Gödel’s confusions— METALOGIC A

A1—Gödel and sound sets
A2—Gödel and sound numbers

A3—Gödel and the ‘paradoxes’

A4—The Return of the Gödel

Introduction

173. The more I drive into the books on modern logic as taught at universities, the more I am convinced that it is a dubious, authoritarian superstition built upon sand. The books are widely ‘contradictory’, to each other, and nobody really seems to notice. The foundations of logic have been deeply, empirically unsound since the time of Aristotle, and every generation adds more non-sense to what is, essentially, a muddled and unstable house of cards.
174. The ‘logic’ maybe an interesting but inferior game of chess for shamans, primarily designed to impress or confuse, but the core is rotten and the king has no clothes.
175. In the document at this site, why Aristotelian logic does not work, I have listed under half a dozen or so headings the prime simple errors in the Aristotelian system. Here I am in process of carving up various areas of the received ‘logic’ as taught at our universities, often under the heading of ‘philosophy’ or ‘mathematical’ ‘logic’. These areas of ‘logic’ are essentially new problems, which have been piled upon the errors of Aristotelian ‘logic’. They should not be attributed to Aristotle, but to those who built upon Aristotle without first examining Aristotle’s foundations for cracks.
176. For approximately 2000 years, from Aristotle, people—often called ‘philosophers’—discussed whether a large and heavy object would fall at a different rate than a small or light object. Galileo went up the leaning tower of Pisa to try it out by dropping various weights and observing the results; and much joy did it bring him. I am pursuing the ‘same’ method as Galileo with regard to Aristotle’s ‘logic’: I am testing it against the real world, not against other muddled arguments. This process is usually called empiric method and is at the heart of scientific advance.
177. The language structures called ‘paradoxes’ have likewise been discussed since the time of greek civilisation; I am approaching these problems with the tools of empiricism. As far as I can ascertain, this approach will systematically and predictably eliminate all so-called ‘paradoxes’ from language usage, just as Galileo’s approach removed the interminable discussions on falling objects. In my view, the ‘paradoxes’ are just muddled usage of the communication system called language. ‘Paradoxes’ are nothing particularly esoteric or surprising; as the programmers sometimes say, it is a matter of ‘garbage in, garbage out’. As long as communication is careful and is focussed upon reality, instead of upon the unsound ‘rules’ of ‘logic’, I think that all such ‘paradoxes’ just fade away.
178. Whilst in the middle of writing this work on Gödel, I decided that it was important first to clear the ground by discussing the so-called ‘paradoxes’ in some detail. This discussion, I have placed in this separate document in order to maintain greater clarity. I think these so called ‘paradoxes’ would be better called ‘muddled nonsenses’. These ’paradoxes’ are misunderstandings intimately involved in the work of Gödel and others, when they are discussing some elements of modern ‘logic’ or establishing ‘proofs’. Let me say at this point that I have more than a sneaking admiration for Gödel who, in a sense, took what was simple muddle and built a surreal nonsense out of it.
179. Modern ‘logic’ delights in masses of symbolism, which varies wildly and almost randomly from work to work. I take the view that anything that can be presented in obscure symbols can better and more accessibly be expressed in plain English. The symbolism is constantly justified as improving clarity. Whereas, it is more usually, in my view, an egotistic attempt to give credence to an obscure and muddled pseudo-religion and to surround a considerable amount of mumbo-jumbo in an obfuscating veil. Then ‘outsiders’ do not come to realise that there is very little substance behind the curtain, and thus the priestly sinecures can be maintained. It has been said that the purpose of “culture is as an instrument wielded by professors to manufacture more professors”[1]. There is much too much truth in this in the area of ‘logic’.
180. In the second document on Gödel, Gödel and sound numbers, I discuss what I regard as severe empiric problems with Cantor’s mapping procedures. In this document, I first intend to to return to Cantor to comment upon his diagonalisation procedure, a procedure I regard as bordering upon the irrational! (See also Logicians, 'logic' and madness)

Cantor’s diagonalisation

181. Cantor’s techniques have the tendency to look interesting, and even simple, until one starts to think about them critically. In recent years, they have even been taught in schools, often under the heading of ‘new mathematics’. This is fine and dandy as long as you don’t look too closely, and you regard work from the early 1870s, based on ideas going back to Galileo in 1638, as ‘new’.
182. The fundamental idea of diagonalisation is very simple to grasp, until as I stated above, you look closely and find what a Pandora’s box of confusions lurks beneath the surface.
183. It is my experience that, if something appears extremely difficult to understand despite patience and careful effort, it is often wise to begin to suspect that the teacher (or the writer of the book) does not understand the subject themselves, or that they are not very good at teaching, or even that they are talking nonsense. You will note that I used the words begin to suspect[2], for I have also noticed that dull people, when they cannot understand something immediately, tend to leap to a conclusion that a person they cannot understand is stupid!
184. Many a great thinker has been shown to be incorrect as knowledge advances, that does not in any manner indicate that they were stupid in the context of the information available in their own time. Likewise, the reality that a child will often make mistakes due to lack of information is not in any sense convincing proof that the child (or adult) is an idiot; in fact, often they may be unusually adventurous and curious, and perhaps over impulsive.
185. Consider 4 decimal fractions arranged so they are in a block of 4 by 4 digits.
     
