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The above arguments are now developed, and expressed formally, in my forthcoming book [An20] (link below), where I seek to highlight the necessity of distinguishing between what is believed to be true, what can be evidenced as true, and what ought not to be believed as true.<br /><br />Sincerely,<br /><br />Bhupinder Singh Anand<br />Mumbai<br />bhup.anand@gmail.com<br /><br />References<br />==========<br />[An20] Bhupinder Singh Anand: The Significance of Evidence-based Reasoning in Mathematics, Mathematics Education, Philosophy, and the Natural Sciences.<br /><br />https://www.dropbox.com/s/gd6ffwf9wssak86/16_Anand_Dogmas_Submission_Update_3.pdf?dl=0<br /><br />(Current update of book; 7.4Mb, 702p as of now; under final revision/editing/indexing; scheduled for release mid-2021)Bhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-25444579496590362642021-01-20T10:25:54.310+00:002021-01-20T10:25:54.310+00:00Hi, lovely post i would like to share this because...Hi, lovely post i would like to share this because its very helpful for me keep it up & please don't stop posting thanks for sharing such kind of nice information with us.<br />Anyone can have a <b><a href="https://www.emailexpert247.com/centurylink-email-login-problems/" rel="nofollow">CenturyLink Email login problems</a></b> and account and then can have access to its multiple services. 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1. The modern notation that you prefer...Dear Jeff,<br /><br />1. The modern notation that you prefer seems suited to argumentation in the language of a set theory such as ZF, which defines functions and relations as sets. By the axiom of extensionality, two functions or relations are identical if they define the same set. My AICB/IACAP 2012 paper---and the paper that I have just submitted to ICLA 2013---aim to highlight a curious limitation of such a language. <br /><br />2. I find that the classical notation followed in Mendelson's `Introduction to Mathematical Logic' and Kleene's `Introduction to Metamathematics' is not similarly limited, since they treat function/relation symbols in a formal language (such as `$R$' in PA or `$prf_{PA}$' in Primitive Recursive Arithmetic) as part of the alphabet only for constructing the formulas that denote functions and relations in the language (such as `$R(x)$' in PA or `$prf_{PA}(x, y)$' in PRA).<br /><br />3. The distinction is convenient when I argue that there are number theoretical relations / functions that are not computationally identical, even though their corresponding relations / functions over the ZF ordinals may define the same set.<br /><br />4. More precisely---and expressing it for the moment in the notation that I have been using---if $[(Ax)R(x)]$ is Goedel's undecidable PA formula, then there is a primitive recursive number theoretic relation $q_{PA}(x)$ in PRA (clarified further in 7 below) such that, for any natural number $n$ and numeral $[n]$, we have the metamathematical equivalence:<br /><br />The PRA expression denoted by $\neg q_{PA}(n)$ evaluates as true in $N$ iff the PA formula denoted by $[R(n)]$ interprets as true in $N$ under any sound interpretation of PA.<br /><br />5. However, I show that whereas there is an algorithm that will give evidence to show that any member of the denumerable sequence of PRA expressions denoted by $\{\neg q_{PA}(1), \neg q_{PA}(2), \ldots \}$ evaluates as true in $N$, there is no algorithm that will give evidence to show that any member of the denumerable sequence of PA formulas denoted by $\{[R(1)], [R(2)], \ldots \}$ interprets as true in $N$ under a sound interpretation of PA.<br /><br />6. In the terminology of my paper, whilst the PRA relation $\neg q_{PA}(x)$ is algorithmically computable as always true in $N$, the (metamathematically) instantiationally equivalent PA relation $[R(x)]$ is algorithmically verifiable, but not algorithmically computable, as always true in $N$ under a sound interpretation of PA.<br /><br />7. As to your final query, I think Wikipedia refers to the argument involved in this case (i.e. Goedel's argument) as `indirect self-reference'. Perhaps I should have expressed the metamathematical interpretation of the primitive recursive relation $q_{PA}(x,y)$ unequivocally by writing:<br /><br />`$q_{PA}(x,y)$ ($x$ is the GN of a PA-proof of the PA-formula $[\phi]$---whose GN is $y$---when we replace the variable `$y$' in the formula $[\phi]$ (whose GN is $y$) with the numeral $[y]$ that denotes the GN $y$ in PA.'<br /><br />I am not sure if there is any `diagonalisation' involved in the above in the sense of your remarks, since $[\phi]$ is not necessarily a formula in a single variable. The $[\phi]$ considered in Goedel's argument is actually a formula $[\phi (x, y)]$ with two variables. <br /><br />Thus, in his 1931 paper (as translated in `The Undecidable' edited by Martin Davis) Goedel's original definition of the primitive recursive relation `$\neg q_{PA}(x,y)$' is expressed as:<br /><br />$\neg xB_{\kappa}[Sb(y \scriptsize \begin{array}{c} 19 \\ Z(y) \end{array})]$<br /><br />Kind regards,<br /><br />BhupBhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-23879210372057465262012-08-15T21:13:32.231+01:002012-08-15T21:13:32.231+01:00Hello Bhup,
