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
Gödel: the final showdown

Index

ON GÖDEL

285. For a less formal expression of some of the ‘points’ below, see Why Aristotelian logic does not work. This will allow adaptation to these concepts, (which concepts at times may be found unfamiliar) by virtue of experiencing the ideas in differing contexts.
286. Gödel was either highly confused, or he took a very sloppy system and produced a non-sense which showed the system to be non-sense. Whether Gödel was himself confused or not, I cannot tell because I have no access to his mind. I do know that the system, and the non-sense construction he assembled from that system, are to be expected by reason of the multiple levels of non-sense buried within that system.
287. I have written three other documents disassembling the various levels of confusion and non-sense inherent in that system. It is important to focus on the fact that mathematics is only a subset of language, no more or less. I find very few people surprised at the fact that idiotic things can be stated in ordinary language, yet they often seem thrown by the fact that non-sense may be stated in numbers. And this, despite a constant flow of such non-sense in the form of contrived ‘statistics’ being available in every ‘news’-paper, from every politician, and from many another source.
288. I will now deal with what Nagel and Newman call the “heart of Gödel’s argument”. This argument is supposed to be of great subtlety; but, to me, it looks rather trivial, once the groundwork is laid and the argument is reduced to its simple components.
289. Gödel’s ‘theorem’/‘theorems’ are essentially constructed upon the liar paradox of Epimenides. I have mentioned some problems with this in the first three metalogic documents A1, A2 and A3. I shall now gather the issues together as I go through the Gödel argument, in order that they are in context, but at the expense of some repetition.
290. To emphasise once more: I cannot guess which bits Gödel, or any other individual, might regard as cogent, or even central to the waterfall of confusions. Yet again, you must realise such judgements are purely individual. There is no Rosetta stone or bible to give you some authoritative set of ‘answers’. Authority is an illusion that only comes to have real effects by the individual adherents acting on their particular beliefs. Lemmings, driven by their migratory instinct, appear to think there is advantage beyond the water edge, so they swim rather blindly onwards. They often swim until they can swim no longer, then they drown. Their actions do not ratify their beliefs.
291. That large numbers of humans are inclined to believe strange things, such as astrology and words ‘referring to themselves’, does not make those beliefs sound, optimal or even rational in everyday context. If a system of thought proves profitable, it is common for that belief system to persist far beyond the areas in which it was developed and the times at which it was profitable. Such is the nature of ‘religion’ and strange taboos. Mathematics, despite its immense usefulness, now carries many of the characteristics of such redundancy-rich belief systems. It is my view that the structures that end in Gödel’s demonstrations comprise such a situation.

292. “I am now lying”, often called ‘the liar’ or ‘the Cretan liar’ or ‘Epimenides[2] paradox’, is a Greek semantic puzzle, possibly invented by Eubulides of Miletus, a contemporary of Aristotle, around the mid-4th century BC.
293. Eubulides took the lead in Megarian criticism of Aristotle’s doctrine of categories, his definition of (and belief in) movement, and his concept of potentiality. For Megarians, only what is now actual is possible. As you might imagine, I feel more than a little sympathy for this kind of point of view. I am told that some passages in Aristotle’s writings are probably retorts to Megarian criticisms. See also Time 2 box.
294. Another form of this puzzle is, “this sentence is untrue”.
295. As early as the 6^th century BC, the Cretan Epimenides allegedly observed that, “all Cretans are liars”, which, in effect, means that, “all statements made by Cretans are false”. Since Epimenides was a Cretan, the statement made by him is false. Thus the initial statement is potentially self-contradictory.
296. Paul of Tarsus[3] (sometimes called ‘saint’ Paul), misunderstanding the original logical-puzzle nature of this statement, castigated the Cretans as liars.[4] A little learning is sometimes said to be a dangerous thing!
297. Another form was given by an English mathematician, P.E.B. Jourdain, in 1913, when he produced a card on one side of which was printed:
     “The sentence on the other side of this card is TRUE.”
     On the other side of the card the sentence:
     “The sentence on the other side of this card is FALSE.”
298. Yet another form of this puzzle is:
     “The sentence below is TRUE.”
     “The sentence above is FALSE.”
     (For more on this form, see a programmers approach below.)
299. Several of these puzzles have been built upon this foundation of the Cretan liar puzzle, as has been explored in detail in Gödel and the ‘paradoxes’. As will be seen, it was upon a careful adaptation of this ancient puzzle that Gödel constructed his demonstrations.