     
     
     
     We can make up a new fraction in a rather strange way.
     We can take the first digit of the first fraction and add 1 to it, that is 3+1 = 4;
     for the next digit we can take the second digit of the second fraction and add 1, that is 1+1 = 2, and so on.
     Thus, we form a new fraction by stepping down the diagonal (top left to bottom right) of the block, hence diagonalisation. The fraction obtained is:
     .4204
186. Now, we know that this number cannot be the same as any of the four numbers we already had, because it is different in at least one of its digits (numbers) from each of the other numbers; and that is how incredibly simple the diagonalisation argument of Cantor is—at first!!
187. Consider a number made up of 1 digit: there are exactly 10 such numbers possible if you include 0 as a number. They are 1,2,3,4,5,6,7,8,9,0. With 2 digits, you can make the numbers 00 to 99, that is a total of 100 combinations, 1 to 99 and again 0. With 3 digits, the number goes up from 100 to 1000 numbers and with 4 digits, up to 10,000 possibilities. So there are 10,000 possible decimal fractions in the example above (paragraph 185). That is, with one digit there are 10¹ possible combinations, with 2 digits 10² possible combinations and with n digits there are 10ⁿ possibilities (in a ten-based number system).
188. Consider that we wrote out a few of the possible 4-digit fraction examples on a piece of paper and then applied the trick of increasing by one each digit on a diagonal. Sure we would have a number, but that number could already be elsewhere in the list, we can not now guarantee to have constructed a new number by the diagonalisation process.

     

     Indeed, if we had a list of all the 10,000 possible 4-digit fractions when we did the diagonalisation trick, we could in fact be certain that our newly formed number was already in the list.

189. We could only guarantee obtaining a new number by extending all the 10,000 4-digit fractions out to 10,000 digits each, because now we would have only a very small fraction of the possible numbers with 10,000 digits (remember, for 10,000 digits there is a possibility of 10^10,000 fractions, vastly greater than a googol! - see paragraph 187 above), and our new number would again differ from each one of the 10,000 digits in at least one place (the place where we added one to that digit).
190. All very interesting and rather straight forward. But Cantor had something rather different in mind. Cantor concerned himself with numbers which ‘go on for ever’! Now, we already know there are severe problems with this notion of ‘for ever’ from Gödel part 2: Gödel and Numbers. But let’s humour Cantor, and many a modern mathematician, for a little while longer.
191. Consider the fraction 1/3. To convert it to a decimal, you divide 3 into the 1 and you end up with .333333… …—yes, that’s it, it’s ‘going on for ever’!!
     Oh damn, hold on a minute. Let’s just take the first three of the 3’s after the decimal point and turn them back into a fraction—that is 333/1000. Now, you will notice that if you add three lots of 333/1000, you end up with 999/1000. We’ve lost 1/1000 in the process. This problem will continue to exist however many 3’s we take after the decimal point, albeit the problem is growing ever smaller until, we hope, it will disappear up it’s own whatnot and go away.
192. Unfortunately, mathematicians have rather a hang-up about things being what they imagine to be ‘exact’. Hence, they have this notion of numbers that ‘go on for ever’. To be unkind for the moment, such an idea is nuts. But at least it makes the sums sort of come out ‘right’. So the idea is that, if the three’s go on ‘forever’ after the decimal point and if you add three of these monsters together, you will, in confused imagination, return to the original 1. And then everyone, or at least the mathematicians playing this strange game, will be ecstatic.
193. The numbers like 1/3 and 333/1000 are called rationals[3] because they form the ratio of two natural (‘ordinary’) numbers like 1,2,3 and 333 and 1000 and so on; that is, no untidy fractional bits hanging on the end , let alone bits ‘going on for ever’.
194. ‘Numbers’ like .33333..., ‘going on for ever’, some mathematicians call irrational. Now they tell me that they call them irrational because they cannot be formed from the ratios of natural numbers, but you and me both know the real reason don’t we? Or perhaps they think to call them irrational because they won’t behave the way they want them to! See also continuous and zeno’s  ‘paradox’.
195. Cantor, who was of a mystical turn of mind, was rather interested in ‘things’ which ‘went on forever’[4] so he spent a lot of time discussing what different ‘kinds’ of infinity ‘existed’. Part of the means he used for this was a far-fetched version of the diagonalisation process that I have discussed above.
196. What Cantor proposed was that, as the numbers went on ‘forever’ (as long as you forget the systematic relationship discussed above!), you could always make up another one by use of the diagonalisation method. Just exactly how he, and the mathematicians who gleefully seized upon this notion, were going to actually manufacture one of these, ‘going on forever’, numbers has yet to be revealed. But worry not you unbelievers; just trust in the miracle as performed by the current high priests. They really really do have the power, so I am told.