Ah, now I see what you mean when you...Hello Bhup, <br /><br />Ah, now I see what you mean when you write, "on the primitive recursive relations $prf_{PA}(x,y)$ ...". Normally, one should simply write "on the primitive recursive relations $prf_{PA}$ ...". Here $prf_{PA}$ is a relation, i.e., a subset of $\mathbb{N}^2$. Strictly speaking, "$prf_{PA}(x,y)$" is a sentence of the meta-language, containing variables "$x$" and "$y$". Similarly, it is better to say "the function $f$ ..." It would be a bit misleading to say "the function $f(x)$ ..." Normally, $f(x)$ is the value of the function $f$ on argument $x$ (and "$f(x)$" is a singular term denoting this value). It's important to distinguish a function $f$ from its value $f(x)$; or, analogously, a relation $R$ and the entity $Rxy$ (which is, technically, a truth value, given $x$ and $y$) or the meta-language sentence "$Rxy$". <br /><br />"$q_{PA}(x,y)$ ($x$ is the GN of a PA-proof of the PA-formula---whose GN is $y$---when we replace the variable ‘$y$’ in this formula with its GN, i.e. with the value $[y]$)."<br /><br />I don't quite get this? What formula does "... in this formula ..." refer to? I think you intend to refer to some sort of diagonalization? <br />The usual definition is this. If $\phi(x)$ is a formula with $x$ free, then the diagonalization of $\phi(x)$ is $\phi(\ulcorner \phi \urcorner)$. <br />So, your relation $q_{PA}$ is the diagonal relation?<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-32513681593114333282012-08-15T00:47:56.368+01:002012-08-15T00:47:56.368+01:00Dear Jeff,
You’re right, the fixed point $G$ (wit...Dear Jeff,<br /><br />You’re right, the fixed point $G$ (with which I am not familiar except through its definition only) for the proof predicate $Prov_{PA}(x)$ cannot be $[\forall x R(x)]$.<br /><br />I was wrongly conjecturing the relation of $G$ to Goedel’s original argument in his 1931 paper.<br /><br />This argument was based on the primitive recursive relations $prf_{PA}(x, y)$ ($x$ is the GN of a PA-proof of the PA-formula whose GN is $y$) and $q_{PA}(x, y)$ ($x$ is the GN of a PA-proof of the PA-formula---whose GN is $y$---when we replace the variable ‘$y$’ in this formula with its GN, i.e. with the value $[y]$).<br /><br />Returning to your original query, if $[Q(x, y)]$ expresses $\neg q_{PA}(x, y)$ in PA, and $p$ is the GN of $[\forall x Q(x, y)]$, then $[R(x)]$ is the PA-formula $[Q(x, p)]$ (to which Goedel refers by its GN ‘$r$’), and Goedel’s original undecidable proposition in PA would be the formula $[\forall x R(x)]$ (whose GN Goedel denotes by ‘$17Genr$’).<br /><br />The reason I use square brackets is to be able to distinguish clearly between the natural number $y$ and the numeral $[y]$ in an argument such as the one above.<br /><br />Regards,<br /><br />Bhup<br />Bhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-90963941835587908562012-08-15T00:43:27.998+01:002012-08-15T00:43:27.998+01:00This comment has been removed by the author.Bhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-6218088930564964592012-08-14T19:55:32.544+01:002012-08-14T19:55:32.544+01:00Hello Bhup,
"so $PA \vdash [G \leftrightarro...Hello Bhup,<br /><br />"so $PA \vdash [G \leftrightarrow \forall x \neg Proof_{PA}(x,\ulcorner R(x)\urcorner)]$ would follow trivially."<br /><br />This is not right. Rather, what I think you have in mind is that $R(x)$ is the formula $\neg Proof_{PA}(x, \ulcorner G \urcorner)$, and $G$ is the formula $\forall x R(x)$. <br />Then we have:<br /><br />$PA \vdash [G \leftrightarrow \forall x \neg Proof_{PA}(x,\ulcorner G \urcorner)]$<br /><br />It's unclear even what your version means, but if it means what I think it means, then its right-to-left direction, i.e., $\forall x \neg Proof_{PA}(x,\ulcorner R(\dot{x}) \urcorner) \rightarrow \forall x R(x)$ is not provable in $PA$. This is not what a fixed point means.<br /><br />$G$ is a fixed point of the undedicable *provability* predicate $Prov_{PA}(x)$ (which is a $\Sigma_1)$ formula), not the proof predicate $Proof_{PA}(x, y)$, which is decidable.<br /><br />"... so as to distinguish them from expressions that denote interpreted relations and/or functions that are not formulas of L."<br /><br />I don't quite get this ... these are *expressions* of English? Why are there any expressions of any language except $L$ involved at all? Why not just write $R(x)$ to mean some formula of the object language $L$, with $x$ free?<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-87984427945318561762012-08-13T16:21:36.547+01:002012-08-13T16:21:36.547+01:00Dear Jeff,
Something odd here!