Saying silly things

300. It is all too easy to say silly things in language. It might be thought that, for most people, it is easier to say silly things than to say sensible things.
301. Consider the sentence, “This is a horse”, the speaker meanwhile pointing at a tall building. Or perhaps the speaker is pointing at a duck and saying, “Look at that tiny elephant”.
302. To extend the idiocy still further, we could say, “This sentence is a unicorn”. Clearly a sentence cannot be a horse, let alone a ‘unicorn’.
303. However, one could write, “This sentence is a sentence”, and people would likely be content. Such easy acceptance is, however, potentially a problem. Consider the question, “Which sentence is a sentence?” If you examine the sentence with care, you should start to wonder just to which sentence the phrase, is a sentence, refers.
304. Returning to “this sentence is a sentence”, perhaps the phrase is a sentence refers to the beginning words this sentence, but “this sentence” is hardly a meaningful sentence (for detail, see paragragh 50 and 51).
305. As demonstrated by Lewis Carroll, another most able logician, one may construct sentences of incredible non-sense and lack of meaning that can sound as if they are structurally well formed.

     ’twas brillig, and the slithy toves
     Did gyre and gimble in the wabe;
     All mimsy were the borogoves,
     And the mome raths outgrabe.

     “Beware the Jabberwock, my son!
     The jaws that bite, the claws that catch!”
     

306. Assertion:
     a statement can only be said to be ‘true’ if:
     
     1. it refers clearly to areas of external reality that are accessible to the people communicating;
     2. the common meaning of the words accord with that observed reality;
     3. there is sufficient agreement between the parties concerning what is perceived;
     4. the meanings of the terms being used can be established through an iterative process;
     5. the meaning of sufficient is determined and expressed by each party individually.
        
        
307. Returning to the liar puzzle, consider the sentence, “this sentence is true”. Just what does it mean to say that a sentence is ‘true’? By common interpretations within western society, the sentence immediately fails any check against external reality. The sentence is not pointing to any external reality, instead it pretends to ‘talk about’ ‘itself’.
308. In what sense can the sentence ‘itself’ be said to be ‘true’?
     Is it perhaps truly written in ink? The sentence certainly doesn’t claim anything about what it is ‘written in’. One could perhaps say that it was ‘truly written’, but of course one would then complete the meaning of the sentence thus: “this sentence was truely written by a person in ink”; clearly a possibly true statement, but now we are referring to how the sentence was written. That is, it was written by a person standing in a pot of ink—no, not really.
     
309. And I will tell you, the sentence which you see was written in ink. Note very carefully that I have now formed with my voice a new sentence as follows: “the sentence at which I am pointing was written in ink”. This new sentence is not the sentence on the paper at which I am pointing.
310. It is a fundamental error to imagine that sentences say anything. The ink just lies in the shape of letters upon the page. People say things, sentences do not ‘say things’.
311. As the ‘sentence’ upon which Gödel bases his ‘theory’ has no real meaning; the ‘theory’ merely ‘proves’ that it is possible to construct meaningless statements in language. Of course, that is clearly trivial, as is shown by sentences such as, “I rode a unicorn to work this day”.

     Remember the original liar ‘paradox’ does not read “this sentence is true”, but reads “this sentence is untrue”. It could as well take the form, “this sentence is not true”. Now, I have shown you that the sentence, “this sentence is true” is not referring to anything outside ‘itself’, simply because sentences can’t speak.
     
     Naturally, the sentence also cannot tell you what it is not. It cannot tell you anything.
     
     

312. Gödel took the sentence, “this sentence is untrue”. He then constructed a form of this sentence in a version of the symbolism of arithmetic.
313. The form he constructed was: “this sentence is unprovable”. He then went on to define closely, but in my view rather irrelevantly, a definition of ‘proveability’.
     
     

proveability

Gödel’s definition amounts to:
If a mathematician (person), by following the fairly clear rules of arithmetic, can produce a result from the application of the rules, this application of the rules is to be called a proof .
That is, it is a proof of the final result.