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Richard’s ‘paradox’

197. Nagel and Newman [6] claim that Gödel used the paradox of Jules Richard as a model for his opus magnum. They then give a rundown of what purports to be the Richard paradox. But, while trying to find my way through the soup of the description they gave, in my usual picky and thorough manner, I became steadily aware that the description was not exactly limpid. As usual in these circumstances, I started to hunt high and low for various descriptions of the supposed paradox.[7] Eventually I ended up with about eight, but every one was rather scruffy. The more I looked, the more it was clear that the ‘version’ given by Nagel and Newman was not the original or even the Richard’s ‘paradox’ at all at all!
198. As far as I can make out, the Richard’s ‘paradox’ is merely a play on the Cantor diagonal argument written out in words. Effectively, ‘.333’ is merely replaced by point three three three etc, both no doubt ‘going on forever’. As you may know from elsewhere, I state that there is no essential difference between mathematics and English, they are just expressed in variant symbols.
199. So exactly why Richard’s version—or perhaps I should say copy—is called a ‘paradox’ while Cantor’s version is called an ‘argument’ or ‘proof ’, I am not at liberty to reveal. I have suspicions that, when the so-called ‘mathematical’ ‘idea’ is put into words people may then more easily grasp that it looks rather dodgy; despite the oft-repeated claim that mathematics is more clear and reliable than plain English.
200. I have not yet seen anywhere described the problem with the Cantor diagonal argument that I have laid out above, nor the isometric difficulty with Richard. However, it is my strong impression that this stuff is repeated by professors and books one from another, with very few ever understanding it or even attempting to so do. For myself, I regard the ‘two’ versions as muddled nonsense.

Russell’s ‘paradox’

206. In Russell’s barber of Seville ‘paradox’, the magic is rather less disguised; so I shall first use that example as an illustration. The following is the way the game is laid out:
207. The only barber in Seville stated that he shaved everyone in the village who did not shave himself. Russell makes no mention of those who do not shave, or of women who do shave or of women barbers, or even how the barber manages to visit everyone in Seville! This ‘paradox’ is a dressed-up version of the ‘set of all sets paradox’, compare with the error called ‘complete’. By the end of this section, any problem with the set-of-‘all’-sets ‘paradox’ should have disappeared!
208. Now to look at the problem without being too easily distracted.
     On the face of it, the statement concerning the barber in paragraph 206 is a perfectly innocent one until someone asks, “Who shaves the barber?”
     The standard analysis goes like this:
     If he does not shave himself, then the barber is one of those in the village who does not shave himself and so is shaved by the barber, namely, himself. If he shaves himself, he is, of course, one of the people in the village who are not shaved by the barber.
     But that is to be caught in the error of the ‘paradox’, or maybe this analysis is given expressly in order to remove your attention from the hand of the magician.
209. The situation is much clearer if you ignore the flummery and first attend to the two lists that have been assembled:
     
     1. those shaved by the barber, and
     2. those who shave themselves.
     Should you be distracted by the sleight-of-hand, you are invited to imagine that everyone in Seville is supposedly on one or other of these two lists. Now in the real world, either the barber shaves himself or he does not, whence you might think he could be assigned to one or other of the lists. If one is content to assign him to one list or another, one must then reject the original statement highlighted above as clearly being false.
210. Whether or not the purported barber shaves himself is a matter of empiric fact, it is not a ‘matter of logic’. You may ask either the question, “who shaves the barber?” or you may ask whether the original statement is true, or even rational.
     
     A BOOK CATALOGUE VERSION
     For practice, here is another dressed up version concerning books and book catalogues:
     
     First you go to your library and read all your books, whilst making note of which books mention themselves in the text. You make up two lists, one of the books which ‘mention themselves’ and a second list for the rest.
     
     Next you bind the two catalogue lists and make them into books. Then comes the conjuror’s trick question, “In which catalogue should you list the catalogue of books which do not mention themselves?” If you list it in the catalogue of those that do not ‘mention themselves’, then clearly the catalogue is now ‘mentioning itself’, and so belongs in the catalogue of books that do ‘mention themselves’. But as it now ‘mentions itself,’ it is no longer just a catalogue of books that ‘do not mention themselves’.
     
     Returning to the empiric requirements for sets discussed in Gödel and sound sets (paragraphs 16-24), to begin with, the catalogues are used to list the two categories of object. Then and only then, an attempt is made to put the category or ‘set’ ‘in itself’.
     
     Russell tried to deal with this error by what was called the theory of (hierarchical) types. But that suffers from the ‘same’ problem: “Where do you file the type of types which do not contain themselves?”[12]
     
     Russell’s ‘paradox’ is founded on the error of not distinguishing the formation of a new item, that is the forming of the catalogue, from the original items to be catalogued. It is an error of not taking into account the time change between the original classification and the introduction of a new item and, with the time change, the change of conditions.
     
     It is an error of not noticing that the category ‘itself’ is an item in the real world. When you form a ‘set’, you form a real-world box or you mark a chalk circle upon the ground, into which to arrange your collections. These so-called ‘paradoxes’ attempt to pick up the chalk circles or boxes, or little programmes in your head (compare with universals), and put them into themselves. I suggest that you get yourself a box and try it.
     
     In the Turing version, the attempt is to feed the ‘machine’ or algorithm ‘into itself’ (for more see Metalogic B1 – Decision Processes).
     
     
211. As stated above, I cannot read the minds of those who are confused by such ‘paradoxes’, so I am in no position to say just which of the possible approaches will suit you, or Bertrand Russell up there, laughing on his cloud.

