<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class="">Sheri,<div class="">I will try to respond to your letter about the post Goedel structures by first quoting the last part of my previous letter that discusses Goedelian ideas from the point of view of fixed points.</div><div class="">My letter was quite long, and it is possible to not get to the second half.</div><div class="">Note also that the first half is based on a referential situation g —> F where #g ——> Fg is what I call the Indicative Shift of g —> F. This is formal and does not assume anythng other than arrow structure.</div><div class="">With g —> F# we have #g —> F#g making F#g refer to its own name. There is more to say herd and references that I cannot send to the list, so I will get a dropbox for it and further discussion later today.</div><div class="">Best,</div><div class="">Lou K.</div><div class=""><br class=""></div><div class="">##########</div><div class=""><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">It is a very interesting question whether such encoding or such multiple relationships to context occur in biology. Here are some remarks.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">1. In biology is is NORMALLY the case that certain key structures have multiple interpretations and uses in various contexts.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">The understanding of such multiple uses and the naming of them requires an observer of the biology. Thus we see the action of a cell membrane and we see the action of mitosis, and so on.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">2. There are implicit encodings in biology such as the sequence codes in DNA and RNA and their unfoldment. To what extent do they partake of the properties of Goedel coding?<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">3. The use of the Goedel coding in the Incompleteness theorem depends crucially on the relationship of syntax and semantic in the formal system and in the mathematician’s interpretation of the workings of that system. The Goedel argument depends upon the formal system S being seen as a mathematical object that itself can be studied for its properties and behavior.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">When we speak of the truth of G, we are speaking of our assessment of the possible behaviour of S, given its consistency. We are reasoning about S just as Euclid reasons about the structure of right triangle.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">4. In examining biological structures we take a similar position and reason about what we know about them. Sufficiently complex biological structures can be seen as modeled by certain logical formal systems.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">And then Goedelian reasoning can be applied to them. This can even be extended to ourselves. Suppose that I am modeled correctly in my mathematical reasoning by a SINGLE CONSISTENT FORMAL SYSTEM S.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then “I” can apply the above proof of Goedel’s Therem to S and deduce that G cannot be proven by S. Thus “I” have exceeded the capabilities of S. Therefore it is erroneous to assume that my mathematical reasoning is encapsulated by a single formal system S. If I am a formal system, that system must be allowed to grow in time. Such reasoning as this is subtle, but the semantics of the relationship of mathematicians and the formal systems that they study is subtle and when biology is brought in the whole matter becomes exceedingly interesting.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">5. We man not need numbers to have these kinds of relationships. And example is the Smullyan Machine that prints sequences of symbols from the alphabet {~,P,R} on a tape. Sequences that begin with P,~P,PR and ~PR are regarded as meaningful, with the meanings:<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">PX: X can be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">~PX: X cannot be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">PRX: XX can be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">~PRX: XX cannot be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Here X is any string of the symbols {~,P,R}.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Thus PR~~P means that XX can be printed where X = ~~P. Thus PR~~P means that ~~P~~P can be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">By printed we mean on one press of the button on the Machine, a string of characters is printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">IT IS ASSUMED THAT THE SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT PRINTS A MEANINGFUL STATEMENT.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then we have the<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Theorem. There are meaningful true strings that the Smullyan Machine cannot print.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">This is a non-numerical analog of the Goedel Theorem. And the string that cannot be printed is G = ~PR~PR.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">For you see that G is meaningful and since G = ~PRX, G says that XX cannot be printed. But X = ~PR and XX = ~PR~PR = G. So G says that G cannot be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">If the machine were to print G, it would lie. And the machine does not lie.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Therefore G is unprintable.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">But this is what G says.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">So we have established the truth of G and proved the Theorem.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">6. Examine this last paragraph 5. The Machine is like an organism with a limitation. This limitation goes through the semantics of reference. ~PRX refers to XX and so can refer to itself if we take X = ~PR. ~PX refers to X and cannot refer to itself since it is longer than X. In biological coding the DNA code is fundamentally smaller or equal to the structure to which it refers.