tag:blogger.com,1999:blog-4987609114415205593.post2588821065190097388..comments2021-10-22T13:17:56.869+01:00Comments on M-Phi: How Might PA be Inconsistent?Jeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger29125tag:blogger.com,1999:blog-4987609114415205593.post-88208991278338598402021-10-18T17:42:15.395+01:002021-10-18T17:42:15.395+01:00Thank you for posting such a great article. Keep i...Thank you for posting such a great article. Keep it up mate.<br /><br /><a href="https://www.bipard.org/2021/09/e-shram-card-online-registration-2021-e.html" rel="nofollow">E Shram Card Online Registration 2021</a> Yakshitahttps://www.blogger.com/profile/05459704732582443449noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-12553994126838404502021-10-17T19:53:42.634+01:002021-10-17T19:53:42.634+01:00Thank you for posting such a great article. Keep i...Thank you for posting such a great article. 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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.comtag:blogger.com,1999:blog-4987609114415205593.post-25849871677095403132012-08-11T08:59:59.005+01:002012-08-11T08:59:59.005+01:00Dear Jeff,
Yes, of course.
However I would descr...Dear Jeff,<br /><br />Yes, of course.<br /><br />However I would describe them as denumerable (non ZF) sets in order to make clear that they are not formally defined ZF sets.<br /><br />This is not mere pedantry.<br /><br />In my proposed ICLA 2013 submission, I show that Goedel's famous 'undecidable' PA formula $[R(x)]$ is not algorithmically computable as always true, even though it is algorithmically verifiable as always true.<br /><br />Now, $[R(x)]$ is defined as the arithmetical representation in PA of a primitive recursive number theoretic relation $Q(x)$ that, by definition, is algorithmically computable as always true.<br /><br />This means that, for any natural number $n$ and numeral $[n]$:<br /><br />If $Q(n)$ is true then $[R(n)]$ is PA-provable;<br />If $Q(n)$ is false then $[\neg R(n)]$ is PA-provable.<br /><br />Thus $[R(x)]$ and $Q(x)$ are instantiationally ‘equivalent’ arithmetical relations, but the latter is algorithmically computable whilst the former is not!<br /><br />This distinction is not possible in ZF, where we would represent the ranges of $[R(x)]$ and $Q(x)$ as completed infinities (i.e. as sets of ordinals in ZF) that define the same ZF relation by the ZF Axiom of Extension.<br /><br />Regards,<br /><br />BhupBhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-4013807412091245292012-08-10T19:29:46.397+01:002012-08-10T19:29:46.397+01:00Hi Bhup,
Well, $\mathcal{L}$ is the set of all ex...Hi Bhup,<br /><br />Well, $\mathcal{L}$ is the set of all expressions/strings. This is infinite, right? <br />And there are a couple of distinguished subsets, e.g., $Tm(\mathcal{L})$ and $Form(\mathcal{L})$ (both infinite too); and a couple of operations, namely concatenation and substitution.<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-51246665003122800822012-08-10T18:06:42.438+01:002012-08-10T18:06:42.438+01:00Dear Jeff,
Don't quite see how one could view...Dear Jeff,<br /><br />Don't quite see how one could view $\mathcal{L}_{PA}$ as a well-defined ZF formula that is also a set in ZF; but OK I pass ... what's the catch?<br /><br />Regards,<br /><br />BhupBhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-41474967184822667522012-08-10T17:35:28.676+01:002012-08-10T17:35:28.676+01:00Hi Bhup,
So you think that $\mathcal{L}_{PA}$ is ...Hi Bhup,<br /><br />So you think that $\mathcal{L}_{PA}$ is an infinite set with cardinality $\aleph_0$?<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-91403525442775346402012-08-10T07:03:17.770+01:002012-08-10T07:03:17.770+01:00Dear Jeff,
You've lost me there!