Process:
1. Start with the formula: x times y = z, or as that is usually written: xy=z.
2. Substitute for x, the number 329 and for y, the number 421.
3. We then have 329 times 421 = z.
So what is the value of z?
4. First place the numbers, one over the other, as follows
329 x
421
5. then carry out the process learnt in primary school…
6. The ‘answer’ turns out to be 138509.

This intervening process is the proof that xy = 138509, where x is 329 and y is 421.

The answer, often also called a ‘formula’, is ‘proved’ by the list of actions involved in ‘doing the sum’.

Be very careful to note that these rules have been refined and pre-agreed by a consensus of many human individuals over a long period of history.

The list does not call anything anything, the list does not speak.

Looking at the last line of the proof, at some time someone may say; “that formula is provable”. Just as they might look at a tree, feel their muscles and the edge of their axe, and sing out, “that tree is chopable”. (See proveability section in Gödel and sound sets.)

The action of proving requires a list of rules, a series of actions by a person following those rules, and a decision that all the actions are in accord with the rules. The series of actions, I will posit have been recorded upon a piece of paper by writing them down as they are thought of.

For a formula to be said to be provable, all these real-world acts and items must be present, just as with the chopable tree there was a person, an axe and a series of actions over a time period. Provable ‘means’ the total situation, just as chopable did in its place, as previously discussed.

314. Returning to the form of the liar, “this sentence is untrue”, which Gödel changed into
     “this sentence is unprovable”, remember that the result is what we set out to prove in a demonstration or a ‘proof’.
     It is vital that you keep clear in your head the difference between a ‘proof’ and a result that is ‘proved’.
     Something that is ‘proved’ is only the last line in the list; the ‘proof’ is the whole list, including the last line! Without the process of proving, there can be no proof; just as without the process of chopping, there can be no chopped down tree.

consistency

By ‘inconsistent’, Gödel meant that the rules of mathematics were capable of generating ‘contradictions’. That is, it is possible to generate formulae that may be both ‘proved’ and ‘disproved’ by application of the rules of the mathematical (or arithmetical) system. For example, both the propositions 2+2=4 and 2+2#4 could be derived by following the rules. (The sign # means is not ‘equals’.)

It will be seen from the box at the excluded middle that idea of contradictions is not empirically sound. Therefore, mathematics must reduce empiric rigour in order to set up a system incorporating this concept. This reduction of rigour is achieved by introducing the empirically unsound idea of the ‘excluded middle’.

As one cannot be sure what is in the mind of another, it is also necessary for mathematics to relax rigour on this count.

It will also be noted that being 4 is not symmetrical with not being 4. Not being 4 leaves a whole world, whereas being 4 somewhat limits the world available (see the asymmetry of ‘not’ for more detail).

Mathematics proceeds by imagining categories that are fully definable and not confusable. As seen, that cannot be done in the real world. Mathematics has been shown to be part of the real world (see, for instance, paragraph 190). Therefore, mathematics moves on the basis of fictions. The concept of contradiction is founded and reliant upon such fictions.

If one is happy to ignore the fictions in order to play the game of mathematics for pragmatic reasons, or perhaps some attraction to puzzles, that is reasonable. But once that game is taken too seriously, it means thinking in an unsane manner.

complete

By ‘complete’, Gödel meant that any genuine formula that you can write down in accord with the rules of the system should be capable of being found to be ‘true’ or ‘false’ within that system.[5]

Consider a painting. It is possible, in principle, to count the molecules or atoms of paint. It is not possible to count the ‘number’ of positions in which any molecule may lay, because ‘the’ positions are continuous in space.

Many a confusion in maths is due to attempting to mix continuous procedures with those of counting procedures. This is a problem that goes back to the ‘paradoxes’ of Zeno, such as the Achilles and the hare, or Achilles and the arrow. Remember also that separability is a relaxation of rigour.

You may only sensibly talk of ‘complete’ in a system which has a defined number of items in a defined box. Consider a chess board, it has 64 squares and the pieces number 32: there are a definable limited number of arrangements in such an arbitrarily defined system.