The Cretan liar ‘paradox’

212. Now, we return to the Nagel and Newman attempt at explicating the Cantor-Richard soup. Sadly, they appear, in fact, to have given a rather muddy version of the ‘Grelling’ ’paradox’, not the Cantor-Richard game.[8]
213. The so-called Russell paradox is a dressed-up version of the Cretan liar paradox, dating from Greek times.[9]
214. Below I have given what I hope is a clear description of the fundamental, but rather trivial, error at the heart of the ‘Russell’ ‘paradox’. The error also occurs under the heading of the Grelling paradox.[10] I will discuss the Grelling ‘paradox’ after dealing with the simpler muddle of the Russell version. Remember that, as I cannot enter the minds of the proposers of these nonsense games, I cannot be sure just what errors they are making in their own minds.
215. In each case two lists are proposed. The magician’s flummery[11] often disguises this fact in one form or another, whether from the audience or from both the viewer and the conjuror is not entirely clear. Russell, for example, gained some degree of fame through the publication of ‘his’ paradox, and his name remains known widely in association with the ‘paradox’ to this day.
216. Once the two lists have been agreed, a new item is invented and surreptitiously introduced—a name for each of the two lists—and we are then asked to decide into which list this new entity (item) should fit. We are thus encouraged to attempt to file the list ‘in itself’!
     As it is important to understand clearly the issues involved, I will now go into Russells’s error in some detail, complete with illustrative examples.

The Return of the Cretan Liar

212. The original Cretan liar paradox runs as follows:
     A Cretan makes the statement, “All Cretans are liars”.
     Of course, if the statement the Cretan made is true, then his statement must be a lie, or so goes the accompanying magician’s spiel. In order to think clearly about the Cretan liar, it is well to note that …>>
213. there is a hidden assumption in the Cretan liar ‘paradox’ which is that all Cretans lie, at all times, on every ‘occasion’. Our focus should then be not upon the Cretans, but upon the statements made by Cretans! Once again, the magician has distracted our attention by focussing our attention upon the Cretans instead of the statements. Possibly, this is a more subtle trick than Russell’s modern copy.
214. In just the ‘same’ manner as in the barber ‘paradox’, where in reality the barber either does or does not ‘shave himself ’, in the real world Epimenides’s Cretan jokester is either ‘a liar’ or he is not.
     Of course I know people who could be called ‘liars’, but really they much better described as “people who sometimes make statements that appear not to be in accord with ‘facts’”. This is an important distinction (for more, see the errror of the verb ‘to be’ ). This idea of someone ‘who always lies’ is part of the unrealistic absolutism that is fundamental to Aristotelian ‘logic’ and much ‘mathematics’. It contains both the error of ‘each’ ‘lie’ being a discrete ‘event’ and the error that a statement is either a ‘lie’ or it is not!
215. Notice that in the Russell version; the word everyone replaces the hidden every statement in the Greek version.
216. Once a reasonably clear meaning for the word occasion is settled upon, the statement, “all Cretans lie on every occasion”, becomes a matter of empirical verification. If any Cretan then states a verifiable fact, the statement, “all Cretans lie on every occasion”, becomes false. Therefore the statement, “all Cretans lie on every occasion”, if factual, could be made by any person not appearing on a list marked ‘Cretans’.
217. If the statement that “all Cretans lie on every occasion” is currently factual, it remains factual unless and until any particular Cretan makes a truthful statement. The moment a Cretan makes a factual statement, the claim that “all Cretans lie on every occasion” becomes false, even though the statement may have been factual in the past.
218. If the statement made by a Cretan happens to be, “all Cretans lie on every occasion”; this statement then becomes false at some point because, in this case, a Cretan is now in process of telling a truth! You may decide ‘precisely’ at ‘which’ ‘point’ the statement becomes false. Note carefully the change of status of the statement occurring over time as the statement is expostulated. Once again, the magician distracts attention from the changing world.
219. The two lists, on this occasion, are the list containing lies and the list containing truths spoken by Cretans. The fact that the paper headed, ‘truthful Cretan statements’, has no entries until some Cretan makes a true statement, I state to be ‘irrelevant’. The new item is the true statement (‘all Cretans lie’), just as the new item in the Seville barber trick was the introduction of the barber as a subject for classifying on one or other of the two lists. But at the point when a Cretan tells the truth about Cretans, the previous state ceases to be true, that ‘all Cretans lie’.
220. You will note that the above discussion relies upon the relaxations of rigour:
     1. that ‘objects’ may be defined sufficiently clearly and
     2. that statements about those objects can be sufficiently clear to be declared as ‘true’ or ‘false’. (See also the excluded middle.)
     Without these relaxations of rigour two-value (on/off) logic becomes untenable or meaningless.
     

Grelling’s ‘paradox’

221. Grelling’s[16] ‘paradox’ makes two lists:
     1. a list of words that do describe themselves, e.g. short is said to be a short word; and
     2. a list of words that do not describe themselves, e.g. long is not a long word.
222. So far so good—then a new word is made up to describe these two lists. In order to cover the trick more confusingly,
     the words which do not describe themselves are termed ‘heterological’,
     and those that do describe themselves are termed ‘non-heterological’.[17]
223. To remove the confusion of the double negative involved in this nonsense, I will start the play by changing the definitions around so that
     the words which do describe themselves are termed ‘autological’, and
     those that do not describe themselves are termed ‘non-autological’.
224. So now autological, in this version of the ‘paradox’, means ‘words that describe themselves’. Thus, autological describes a characteristic of other words.

     Then comes the usual awkward question; “Is autological, autological”. In order to ask this question about the word autological, you need two copies of the word autological. You need one copy of the word to ask the question about and another when you construct the question, “Is this word ‘autological’?”. Or stated more fully, “Is this copy of the word autological (copy one), autological (copy two)?”.