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Thus the self-reproduction of the DNA is possible since DNA = W+C the convention of the Watson and Crick strand and each of W and C can by themselves engage in an action to encode, refer to, the other strand. W can produce a copy of C in the form W+C and C can produce a copy of W in the form W+C each by using the larger environment. Thus W+C refers to itself, reproduces itself by a method of encoding quite similar to the self reference of the Smullyan Machine.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">7. Von Neuman devised a machine that can build itself. B is the von Neuman machine and B.x —> X,x where x is the plan or blueprint or code for and entity X. B builds X with given the blueprint x.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then we have B,b —> B,b where b is the blueprint for B. B builds itself from its own blueprint. I hope you see the analogy with the Goedel code.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">8. I will stop here. The relationships with biology are very worth discussing.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Before stopping it is worth remarking that the Maturana Uribe Varela autopoeisis is an example of a system arising into a form of self-reference that has a lifetime due to the probabilisitic dynamics of its process.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> ###############</o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class="">Best,</o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class="">Lou Kauffman</o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""><br class=""></o:p></div><div><blockquote type="cite" class=""><div class="">On Nov 15, 2022, at 2:19 PM, Markose, Sheri <<a href="mailto:scher@essex.ac.uk" class="">scher@essex.ac.uk</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div class="WordSection1" style="page: WordSection1; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Dear Louis, dear Colleagues - <span class="Apple-converted-space"> </span><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Louis has given an excellent exposition of Gödel Numbering (g.n) (your point number 2 on coding and semantics is giving me food for thought) , giving example of prime factorization and also of Gödel Sentence as one that states its own unprovability. Unlike statements like "this is false", GS is not paradoxical and in a consistent system it is a theorem with a constructive g.n. The latter in terms of the prime factorization format, it is indeed a Hilbert 10 Diophantine equation with no integer solutions. A remarkable achievement in maths, considering Gödel was only 23 years of age .... But what has this got to do with Biology and novelty production, the objectives of the my FIS discussion ? <o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">In view of brevity and also urged by Pedro, I dropped a couple of paragraphs in my FIS kick off submission as to why we need to exceed Gödel (1931) and couch the Gödel Incompleteness Results and the Gödel Sentence with a fuller understanding of algorithms as encoded instructions and as machine executable codes, of the notion of recursive enumeration (re) and re sets that was developed in the Emil Post (1944). I hope Louis Kauffman can comment on the the application of the fuller Gödel-Turing -Post-Rogers framework mentioned in my FIS note and in my papers cited there. <o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">1. I have found the following statement by Joel Hamkins ( :<span class="Apple-converted-space"> </span><a href="https://urldefense.com/v3/__http://jdh.hamkins.org/wp-content/uploads/A-review-of-several-fixed-point-theorems-1.pdf__;!!D9dNQwwGXtA!WIYX4YX9aKL866Hx0OZG__Wi-adUHMu8_HbQwsiYwwQdgPl9_Do9F9MR5bapZPsYv-tD5GG2QP6uftez$" style="color: blue; text-decoration: underline;" class=""><span style="font-size: 11pt; font-family: Calibri, sans-serif;" class="">http://jdh.hamkins.org/wp-content/uploads/A-review-of-several-fixed-point-theorems-1.pdf</span></a><span class="Apple-converted-space"> </span>) useful as it makes an important observation that the original Gödel (1931) framework permits an encodable proposition to make statements about itself while Second Recursion Theorems (SRT) also called Fixed Point Theorems are needed “to construct programs/algorithms that refer to themselves”. The terms programs and algorithms will be used interchangeably.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">I choose Rogers Fixed Point Theorem of (total) computable functions starting with the staple I have already indicated Diag (g) (RHS of (8) below) is what Neil Gerschenfeld calls ribosomal self-assembly machines in gene expression where the program<span class="Apple-converted-space"> </span><i class="">g builds the<span class="Apple-converted-space"> </span></i>machine that runs g.<span class="Apple-converted-space"> </span><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">II. The first requirement of a system to identify Fixed Points viz. self-referential constructions of algorithms/programs is (8) viz to identify what function/algorithm has altered the Diag (g).<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><span id="cid:image001.png@01D8F917.B36D04E0"><image001.png></span><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">When online gene expression takes place on RHS of (8), viz. these programs have halt commands and builds the somatic and phenotype identity of vertebrates online, the offline record of this is made in the Thymus that can not only represent the Thymic/immune self but also concatenate changes thereof. <span class="Apple-converted-space"> </span><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">I have suggested that the Adaptive Immune System and the Mirror Neuron System have these structures in (8). And the domain of self-halting machines as in (8) are the Theorems of the system and a subset of Post (1944) Creative Set. The non-Theorems have codes say<span class="Apple-converted-space"> </span><span style="font-size: 12pt;" class="">g<sup class="">¬</sup></span><span class="Apple-converted-space"> </span>which cannot halt in a formal system that is consistent. To my mind, the embodiment via the physical self being self-assembled and an offline record of this on LHS of (8) is what fuses syntax and semantics.