If your number system is considered ‘open’, or the length of your formulas are considered as potentially unlimited, you may not legitimately define anything as ‘complete’ when using that system. You may, at times, define a subsection as ‘complete’, such as the number of ‘different’ symbols available. But if you intend to be able to add new copies of the symbols at will, ‘completeness’ once more becomes undefinable.

315. 2+2=4 can be obtained by following the rules, it is therefore said to be ‘provable’.
     2+2#4 is a true formula in that it is written according to the grammar required by a genuine formula, but it cannot be proved by following the rules. By following the rules, you get 4 as the result of the sum 2+2. This ‘contradicts’ the assertion that 2+2 # 4, or a statement such as 2+2=5. The formula 2+2#4is then declared to be ‘false’ or ‘untrue’.
316. Keep in mind, this is a game. When we start to match it to the real world, it is often very unreliable.
     1elephant + 37 mice ‘=’ ~2 tons
     i.e. 1 + 37 ‘=’ ~2
     2 elephants + 7 big mice and a tooth pick ‘=’ ~ 4 tons
     Interpretation matters. Read on.
317. The pound in my pocket is not worth the ‘same’ as the pound in your pocket.
     Look at all those 2s above: each one of them is in fact different, they just look rather alike.
     I have commented upon some of the relaxations of rigour required to progress this far in following the ‘theorems’ of Gödel.
     I have built up various tools in order to approach these difficulties with clarity.
     Using this foundation, I now intend to focus upon the more damaging confusions in this wonderful muddle; a muddle that I have seen referred to enthusiastically as “Gödel’s amazing intellectual symphony”.[6]

Gödel’s main confusions

CONFUSION 1: TRUTH AND proveability

318. You may ask if a sentence is true of the real world. You may ask if a formula is provable in the context of a system of rules. You may ask if I call that object over there, a horse, and I may answer truthfully that I do. If you ask me if the horse is provable, I am bound to wonder what in tarnation you mean. The horse may be rideable, but provable it is not. I may prove, in some sense, to a blind person that the horse is in the pasture by letting them touch or smell or listen, but the horse itself is not ‘provable’.
319. Formulae and language and grammar must be tested against reality in order to have meaning. You may not put any predicate you wish, wherever you wish in a formula or other sentence, and rely upon the resulting words to make communicative sense, no matter what the ‘rules’ of grammar or arithmetic may suggest.
320. Any ‘object’ may change constantly; but the tree sits still enough for me to say, somewhat loosely or casually, “that is a tree”.
321. I cannot rightly say that the tree is chopable without careful thought to my meaning of chopable. For a more detailed explanation see chopability. Neither may I simply state that an isolated formula is ‘provable’. I might as well be as sloppy as to say that a felled tree is ‘chopable’. ‘Provable’ can only be meaningfully applied to the whole process of proving a formula; that process includes the prover and the relevant lists (axioms/rules and formulae).
322. True and provable are not simply interchangeable. They are different real-world acts. The Cretan will not mean the ‘same’ when stating, “I am lying”, as when the Cretan says, “I am proving” or ‘not proving’.
323. The act of proving is external and observable; the act of lying is at least in part, internal; that is, it cannot be examined or verified by another person.

     CONFUSION 2: CHOPABLE AND THE FELLED TREE

324. You may say that the fallen tree has been felled, or even chopped, but you may no longer refer to the past process of being chopable, the world has changed. The felled tree is present and I may say that I truly now see a felled tree, or even that I now have a recorded memory in my head of seeing a tree being chopped.
325. I may wonder whether a formula is provable in the future, I may watch the present process of a formula being proved, I may look at a formula and recall that it was proved in the past. It is important, for clarity, that I do not confuse these issues. I assert that Gödel has fatally confused them.
326. It may be true that I can say that I have seen a formula proved,
     but I cannot meaningfully state that:
     It may be proved that I can say that I have seen a formula trued.
327. Provable is a complex action; true is an observation that an ‘object’ accords with a picture, or an algorithm, stored in my head. You may wish to draw an analogy or mapping between the two processes, but you must ever remain clear upon which processes you are focussing. The sitting or standing Socrates can also be discussed in like manner, as can any chosen object or phenomenon. ( See also actions are objects.)
328. Or perhaps the provable sentence is 'the foot is the horse'! [7]