225. So is copy one autological? Well, the word autological tells you something about other words which it describes (that is, whether they ‘describe themselves’). The word autological (copy one) is used by the person categorising the words to call those other words autological. The word autological (copy one) does not call the other words anything; the person using the words uses it to label other words.
226. Now this was allowed to slip by in the original statement (paragraphs 222-224 above) concerning the words. Words do not ‘describe themselves’; they are used by humans to describe things, in this case, other words. It was an error to blithely accept, in the first place, that a word could ‘describe itself’.
227. Copy two can be used to describe copy one, but no word can ‘describe itself ’.

Humans describe things; words do not

228. Humans make lists, and decide what to put into the lists and what not to put into the lists.
229. Barbers and shavers are real-world objects that can be described by words, so are instances of lies told by Cretans. Words may also be described by other words. Lies are, of course, combinations of words that are being listed as so, or not so—true or not true. But the word lie, which refers to the untrue communication, is not itself the untrue communication ; any more than the Cretans, referred to by the word Cretans, are the word Cretans typed upon a piece of paper.
230. If you are to use a word to describe ‘itself’, you need a copy of the word to describe it. For example, I can say, “Look at the word¹ word² upon your screen”. The fact that the two occurrences of the word word may look rather alike does not mean that the two versions are performing the ‘same’ task. The first one is pointing at the second and, of course, the second one is sitting still being pointed at. Remember that some person is waving the first one word¹ around to do the pointing; the word¹ is not doing the pointing all on its ownsome without human intervention. (Remember chopability.)
231. So again, is the word autological autological, in other words does the word autological ‘describe itself ’? It really depends upon what you, the wielder of the word or the chopper, decide. The word sure ain’t going to decide for you, and neither am I. perhaps it is time to have a think about what ‘meaning’ means (see also essence, meaning and empathy). I will return to this in a while (see Berry and Metalogic B1 – Decision Processes).

recap

232. Remember; in the original version of Grelling’s ‘paradox’, ‘not-heterological’ is taken to mean a word which does describe itself; while ‘heterological’ means a word which does not describe itself. As stated earlier, this is just another trick designed to confuse matters further, so that you do not look too closely at the real card being palmed by the magician. It has no bearing upon the real game that is going down.
233. Paradoxes are rather like magician’s tricks, they often rely upon sleight-of-hand—the real problem is often obscured by obfuscation. These ‘paradoxes’ are not really ‘problems of logic’, they are confusions of communication and word usage. You must keep your eye on reality, not be distracted by the magician’s waffle, even when they are also confusing themselves with the patter. Always attend to the real-world marks on paper, or sounds in the air, and ask yourself, “Just exactly which is the sticky label and exactly at what is it really pointing?”

Nagel and Newman

234. The description in Nagel and Newman is a highly confused version of the Grelling ‘paradox’.[18] As stated above, Nagel and Newman claim to be describing the Richard’s ‘paradox’, which most clearly they are not.
235. Nagel and Newman have also a comment stating, “Up to a point, the structure of his [Gödel’s] argument is modelled as he pointed out, on the reasoning involved in the ‘Richard’s paradox’”.[6] Elsewhere[19], I have seen comments that Gödel’s work is based upon the Cretan liar[29]. I think that Nagel and Newman have, by using a version of Grelling, used the appropriate ‘logic’ in their description of Gödel’s intent, but they have mis-attributed their exposition to Richard.
236. To complicate the issue a little further, Nagel and Newman are using numbers-labels to point at numbers. The numbers to which they are pointing in this way have been generated, assembled, ordered and expressed in words in the manner similar to that of Richard.[20] Nagel and Newman then go on to treat the labelled word-numbers in the manner in which Grelling treats other words, maybe this is why they have become confused in their attribution.
237. Nagel and Newman, Grelling and Richard all ‘fall into’ other errors of any rational, reality-based logic. These errors go back to the Cretan liar and to Aristotle, and to the modern accretions upon Aristotle.
238. My prime concern is to show how an empiric and rational logic must function; it is not to expend efforts denigrating the work of many fine people who have attempted to increase human reasoning skills. However, I must clear this deadwood, if I am to convince people to change their thinking habits for something more effective (see why Aristotelian logic does not work, particularly the boxes). Nagel and Newman, and others, appear to think that Gödel’s work is founded upon this stuff that I regard as full of errors of sloppy reasoning, but I am yet to be convinced so I will plough on. As you will have seen, I am not dealing with just one logical problem, but whole cascades of them. I have some suspicion that Gödel merely generated a demonstration of the inconsistency,[21] or, most seem to claim, the incompleteness of this ‘system of logic’. Or else, the system has blown up into such a complexity of confusion that nobody has yet been able to pick their way through the generated swamp.
239. It is now useful to examine another ‘paradox’ in order to focus upon an endemic problem with Grelling, and the Nagel and Newman, supposed version of Richard’s ‘paradox’.

