON(P) ⇔ unprovable(P |- ~(x=x)).

That's it. _No_ amount of Gödel's invalid encoding, "expressible in P", "representable in P", or the like of philosophical invalid hand-waving is required.

[Khong Dong]

Post by Khong Dong

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Your "what is it?" is the key for a _valid_ answer.
"We shall write |-F... as an abbreviation for ... is a theorem if F. When no confusion results, we omit the subscript F."
(Shoenfield, Mathematical Logic, pg. 6.)

Rather, "... is a theorem of F."

Post by Khong Dong
- provable(P,F) ⇔ (P |- F)
- unprovable(P,F) ⇔ neg(P |- F)
- undecidable(P,F) ⇔ unprovable(P |- F) and unprovable(P |- ~F)
- CON(P) ⇔ unprovable(P |- ~(x=x)).
That's it. _No_ amount of Gödel's invalid encoding, "expressible in P", "representable in P", or the like of philosophical invalid hand-waving is required.

[Rupert]

Post by Khong Dong

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Your "what is it?" is the key for a _valid_ answer.

The correct answer which I gave him is that it is expressible in P but not strongly representable in P.

Post by Khong Dong
"We shall write |-F... as an abbreviation for ... is a theorem if F. When no confusion results, we omit the subscript F."
(Shoenfield, Mathematical Logic, pg. 6.)
- provable(P,F) ⇔ (P |- F)
- unprovable(P,F) ⇔ neg(P |- F)
- undecidable(P,F) ⇔ unprovable(P |- F) and unprovable(P |- ~F)
- CON(P) ⇔ unprovable(P |- ~(x=x)).
That's it. _No_ amount of Gödel's invalid encoding, "expressible in P", "representable in P", or the like of philosophical invalid hand-waving is required.

This doesn't address the question he was asking.

[Khong Dong]

Post by Rupert

Post by Khong Dong

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Your "what is it?" is the key for a _valid_ answer.

The correct answer which I gave him is that it is expressible in P but not strongly representable in P.

Post by Khong Dong
"We shall write |-F... as an abbreviation for ... is a theorem if F. When no confusion results, we omit the subscript F."
(Shoenfield, Mathematical Logic, pg. 6.)
- provable(P,F) ⇔ (P |- F)
- unprovable(P,F) ⇔ neg(P |- F)
- undecidable(P,F) ⇔ unprovable(P |- F) and unprovable(P |- ~F)
- CON(P) ⇔ unprovable(P |- ~(x=x)).
That's it. _No_ amount of Gödel's invalid encoding, "expressible in P", "representable in P", or the like of philosophical invalid hand-waving is required.

This doesn't address the question he was asking.

Of course it is. Read his "or what is it?".

[Rupert]

Post by Khong Dong

Post by Rupert

Post by Khong Dong

Post by Antonio Speltzu
Is unprovability expressible in P, representable in P or what is it?

Your "what is it?" is the key for a _valid_ answer.

The correct answer which I gave him is that it is expressible in P but not strongly representable in P.

Post by Khong Dong
"We shall write |-F... as an abbreviation for ... is a theorem if F. When no confusion results, we omit the subscript F."
(Shoenfield, Mathematical Logic, pg. 6.)
- provable(P,F) ⇔ (P |- F)
- unprovable(P,F) ⇔ neg(P |- F)
- undecidable(P,F) ⇔ unprovable(P |- F) and unprovable(P |- ~F)
- CON(P) ⇔ unprovable(P |- ~(x=x)).
That's it. _No_ amount of Gödel's invalid encoding, "expressible in P", "representable in P", or the like of philosophical invalid hand-waving is required.

This doesn't address the question he was asking.

Of course it is. Read his "or what is it?".

I re-read that part of his post, and still remain currently unconvinced of your claim.