     CONFUSION 3: A TALKING SENTENCE

329. I have discussed talking sentences in more than one place.
330. The reality basis of ‘a provable sentence’ is discussed as: the person, the rule lists and the person performing the proof over a period of time, (see proveability). The very silly notion of a sentence which ‘points’ to ‘itself’ or talks to the world is analysed (see saying silly things section). And earlier, I discussed the meaning content of the sentence ‘this sentence is true’ or ‘untrue’ as the case may be (see paragraph 318).
331. A ‘provable sentence/formula’ must be clearly distinguished from a ‘proved sentence/formula’. It is my conviction that this distinction is not properly analysed in what I understand of the Gödel contentions.

     CONFUSION 4: NOT A TALKING SENTENCE

332. There is an outline of the problems with ‘not’ in the asymmetry of ‘not’.
     If the provable sentence is the subject of discussion, then a sentence that cannot be proved must be clearly established, by showing a sentence in the context of an attempted proof. It is far from adequate to merely assert that a non-provable sentence can be assumed by writing down ‘this is the negation of a provable sentence’, by just writing ‘not’ in front of a previously ‘proved sentence’.
333. A statement, that is said to be proved, is said to be proved by a person; the ‘sentence’ is just marks on paper. It is not a ‘proved sentence’ by virtue of some mystical inherent quality of the written marks upon the paper. Neither is it to be declared as ‘proved’, prior to going through the actions of proving the said formula.
334. As of this date (12/02/01) I think that the Goldbach conjecture and the Riemann hypothesis remain ‘unproved’. There is a last line in both these ‘theorems’, but whether these last lines are to be called ‘unprovable statement/formula’ we have yet to learn. The range of such non-statements is not easily defined or discussed. Statements such as 2+2=5 are also unprovable; they are, in fact, disprovable within the rules of the game. Such unproved or disprovable formulae are clearly not ‘the opposite’ [8] of the Gödel formula that asserts ‘this formula is provable’. Also see paragraph 354.
335. The Gödel version of the statement opposing the statement, “this formula is provable”, is “this (particular) formula is unprovable”. Gödel has shown what he defines as ‘a provable formula’, but he has not clearly defined the class of unprovable formulae by example or otherwise. This mere assertion of the ‘unproveability’ of his new sentence makes no solid sense that I can perceive; as we have yet to see a defined example in the real world of an unprovable formula. I am unsure whether Gödel thought that his negated ‘sentence’ was a unicorn, some imagined unchopped tree, a previously chopped tree, or perhaps some fourth category added to those of proved, unproven and disproved statements. I am even unsure that it was entirely clear to Gödel. I am unconvinced that it is sufficient for Gödel, in a cavalier manner, to just write down “this formula is unprovable” without having any unprovable formula to point to; as if an ‘unprovable statement’ is somehow an ‘opposite’ of a ‘provable formula’ with no further ado.
336. The sentence, “This sentence is provable”, is in fact written on the basis of a definition of ‘proveability’, it is not merely the result of following rules. The sentence therefore, is, not ‘just a formula’; it also contains a definition of ‘provable’ within its structure, just as the term ‘chopable’ has an external process meaning. Where is an unchopable tree? The problems with ‘not’ and with the excluded middle—or should that be the excluded muddle—are coming home to roost.
337. To state that “this tree is unchopable” has no meaning, because chopable is a real-world process that has not yet occurred. ‘Not chopable’ certainly does not mean that such a process cannot be made to occur. Likewise, or worse, the concept of ‘unprovable’ is not a statement of any real-world fact, but is the absence of a fact; the type of which absence is far from defined with any reasonable clarity.
338. At the very heart of all this poor reasoning is a failure to note time or change elements, in an ordered and tidy manner. There is a confusion of the process, ‘chopability’ or ‘proveability’, with the end fact of ‘chopped’ or ‘proved’.