Berry ‘paradox’

240. Our next beauty is called the Berry[16] ‘paradox’ and it goes like this. Consider the expression, “The least natural number not nameable in fewer than twenty-two syllables”. Note that the expression takes up twenty-one syllables;[22] notice the usual ’not’ to confuse the issue. This so-called paradox is trivial in the extreme, but provides a theme for useful discussion.
241. Remember the googolplex? It is possible to name a large assemblage of ‘parts’ as ‘motor car’. There is no number that cannot be named in a single word, you only need to make the word up for the purpose, as was done in the Grelling ‘paradox’.
242. I think it possible that Berry had in mind playing with the numbering method of Richard. Whereas, in as much as I understand Richard’s intention, Richard was attempting to mirror ‘numbers’ in ‘words’, in order to apply Cantor’s diagonal argument to language (remember most mathematicians struggle under the misapprehension that mathematics is somehow ‘different’ from language!).
243. Cantor seems to be claiming that the reals cannot be ordered definitively, a claim that I regard as dubious, as you will see from paragraph 186 onwards. But once you start adding ‘definitions’ of ‘numbers’, such as the one chosen by Berry, or the similar confusion introduced in the Nagel and Newman discussion, or even Cantor’s ‘on for ever’ notion, ordering does indeed become a fraught problem.[23]

More on ordering [24]

244. In a list of words purporting to be ‘numbers’, suppose that those ‘numbers’ are ordered—as they are claimed to be ordered by Richard and also by Nagel and Newman. Then suppose that a new ‘number’ is invented, as is the case with the supposed ‘number’[25] suggested by Berry and also, in effect, by Nagel and Newman.[26] Remember, of course, it is always possible to invent a new supposed ‘number’ in such a manner, with the application of a very small amount of ingenuity (see paragraph 241).
245. Then any ordered system of labelling applied to the list of ‘numbers’ becomes disrupted. This matters when the labels are supposedly ‘logically’ related to the ’number’ listed alongside. This situation definitely hovers at the edge of the discussion by Nagel and Newman, but is not considered by them. Advanced programmers will recognise this as the problem with absolute labels when another line of code is inserted above the label. In high-level computer languages, the labels are relativised and the software (compiler) sorts the problem out for you.
246. This rather echoes Cantor’s idea of numbering, where each diagonal number or extended number is taken to be a new invented number, and thus disrupts the ordering of the reals.
247. By the nature of Nagel and Newman’s discussion, inserting the new item for the list anywhere but at the end is very likely to disrupt the coherence of the list. If Richard did have in mind a similar list to that suggested by Nagel and Newman, such a list would also be disrupted by an invention such as the one by Berry. It is clear that there is very little limit to the number of such definitions that can be generated by creative humans, beyond those defined by the intrinsic structure of machines or visual acuity. (For further discussion, see note on Landauer and the comments on a perfect circle in why Aristotelian logic does not work). Thus eventually, ordering becomes an issue of physics and iterated ‘agreements’.

More on lists [27]

248. We must be very clear on just what lists we are making and which items are in those lists at any particular time. We must decide when we are about to add an item to our lists and whether that item makes sense as a ‘member’ of the list or ‘set’. These matters are matters of definition and decision, they are not somehow pre-defined upon high.
249. Returning to Grelling’s ‘paradox’: the decision, whether to call an item ‘autological’ or not, is hardly closely defined—is short a short word? Well, is 5 letters ‘short’? Or is the word short in fact high, because the letter ‘h’ in short stands higher than short letters? Whereas massive would be a short word because it keeps so low. Or is short a rather long word, because it may take more chalk to write it on a blackboard than does a word like icicle? Each decision is made by an individual; it is quite likely that at times different people will make different decisions.
250. I left the discussion previously by asking you to decide whether autological is autological, or not. Was Aristotle ‘tall’? To decide, apart from knowing the height of Aristotle, it is well to note that ‘tall’ is a relative term. Clearly, Aristotle would hardly be thought of as tall, if compared with a giraffe or the Empire State Building.
251. Is autological a relative term? Is it autological relative to previous decisions made when compiling your list of autological terms?
252. Remember ‘the asymmetry of not’. The Grelling paradox requires you limit yourself to ‘exactly’ two boxes. Perhaps you would prefer extra boxes for words that you were not sure about: for example, you could have a box labelled, “almost, but is it really?” Remember as well, when I was discussing Russell’s ‘set of all sets’(paragraph 206 onwards), that I ended up concluding that he was attempting to put the box into the box; that is, picking up the box and attempting to put it into itself! Likewise, Grelling manages this trick as well, by attempting to pick up the word autological at the top of one of his lists and then discussing into which list it should be assigned. So now he has no heading on one list, in order to know what to put in that list.
253. Grelling even suggests taking the word over to the other list and seeing whether it belongs there. But look, that list may also have a copy of the word autological as part of the term non-autological at the head of that list. You surely do need to attend to which copy you are going to call the word, and which you will call a copy. If you have several copies, it might be advisable to number them to make them clearly different.
254. The central error comes from assuming the word can talk, or point, of its own volition. The word is but marks on paper (or some such); it does no pointing, it just sits there looking pretty. Short does not ‘describe itself’, it just hangs around waiting for some idiot to ask it if it is short; and then it tends not to answer. When said idiot asks it if it is ‘short’, that idiot uses another copy of the word short in order to pose the question. Then the idiot ‘thinks’ to itself, “Is this word short?” The question demands all manner of real-world data to do the comparison, although it is likely, in this case, to be a comparison with other words, rather than with giraffes. But who knows, or can predict, the comparisons that any particular individual will choose?
255. Again I could ask, “do you think the word autological is autological?” Perhaps you would be better to say “No, it is marks on paper”, or, “It depends on just what you mean by ‘autological’, maybe you would first explain to me clearly what you do mean by ‘autological’?”
256. A word can no more describe without human assistance, than a axe can get up and chop without human intervention. A word can no more describe itself than a axe can axe itself, with or without human assistance. To describe a word, you need another separate word, which you may decide to call a copy of the original word if it helps you, but you must remain aware that the new word is different, despite outward similarities to your eyes.
257. Any similarities are due to matching it against the blueprint of ‘that’ word contained in your head, and to your lack of attention to the real differences between the various copies.
258. You may pick the words up and use them to point at other words and other things, but the words themselves can never up sticks and go do any pointing all on their ownsome. Of course, if you should doubt me, you can always go and ask the word short just exactly how short it is.
259. If you are going to use the words to point, or attempt to follow the pointing of other people, first you must focus your real eyes upon just at what ‘bit’ of real matter they are pointing. When Russell refers to the ‘set of all sets’, first ask him to show you the box with the items inside. And, just in case, check up his sleeves.
260. Because mathematicians want to be important and sound impressive, they tend to make up fancy words for simple ideas. ‘Words that talk about themselves’ did not sound quite portentous enough, so they called such words self-referencing. You’ve gotta admit they have a point: discussing words talking about themselves could so easily draw the attentions of men in white coats.
261. No person can believe that words ‘talk to themselves’ or ‘refer to themselves’ without being out of touch with reality. This is also true of sentences.
262. Words and Sentences do not ‘talk to themselves’.