I would have tho...Dear Jeff,<br /><br />Something odd here! <br /><br />I would have thought that, in modern notation, $[R(x)]$, would be $\neg Proof_{PA}(x, \ulcorner R(x) \urcorner)$!<br /><br />We would then have that $[G]$ is $[\forall x R(x)]$, and so $PA \vdash [G \leftrightarrow \forall x \neg Proof_{PA}(x, \ulcorner R(x) \urcorner)]$ would follow trivially.<br /><br />Perhaps I need to go back to first principles and retrace Goedel's original argument.<br /><br />Regards,<br /><br />Bhup<br /><br />Notation: Although I forgot to do so consistently in my previous post, I try to use square brackets to enclose expressions that denote formulas (uninterpreted strings) of a formal language $L$, so as to distinguish them from expressions that denote interpreted relations and/or functions that are not formulas of $L$.<br />Bhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-47550684405727246092012-08-13T16:19:50.069+01:002012-08-13T16:19:50.069+01:00This comment has been removed by the author.Bhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-7921251363375205072012-08-12T13:02:34.196+01:002012-08-12T13:02:34.196+01:00So, $[R(x)]$ is, in modern notation, $\neg Proof_{...So, $[R(x)]$ is, in modern notation, $\neg Proof_{PA}(x, \ulcorner G \urcorner)$, where $G$ is such that<br />$PA \vdash G \leftrightarrow \forall x \neg Proof_{PA}(x, \ulcorner G \urcorner)$?<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-23577912738726664112012-08-12T04:12:19.461+01:002012-08-12T04:12:19.461+01:00Dear Jeff,
Yes, we agree that $\mathcal{L}$ is an...Dear Jeff,<br /><br />Yes, we agree that $\mathcal{L}$ is an infinite set.<br /><br />The formula $[R(x)]$ is the one with Goedel-number $r$ defined by Goedel (in his seminal 1931 paper on formally undecidable arithmetical propositions), for which he first proved that the (fixed point) formula $(\forall x)R(x)$ with Goedel number $17 Gen r$ is not provable in the second-order Peano Arithmetic P (also specifically defined by him in the 1931 paper) if P is consistent; and then proved that the formula $\neg(\forall x)R(x)$ with Goedel number $Neg(17 Gen r)$ is also not provable in P if P is further assumed to be $\omega$-consistent.<br /><br />Goedel constructed $[R(x)]$ such that, if $[R(x)]$ interprets as the arithmetical relation $R*(x)$ then, for any natural number $n$ and numeral $[n]$:<br /><br />If $R*(n)$ is a true arithmetical sentence then $[R(n)]$ is not PA-provable.<br /><br />Regards,<br /><br />BhupBhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-47947332545349276972012-08-11T20:47:12.965+01:002012-08-11T20:47:12.965+01:00Hello Bhup,
So I think you agree that $\mathcal{L...Hello Bhup,<br /><br />So I think you agree that $\mathcal{L}$ is an infinite set.<br /><br />I'm not sure what your formula $[R(x)]$ is meant to be and how it is related to the fixed point $G$ of $Prov_{PA}(x)$.<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.com