     CONFUSION 5: NOT NOT A TALKING SENTENCE

339. Because of the multiple confusions of the above, not ‘not a provable sentence’ is not clearly shown to be ‘a provable sentence’. From the previous section it will be seen that; not ‘not a chopped down tree’ could nearly as well be a horse as it could be a still standing tree, for a horse is most certainly not a chopped down tree. While not a horse may as well be an elephant, therefore it becomes clear, not not a provable statement is very probably an elephant. Or, now to give the full Brouwer quote, “Absurdity of absurdity of absurdity is equivalent to absurdity”,[9] unfortunately such absurdity has now become the modern fashion. See also Logicians, 'logic' and madness.

     CONFUSION 6: PLACING A COPY INSIDE A COPY!

340. By now the situation is in such disarray that going on is almost churlish, but in karate one is always taught to kick the fellow when he is down in order to stop any inclination on his behalf to get up again.
341. Gödel constructs his not ‘not a sentence’, in part by taking a copy of his (not!) sentence and putting it inside his (not) sentence, a simple example of the Russell inclination to put the ‘new’ set within the original set (see paragraph 206). Somehow Gödel appears to convince himself that the sentence he inserts is ‘the same’ as the one from which he copied it, instead of being a new entity. Perhaps this flows from the Gödelian Platonist who used the numbers on that great skyhook where god had conveniently placed them—a sort of number tree that fruited in some kind of perpetual motion, a place where he could obtain endless labels that were ‘vaguely exactly the same’ as one another without any effort of printing or examining them (see the word factory).

A programmers approach

342. In the next document (Metalogic B1 – Decision Processes) I switch my attention to Turing’s stopping problem, also known as the entscheidungsproblem. Before so doing, I will take a look at the liar’s paradox as a programming problem. The form of the liar that I will use is that previously mentioned as attributable to the English mathematician, P.E.B. Jourdain, in 1913:

“The sentence below is TRUE.”
“The sentence above is FALSE.”

343. Consider a computer program or a ‘machine’ that is designed to analyse such statements. Be clear that the machine is separate from the sentences it is going to analyse.
344. The machine is instructed that:
     all sentences, in order to be meaningful, must refer to something in the real world,
     and further that
     it must test the sentence to see whether what it ‘says’ about the real world is ‘true’ or ‘false’.
345. Upon examining the first sentence, the machine will be content because the sentence does indeed refer to something outside itself, so the first condition is met. The machine would normally expect to be given a sentence like, “it is raining”. In response, it would look out the window with its super sensors and decide one way or another. It will then return an answer, “true”, or, “false”, and then switch itself off until someone asks it a new question.
346. When the machine is given a more tricky question like the first sentence above—which ‘states’ that another sentence is ‘true’—providing the programme is complex enough, it will not be phased by being asked to see if another sentence is in fact ‘true’.
347. The worst the machine will do is say, “Oh damn, more work”, and proceed to check out the next sentence. If the second sentence (‘the sentence below’) makes a reasonable claim about the world, the machine will then go and check that out. If it finds the facts to be in accord with the claim of the sentence, it will then mark the statement ‘true’ or ‘false’. Then the machine will unwind and report this result to the first sentence.
348. If the second sentence was indeed factual and the first statement ‘said’ it would be factual, the machine would then mark the first question ‘true’ and stop as before. If the second statement had proved to be untrue, it would of course come back and mark the first sentence as incorrect.
349. But in our example above, when it came to the second sentence, the machine would be faced with the further nuisance of going back and checking the first sentence. When the machine got to the first sentence, it would be returned to the second sentence, and so on until it ran out of electricity, or it wore out. In computerese terms, the machine would ‘go into a loop’, a very familiar circumstance to any programmer who gives a computer clumsy instructions.
350. Any half-competent programmer would immediately trace the problem and solve it by checking whether the machine had returned to the original statement, in which case the machine would print an error message and then stop. Of course, there are loops, which appear to go on interminably, that are not so easily detected, such as the instruction, “calculate the value of pi”. But given such a simple situation as the Jourdain version of ‘the liar’, the solution would be trivial.
351. Of course in the simple form of ‘the liar’, the strange ‘idea’ of ‘a sentence referring to itself’ is part of the magician’s wind-up. But, as has been copiously shown, such a form means nothing because the very phrase ‘refers to’ directs your attention to something other than the referring sentence. Sometimes people seem to be confused between the sentence doing the referring and the copy of that sentence which is being referred to, but I doubt that you will make that error after reading this far!
352. As seen above(Confusion 6: placing a copy inside a copy!), Gödel does indeed put a copy of the original sentence ‘inside’ the original, which assemblage then duly ‘goes into a loop’ in a very similar manner to the Jourdain version, if it were to be interpreted by a simple programme or a simple mindset. As you will know, it is an attempt to put one item ‘inside itself’that is a major generator of ‘paradoxes’.
353. Remember that every word depends upon individual definition. By taking the definition of ‘paradox’ to cover the situation in paragraph 352, you might conclude that Gödel has merely generated a ‘paradox’, but see also paragraph 361.
     