The word factory

263. Words, when correctly used, are labels; they are nothing more.
264. Consider a warehouse for storing labels. Each label has a single word upon it; the labels are stored in boxes upon shelves in the warehouse. For the convenience of the warehouse workers, each box has a copy of the label with the word printed upon it attached to the outside of the box.
265. I expect they would label the boxes by taking one of the labels from inside the box and attaching it to the outside of the box. Strangely, that would leave one less label in the box. That single label would now be on the outside!
266. Very likely, the warehouse operatives will store the boxes in ‘alphabetical order’ as a means of making their lives easier; but this last is a convenience, not an essential to understanding the warehouse system.
267. When someone wants a copy of a word to put into a book, they go and take a copy from the appropriate box. Those running the warehouse keep an eye on the number of labels in each box, and arrange for more words to be printed when any box starts to run low.
268. The factory has procedures for printing the labels.
269. Returning yet again to the word autological, just how is the magician confusing the issue? First, the magician sets up two lists: the list of autological words which you are told ‘describe themselves’ and another list for the remainder. This second list is a not box and by now we know that we must be particularly cautious with not boxes.
270. Then the magician gives out with the distracting spiel. “For example”, s/he broadcasts to the audience, “The word short is short so that word clearly ‘describes itself’! and then there is the word English which is certainly an English word and the word noun which is also a noun.” And all innocent, we are expected to swallow this baloney. If you look carefully, you will discover that each of these chosen ‘relationships’ is different![28]
     Oh dear, another useful confusion.
271. So let’s go down to the warehouse and buy a copy of the autological label, take it home, and have a look at the instructions before we actually open the little box containing the label.

INSTRUCTIONS

272. The word autological ‘means’
     “take any word, see what the word ‘means’, and then see if the word on the printed label looks anything like what the word describes. For example, is the word ‘short’, or ‘English’, or a noun, or anything else in any way like what the word describes?”
273. Right, so first I have to look at two things,
     1. what the word describes and
     2. what the word ‘looks’ like.
274. Then I have to determine whether I can see any ‘similarities’.
275. So, first the word describes other words. There it sits on the table. Is it describing any other word right now? I don’t think so.
     Is it similar to other words? Well yes, it is constructed with paper, and ink printed into shapes called characters.
     Do the other words also describe things? No, words do not describe. People do most of the describing, unless we are diverted into questions about whether bees describe to other bees where to find honey.
     Can the word be used to describe things? Yes, it can be used to describe whether other labels ‘point’ at things which bear some relationship to some aspect of the word itself, such as shortness.
276. Now, this label can be used to indicate things and it can be indicated. “Oh look, missus! There’s a word on the table that can be used to indicate things.” So the word can both indicate and be indicated (pointed at).
277. Now is that a similarity within the original meaning of the word? Well, I had asked you to decide before; about here, it might be neat to ask you to decide once more!
278. So, now we are closing in on the problem. You can go on ‘forever’ being creative and asking various questions about whether this word has components which match ’their’ use when used to describe other words. When you decide you have a match is entirely a matter for you. I might not be creative enough to find a match of my own; you may convince me that your match makes sense, as when originally you were urged into accepting that the word short was ‘itself’ ‘short’.
279. Only upon the outcome of such discussion and feedback can we decide to agree whether there ‘is’ a match. And then some other lunatic will come along and say; “I don’t see it” or “I disagree”, as if these things are facts in the real world, existing outside each of our individual heads!
280. So, is the word autological autological? Or is the word short short? You tell me! After all, it is your decision, not mine. Me, I just refuse to decide, or I state that I think it is a silly question, or that discussing such babble is a waste of my time. You discuss it if you want! I have to catch a bus.
281. I assert that all ‘words’ and ‘numbers’ are like this.
282. Likewise, you may keep on making up new words with new meanings. For example, using the words of mathematics, you could make up a word like ‘prime’ to mean any number where you can’t find a smaller number (other than one) which will go into it without leaving a remainder.
283. In like manner, Nagel and Newman made up a new word, Richardian; by which they meant something very similar in type to the term not-autological. In their case, the word Richardian was meant to indicate the lack of a relationship between a number¹ and another number (number²), the second number indicating the first number’s place in a list. They then had another list for not-Richardian numbers where this relationship did apply. They then asked in which box their word Richardian belonged. As the word Richardian was supposed to indicate a relationship between the double entries in the lists—the type of ‘number’ and its place number in the list—that relationship was as ill-defined as it is in the Grelling ‘paradox’. Note the usual confusion of defining the ‘positive’ word to indicate a supposed negative quality.
284. The decision as to where to place the word autological is not easy to make (see Metalogic B1 – Decision Processes) and, very likely, the question is better described as ‘meaningless’. As stated earlier, Nagel and Newman, in the course of setting up their pseudo-Grelling, introduce an ordering issue (see also paragraph 245) which I will not repeat here; as to so do would add nothing to the above analysis.