<span class="Apple-converted-space"> </span><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">II. Once, (8) is in place, the Adaptive immune system has to identify novel negation software function<span class="Apple-converted-space"> </span><i class="">f<sup class="">¬!<span class="Apple-converted-space"> </span></sup></i><sub class=""> </sub>of non-self antigens which is an uncountable infinite possibilities. Hence the close to astronomic search with V(D) J of 10<span class="Apple-converted-space"> </span><sup class="">20<span class="Apple-converted-space"> </span></sup>– 10<span class="Apple-converted-space"> </span><sup class="">30</sup><span class="Apple-converted-space"> </span>) of non-self antigens that can hijack the self- assembly machines as recorded on RHS of (8). Only from knowledge of self can the hostile other, in the case of the AIS, be identified.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">III. Roger Fixed Point assures us that the indexes of the fixed point for<span class="Apple-converted-space"> </span><i class="">f<sup class="">¬!<span class="Apple-converted-space"> </span></sup></i><sub class=""> </sub>be generated. I have cced Guillame Bonfante who I think was among the first (with coauthors, 2006) to suggest how SRT can be used to identify computer viruses. But they do not use the full force of Self-Ref and Self -Rep and only implicitly use Post Creative and Productive Sets. The index of the Godel Sentence for the fixed point will endogenously lie outside of Post listable or recusively enumerable set for Theorems and known non-Theorems.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">IV. From these Gödel Sentences produced in the immune-cognitive systems, the explicit use of Post (1944) Theorems indicates how novel antibodies cannot be produced in the absence of the Gödel Sentence which allows a biotic element to self-report it is under attack.<span class="Apple-converted-space"> </span><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">V. In conclusion, while it has become fashionable for some like Jurgen Schmidhuber to claim that there can be endogenous self improving recursive novelty (he calls them Gödel machines) , the Gödel Logic says that the original theorems and self-codes are kept unchanged/hack free and novelty is produced only in response to adversarial attacks of self codes. So the AIS story is somatic hypermutation so that nothing in the genome changes. As to how the germline itself changes, needs more investigation, in Biosystems paper, I suggest something very briefly. <o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">So thankyou all again for your in depth comments and interest.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Best Regards<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Sheri<span class="Apple-converted-space"> </span><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">-----Original Message-----<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">From: Fis <<a href="mailto:fis-bounces@listas.unizar.es" style="color: blue; text-decoration: underline;" class="">fis-bounces@listas.unizar.es</a>> On Behalf Of Louis Kauffman<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Sent: 08 November 2022 00:13<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">To: "Pedro C. Marijuán" <<a href="mailto:pedroc.marijuan@gmail.com" style="color: blue; text-decoration: underline;" class="">pedroc.marijuan@gmail.com</a>><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Cc: fis <<a href="mailto:fis@listas.unizar.es" style="color: blue; text-decoration: underline;" class="">fis@listas.unizar.es</a>><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Subject: Re: [Fis] A new discussion session<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">CAUTION: This email was sent from outside the University of Essex. Please do not click any links or open any attachments unless you recognise and trust the sender. If you are unsure whether the content of the email is safe or have any other queries, please contact the IT Helpdesk.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Dear Pedro,<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Here are some comments about Goedel numbering and coding.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">It is interesting to think about Goedel numbering in a biological context.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Actually we are talking about how a given entity has semantics that can vary from context to context.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">It is not simply a matter of assigning a code number. If g —> F is the relation of a Goedel number g to a statement F, then we have two contexts for F.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">1. F as a well formed formula in a formal system S.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">2. g as a number in either a number system for an observer of S or g as a number in S, but g, as a representative for F can be regarded in the system S with the meanings so assigned.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Thus we have produced by the assignment of Goedel numbers a way for a statement F to exist in the semantics of more than one context.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">This is the key to the references and self-references of the Goedelian situations.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Lets look at this more carefully. Recall that there is a formal system S and that to every well formed formula in S, there is a code number g = g(S). The code number can be produced in many ways.