Gödel’s lists [addendum]

354. As with each of the ‘paradoxes’, two lists are set up; in this case, a list of ‘provable’ formulae and a list of unprovable ‘formulae’. While Gödel’s category of provable formulae is reasonably well defined, his category of unprovable formulae is a ragbag of everything else. What one imagines to be in that ragbag becomes somewhat a matter of guesswork. I will assume that it only contains formulae that are correctly formed. That still leaves us with formulae that can be disproved such as 2 + 2 = 5, formulae which we have not yet been able to solve such as the Goldbach conjecture and formulae such as Gödel’s, which you may regard as insoluble or uninterpretable (meaningless) according to your taste.
355. Thus, there are several different, apparent reasons for putting a formula into the list marked unprovable formulae. Clearly, unprovable does not then mean ‘impossible to prove at some future date’; but instead it means that we don’t know how to prove a particular formula right now, or that we can prove it false.
356. I don’t think that this is exactly what Gödel had in mind when referring to an unprovable formula. I believe that he meant a formula which was intrinsically unprovable. But just what Gödel meant by that depends entirely upon just what he had in his own mind when setting up his theorems. However, what ever Gödel did have in mind, the two lists remain headed provable and unprovable formulae. Then, by making the many standard errors of reading a meaning into his pseudo-liar construction, the normal problem of list placement occurs. Viewed in this manner, Gödel’s fear that he just had another ‘paradox’[11] would probably be reasonably well justified.
357. To keep matters absolutely clear, you must remember that there remains the intrinsic confusion between a provable formula and proven formula. In this section, I have thus far elided this confusion, I have done this by referring to the lists achieved after the event of either a proved or an unproved formula. Gödel, on the other hand, tries to distinguish between a provable and an unprovable formula, whilst concurrently attempting to construct an unprovable formula. Therefore, in fact, he appears to be setting up a definition of an unprovable formula, in the very act of constructing his not not a provable formula. He does not first show a formula and then show it to ‘be’ ‘unprovable’, nor does he first clearly define ‘unprovable’ and then construct an ‘unprovable sentence/formula’. Once again, the time element is overlooked. This all gets rather sloppy for my taste.
358. The definition of a provable formula, we are now offered and told is unprovable. But nowhere does seem to be any definition of how we ‘unprove’ it! Always keep in mind that proveability is a process, whereas proved is a categorisation after the event.
359. Unlike with other paradoxes, with Gödel it can become exceedingly difficult to decide just what is meant by the verbiage. If we stop at the categorised formulae; well, Gödel has just introduced a new formula and defined it as ‘unprovable’. He has merely shown that he can write down something silly in metalanguage and sort of mirror-image that in arithmetic by the process that he has defined.
360. If Gödel had proceeded in the manner of the other paradoxes, he would have produced the formula, and then posed the question, “Is this formula provable or unprovable?” As this formula is constructed to make a statement about proveability, he would, of course, have been asking a question analogous to the questions posed in other paradoxes about actual lists, and then been seeking to put his new formula into one or other of the predefined lists: proved formulae and other formulae. That is, he would have been using the standard magician’s flummery.
361. It seems to me that Gödel lost his way, by confusing ‘proveability’ with the resultant ‘proof’. He did not end up, as he apparently feared, with a paradox[12] or even with any clear contradiction, but merely with a definition: that loops in arithmetic logic were to be defined as ‘unprovable’ (see also a programmers approach above). And Gödel managed to give an example of such a loop with a clumsy, or rather artificial, version of ‘the liar’. Even to regard his formula as a loop, as has been copiously demonstrated, requires several dubious or over-simple assumptions to have been previously implicitly accepted.
362. I think Gödel’s example is of not much more interest than a simple programmer’s loop. However, I have tried to use it, with all its confusions, as an instructive test-bed in order to show how language can be made clearer, and to show the care necessary in order to achieve that objective.