Next week:
The Return of the Gödel

(Continue Gödel’s confusions with A4—The Return of the Gödel)

Endnotes

1. Quoted from Simon Weil, Barrow, p.156.
   
   
2. Words in italics are words being spoken of, as the word used is being used here. Words in inverted commas indicate a rather loose, or even suspect, usage of that word. See C O L O U R K E 'Y for full details of word coding on this site.
   
   
3. By the way, the rationals are said to include the integers, because you can take any integer and convert it to a rational by putting it over 1, e.g. 3 is the ‘same’ as 3/1, 29 is the ‘same’ as 29/1 etc.
   
   
4. Note the comparison with the ‘religious’ notions of eternal gods that are ‘everywhere’ and go on ‘forever’, and ideas of ‘eternal’ ‘life’ (see stopping and changing). I am not joking when I compare modern mathematics with ‘religion’. See also MetalogicA-supplement, currently in preparation.
   
   
5. Pseudo-irrational: My term for conceiving an irrational which is limited at some arbitrary number of digits in order to clarify discussion or just to keep the party clean.
   
   
6. Nagel and Newman, p.60
   
   
7. I have not yet accessed the original description, which was communicated in a letter in 1905
   
   
8. Nagel and Newman, pp.60-63
   
   
9. See exposition in Kneale and Kneale pp 227-8. The Liar ‘paradox’ is often attributed to Epimenides.
   
   
10. See, for instance, Kneale and Kneale p. 656
    
    
11. Sleight-of-hand or misdirection would be other terms.
    
    
12. See also Kahane and Tidman, p.226
    
    
    
13. Note this use of the word are suffers from the problem with the verb to be.
    
    
14. Britain is a small island and parking often causes problems. Our friendly government has found a solution to this problem: they have a system of painting yellow lines alongside the kerb of the roads. A broken yellow line means ‘no parking’, a continuous yellow line means ‘no parking at all’ and a double continuous yellow line means ‘no parking at all at all’.
    
    
15. Quoted in Barrow, p.216.
    
    
16. See Metalogic-A-supplement (in preparation) for biographical information.
    
    
17. Kneale and Kneale p. 656. Hofstadter p. 21, use two separate words, autological ‘meaning’ ‘self-descriptive’ and heterological ‘meaning’ ‘not-self-descriptive’. I have used a different combination for best exposition and clarity; that is, I have used the terms autological and non-autological.
    
    
18. Nagel and Newman do also have a brief unattributed description of the Grelling ‘paradox’ in a footnote on p. 61—strange.
    
    
19. For more detail see Metalogic A4—The Return of the Gödel.
    
    
20. To add still further confusion, it is in my view probable that the ‘numbers’ intended by Richard are not the ‘same’ as ‘those’ envisaged by Nagel and Newman! Another example with a too easy assumption of ‘equality’.
    
    
21. Hilbert seemed to plump for inadequacy in the methods used by Gödel ( Barrow, p. 122), probably because other approaches would have been more damaging to Hilbert’s dream. Emil Post however seems to have come nearer to understanding the problem, by regarding it as a problem involving human psychology (Barrow, note on p. 121).
    
    
22. Kleene p. 39, where it is tacked confusingly on the end of an over-wordy description of the Richard ’paradox’. In Kneale and Kneale, p. 656, the wording is ‘the least integer not nameable in fewer than nineteen syllables’. Berry’s original version was different again.
    
    
23. Again this is something which Nagel and Newman seem to miss when they start to discuss their made-up word, Richardian. In their discussion, the word Richardian is very similar in value to the word non-autological as used while discussing the Grelling ‘paradox’, see the main text for details.
    
    
24. See also comparing predicates ... box
    
    
25. ‘The least natural number not nameable in fewer than twenty-two syllables’.
    
    
26. Where they raise the supposed definition of a new invented type of number, which they term ‘Richardian’ and which ‘number type’ they also suggest to be definable in analogously to the term ‘a prime number’. This false analogy they regard as related to the error of reason in their version of what they call the Richard paradox. But they overlook the ordering issue that they have also conjured up. See also discussion on the status of the term ‘autological’ in paragraph 230.
    
    
27. See also paragraph 24.
    
    
28. Be careful out there. All relationships are different! All ‘things’ are different.
    
    
29. Godel’s work, in fact, uses two methods:
    1 the Cretan Liar, and
    2 Cantor’s diagonalisation argument.
    See also Decision processes.

Bibliography

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