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">For example, one can assign different index numbers n(X) to each distinct generating symbol in S. Then with an expression F regarded as an ordered string of symbols, one can assign to F the product of the prime numbers, in their standard order, with exponents the indices of the sequence of characters that compose F. For example, g(~ x^2 = 2) = 2^{n(~)} 3^{n(x)}5^{n(^)}7^{n(2)}11^{n(=)}13^{n(2)}. >From such a code, one can retrieve the original formula in a unique way.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">The system S is a logical system that is assumed to be able to handle logic and basic number theory. Thus it is assumed that S can encode the function g: WFFS(S) —> N where N denotes the natural numbers.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">And S can decode a number to find the corresponding expression as well. It is assumed that S as a logical system, is consistent.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">With this backgound, let g —> F denote the condition that g = g(F). Thus I write a reference g —> F for a mathematical discussion of S, to indicate that g is the Goedel number of F.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Now suppose that F(x) is a formula in S with a free variable x. Free variables refer to numbers. Thus if I write x^2 = 4 then this statement can be specialized to 2^2 = 4 with x =2 and the specialization is true.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Or I can write 3^2 = 4 and this is a false statement. Given F(x) and some number n, I can make a new sentence F(n).<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Now suppose that<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">g —> F(x).<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then we can form F(g) and this new statement has a Goedel number. Let #g denote the Goedel number of F(g).<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">#g —> F(g).<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">This # is a new function on Goedel numbers and also can be encoded in the system S. I will abbreviate the encoding into S by writing #n for appropriate numbers n handled by S.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then we can consider<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">F(#x) and it has a Goedel number<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">h —> F(#x)<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">And we can shift that to<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">#h —> F(#h).<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">This is the key point.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Now we have constructed a number #h so that F(#h) discusses its own Goedel number.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">This construction allows the proof of the Goedel Incompleteness Theorem via the sentence B(x) that states<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">B(x) = “The statement with Goedel number x is provable in S.” (This can also be encoded in S.)<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">We then construct<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">h—> ~B(#x)<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">and<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">#h —> ~B(#h)<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">and obtain the statement<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">G= ~B(#h).<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">G states the unprovability of the Goedel decoding of #h.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">But the Goedel decoding of #h is the statement G itself.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Thus G states its own unprovability.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Therefore, S being consistent, cannot prove G.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">By making these arguments we have have proved that G cannot be proved by S.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Thus we have shown that G is in fact true.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">We have shown that there are true statements in number theory unprovable by system S..<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">##########################<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">The above is a very concise summary of the proof of Goedel’s Incompleteness Theorem, using Goedel number encoding.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">It is a very interesting question whether such encoding or such multiple relationships to context occur in biology. Here are some remarks.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">1. In biology is is NORMALLY the case that certain key structures have multiple interpretations and uses in various contexts.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">The understanding of such multiple uses and the naming of them requires an observer of the biology. Thus we see the action of a cell membrane and we see the action of mitosis, and so on.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">2. There are implicit encodings in biology such as the sequence codes in DNA and RNA and their unfoldment. To what extent do they partake of the properties of Goedel coding?<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">3. The use of the Goedel coding in the Incompleteness theorem depends crucially on the relationship of syntax and semantic in the formal system and in the mathematician’s interpretation of the workings of that system. The Goedel argument depends upon the formal system S being seen as a mathematical object that itself can be studied for its properties and behavior.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">When we speak of the truth of G, we are speaking of our assessment of the possible behaviour of S, given its consistency. We are reasoning about S just as Euclid reasons about the structure of right triangle.