To read, or re-read, the earlier pages on Gödel’s confusions:
A1—Gödel and sound sets
A2—Gödel and sound numbers
A3—Gödel and the ‘paradoxes’

Reference material and bibliography

I have used Nagel and Newman as my main outline guide for Gödel’s results.

The prime logic reference books that I use are:
Kahane and Tidman’s Logic and Philosophy: A Modern Introduction and
Kleene’s Introduction to Metamathematics.
In each case, these are the best primers I know of in their respective areas.

I have referred to many other works for minor details and cross checking, but they are of much lesser relevance or importance.

Bibliography

[1st ed. 1872]
1993, Everyman, pbk, 0460873598, $3.55
1993, William Morrow & Co, hbk, 0688120504 $25.60
1998, OUP, pbk, 019283374X, £2.99
1998, Macmillan, hbk, 0333722728, £17.00

1989 [1st ed. 1979] Vintage Books, NY, 0394756827
Currently listed:
1999, Basic Books, pbk, 0465026567, $16.00
2000, Penguin Books, pbk, 0140289208, £11.99

1995, Wadsworth Publishing Company, Belmont, 0534177603, $59.95

Introduction to Meta-Mathematics

1991 [1st ed. 1952] Wolters-Noordhoff Pub, Groningen
Currently listed:
North-Holland Pub. Co., Amsterdam 0720421039, £61.50
1996 reprint, 0720421039 $114.00

The Development of Logic

1994 [1st ed. 1962] Clarendon Press/Oxford University Press
Currently listed:
Clarendon Press/Oxford University Press, pbk, 0198247737,
1984, £24.00 / 1985, $49.95

Gödel’s Proof

1976[1st ed. 1959] RKP Ltd, 0710070780
Currently listed:
1983, New York Univ Pr., pbk, 0814703259, $12.50

Who got Einstein’s office?

1987, Penguin Books, 0140149236
Currently listed:
1988, Perseus Pr/Addison Wesley, pbk, 0201122782
$17.50 / £11.98

endnotes

1. See also the Cretan Liar ‘paradox’.
2. We know so little of Epimenides that he is regarded as near legend.
3. By 67 BC, Tarsus, now in southern Turkey, was part of the Roman empire. A university was founded there and existed at the time of Paul. The university was known for its school of Greek philosophy.
4. Titus, ch.17, 12-13:
   One of them, a prophet of their own, once said, “Cretans have always been liars, vicious beasts, and lazy gluttons”.
   That testimony is true. Therefore, admonish them sharply, so that they may be sound in the faith.
   To irritate the modern tender conscience, one might suggest that Paul was carried away with ‘racism’!
5. See, for instance, Nagel and Newman, p.94.
6. Nagel and Newman, p.94-5.
7. I have been told that it would be wise to spell this out to non-technicians who are having trouble threading their way through the spaghetti. The foot of a proof, that is the last formula in the proof, is the proved formula. A provable formula is not provable of itself, outside the context of it being proved. Thus, Gödel is attempting to convince that the horse’s foot is the whole horse, complete with rider having a gallop through the Bois de Boulogne.
8. Keep also in mind the idea of the excluded middle, and the relaxations of empiric rigour that are required in order to live with that fiction. The required relaxations of rigour are
   1. Separation, and
   2. Inattention to the real differences (for more detail, see paragraphs 102 and 103).
9. Barrow, p.216.
10. Gödel’s work then goes on to show that, by adding a new axiom to the system to resolve this nuisance, that immediately another like nuisance may be crafted. I am not concerned with this further work.
11. Regis, p.66 – A quote from Stanislaw Ulam:
    Gödel suffered from “a gnawing uncertainty that maybe all he had discovered was another paradox à la Burali forte or Russell”.
    But also see paragraph 87.
12. Other than the one he deliberately constructed. See also end note 8 and paragraph 366.

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