Cardinal nu...Dear Jeff,<br /><br />You've lost me there!<br /><br />Cardinal numbers are defined specifically as equivalence classes in a formal set theory such as ZF, which contains a sub-set of finite ordinals that, meta-mathematically, can be effectively put into a 1-1 correspondence with the natural numbers, and whose cardinal number is defined in ZF as $\aleph_{0}$.<br /><br />PA is a specific, recursively defined, first order Peano Arithmetic, whose domain contains the numerals, which meta-mathematically can also be effectively put into a 1-1 correspondence with the natural numbers.<br /><br />$\mathcal{L}$ seems to be the informal language of Peano Arithmetic in which the Standard interpretation of PA is defined, and whose domain is that of the of the natural numbers.<br /><br />So what exactly do you mean by the cardinal number of $\mathca{L}$, or the cardinal number of something that I believe in?<br /><br />And how exactly would a 'belief' be relevant, or even useful, here?<br /><br />After all, I may choose to believe that Pegasus exists in a world of my conception in the same way as I choose to believe that 1+1=2 in the same world. <br /><br />Prima facie, that should have less significance than my being able to convincingly demonstrate to others---with whom I share a common lingua franca---that their belief that Pegasus exists, or that 1+1=2 in their conception, would not 'conflict' with my beliefs or conceptions. <br /><br />Regards,<br /><br />BhupBhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-81412932264047021092012-08-09T22:02:54.119+01:002012-08-09T22:02:54.119+01:00Hi Bhup,
But I'm just asking for the cardinal...Hi Bhup,<br /><br />But I'm just asking for the cardinality of something you believe in.<br />You believe that there is some "formal" thing, which you call "$PA$", in some language $\mathcal{L}$. So, what is the cardinality of $\mathcal{L}$?<br /><br />Cheers, JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-11881796707857988442012-08-09T20:20:52.796+01:002012-08-09T20:20:52.796+01:00Dear Jeff,
Let me respond obliquely to your quest...Dear Jeff,<br /><br />Let me respond obliquely to your question.<br /><br />The philosophical position underlying the argumentation of my posts is that we may need to explicitly recognize the limitations on the ability of highly expressive mathematical languages such as ZF to communicate effectively (unless we can offer a finitary interpretation of ZF); and the limitations on the ability of effectively communicating mathematical languages such as PA (which can be shown to have a finitary interpretation) to adequately express abstract concepts---such as those involving Cantor's first limit ordinal $\omega$.<br /><br />For instance, in an unpublished critical examination of the proof of Goodstein's Theorem (link below), I argue that we cannot add a symbol corresponding formally to the concept of an `infinite' mathematical entity---such as is referred to symbolically in the literature by `$\aleph$' or `$\omega$'---to the first-order Peano Arithmetic PA without inviting inconsistency; and that no model of PA can admit a constant term `greater than' any natural number (which I would term as a completed infinity).<br /><br />http://alixcomsi.com/10_Goodstein_case_against_1000.pdf<br /><br />Regards,<br /><br />BhupBhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-79961766365492845812012-08-09T18:47:18.936+01:002012-08-09T18:47:18.936+01:00Hello again, Bhup,
What would you say the cardina...Hello again, Bhup,<br /><br />What would you say the cardinality of $\mathcal{L}$ is?<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-7761263491464682582012-08-09T05:47:26.919+01:002012-08-09T05:47:26.919+01:00Dear Jeff,
Sceptic! Difficult to see myself in th...Dear Jeff,<br /><br />Sceptic! Difficult to see myself in that light, though I do prefer ‘convincing’ to ‘satisfactory’ in mathematical argumentation (which is why I have difficulty conceiving of any interpretation that admits completed infinities without qualification).<br /><br />I assumed that the ‘fixed’ interpretation you referred to in your post was the Standard interpretation of PA over the structure of the natural numbers (I presume that it is this structure that you refer to by $(\mathcal{L}, \mathbb{N})$).<br /><br />If so, then---even if the above is the intended interpretation (under Tarski’s classical definitions) that may initially motivate a human intelligence in the formalization that is defined as the first order Peano Arithmetic PA---I have essentially argued in my recent AISB/IACAP 2012 paper ‘Evidence-Based Interpretations of PA’ that (and suggested why) this interpretation (of PA) cannot be seen as sound; in the sense that the Axioms of PA are not seen to interpret as objectively true under the interpretation, and that the rules of inference are not seen as preserving such truth objectively under the interpretation.<br /><br />However, I have further argued in the paper that there is an objectively definable, algorithmic (hence finitary), interpretation of PA (which could be seen as reflecting the way a machine intelligence would interpret PA over the numerals) which is sound; in the sense that the Axioms of PA can be shown to interpret as objectively true under the interpretation, and that the rules of inference can be shown to preserve such truth objectively under the interpretation.<br /><br />http://alixcomsi.com/34_Evidence_Based_Interpretations_Of_PA_Final_AISB_IACAP_2012.pdf<br /><br />In a follow-up paper, ‘Some Consequences of Evidence-Based Interpretations of PA’ (link below) that I am currently finalising, I argue further that the Standard interpretation of PA can be shown to be unsound (a conclusion that may perhaps lie implicitly at the heart of the argument that led Ed Nelson to conclude that PA is inconsistent), and suggest why this interpretation cannot validate the PA Induction Axiom schema.<br /><br />http://alixcomsi.com/39_Consequences_Evidence_Based_Interpretations_Of_PA.pdf<br /><br />This is the sense in which I remarked ‘… that it is this subjectively-imposed constraint on a 'fixed' interpretation that has perhaps prevented a satisfactory perspective concerning a proof of consistency for PA’.<br /><br />Regards and thanks for your prompt response,<br /><br />BhupBhupinder Singh Anandhttps://www.blogger.com/profile/13505076032940030790noreply@blogger.com