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">4. In examining biological structures we take a similar position and reason about what we know about them. Sufficiently complex biological structures can be seen as modeled by certain logical formal systems.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">And then Goedelian reasoning can be applied to them. This can even be extended to ourselves. Suppose that I am modeled correctly in my mathematical reasoning by a SINGLE CONSISTENT FORMAL SYSTEM S.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then “I” can apply the above proof of Goedel’s Therem to S and deduce that G cannot be proven by S. Thus “I” have exceeded the capabilities of S. Therefore it is erroneous to assume that my mathematical reasoning is encapsulated by a single formal system S. If I am a formal system, that system must be allowed to grow in time. Such reasoning as this is subtle, but the semantics of the relationship of mathematicians and the formal systems that they study is subtle and when biology is brought in the whole matter becomes exceedingly interesting.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">5. We man not need numbers to have these kinds of relationships. And example is the Smullyan Machine that prints sequences of symbols from the alphabet {~,P,R} on a tape. Sequences that begin with P,~P,PR and ~PR are regarded as meaningful, with the meanings:<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">PX: X can be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">~PX: X cannot be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">PRX: XX can be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">~PRX: XX cannot be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Here X is any string of the symbols {~,P,R}.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Thus PR~~P means that XX can be printed where X = ~~P. Thus PR~~P means that ~~P~~P can be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">By printed we mean on one press of the button on the Machine, a string of characters is printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">IT IS ASSUMED THAT THE SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT PRINTS A MEANINGFUL STATEMENT.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then we have the<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Theorem. There are meaningful true strings that the Smullyan Machine cannot print.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">This is a non-numerical analog of the Goedel Theorem. And the string that cannot be printed is G = ~PR~PR.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">For you see that G is meaningful and since G = ~PRX, G says that XX cannot be printed. But X = ~PR and XX = ~PR~PR = G. So G says that G cannot be printed.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">If the machine were to print G, it would lie. And the machine does not lie.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Therefore G is unprintable.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">But this is what G says.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">So we have established the truth of G and proved the Theorem.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">6. Examine this last paragraph 5. The Machine is like an organism with a limitation. This limitation goes through the semantics of reference. ~PRX refers to XX and so can refer to itself if we take X = ~PR. ~PX refers to X and cannot refer to itself since it is longer than X. In biological coding the DNA code is fundamentally smaller or equal to the structure to which it refers.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Thus the self-reproduction of the DNA is possible since DNA = W+C the convention of the Watson and Crick strand and each of W and C can by themselves engage in an action to encode, refer to, the other strand. W can produce a copy of C in the form W+C and C can produce a copy of W in the form W+C each by using the larger environment. Thus W+C refers to itself, reproduces itself by a method of encoding quite similar to the self reference of the Smullyan Machine.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">7. Von Neuman devised a machine that can build itself. B is the von Neuman machine and B.x —> X,x where x is the plan or blueprint or code for and entity X. B builds X with given the blueprint x.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Then we have B,b —> B,b where b is the blueprint for B. B builds itself from its own blueprint. I hope you see the analogy with the Goedel code.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">8. I will stop here. The relationships with biology are very worth discussing.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Before stopping it is worth remarking that the Maturana Uribe Varela autopoeisis is an example of a system arising into a form of self-reference that has a lifetime due to the probabilisitic dynamics of its process.<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Very best,<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Lou Kauffman<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><o:p class=""> </o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">_______________________________________________<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class="">Fis mailing list<o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><a href="mailto:Fis@listas.unizar.es" style="color: blue; text-decoration: underline;" class="">Fis@listas.unizar.es</a><o:p class=""></o:p></div><div style="margin: 0cm; font-size: 10pt; font-family: Arial, sans-serif;" class=""><a href="https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=http*3a*2f*2flistas.unizar.es*2fcgi-bin*2fmailman*2flistinfo*2ffis&c=E,1,1A1lgz03IPLQ2hs74kfiKoMaeMUB45427CkA4or9aZPmd25ZxHPE88KK0k9wHh7-Un8A9g25n5WMXHIu8yhAyMhiouezvcso3GGs3inouA,,&typo=1__;JSUlJSUlJQ!!D9dNQwwGXtA!WIYX4YX9aKL866Hx0OZG__Wi-adUHMu8_HbQwsiYwwQdgPl9_Do9F9MR5bapZPsYv-tD5GG2QO45VoPg$" style="color: blue; 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