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Gödel’s confusions—
METALOGIC A





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A1—Gödel and sound sets



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A2—Gödel and sound numbers
A3—Gödel and the ‘paradoxes’
A4—The Return of the Gödel

 

 
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Gödel and sound sets is the first part of the Confusions of Gödel, one in a series of documents showing how to reason clearly, and so to function more effectively in society.

Index

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E 'Y'
SECTION 1: Introduction
SECTION 2: Categorisation 1 – Boxes
SECTION 3: Categorisation 2Contents of boxes
SECTION 4: Lists of instructions
SECTION 5: Freedom
SECTION 6: Internal and External Categories
SECTION 7: Facts
  Asymmetry of negatives and positives
  Context
  Box: Comparing predicates, relational strengths
    More categories
      1. Quality
      2. Ordered
      3. Even-step ordered
      4. Continuous
    Every comparison is a personal choice
SECTION 8: Is a sentence well structured?
  Box: Levels of language
SECTION 9: Chopability, rideability and proveability
  Chopability
  Rideability
  Proveability
  Box: The error of ‘qualities’ or ‘properties’ – this box is for emphasis and focus
Endnotes
Bibiliography
  return to top of the index

SECTION 1: Introduction

I have ‘picked on’ Gödel because he has perpetrated one of the most complex streams of logic in the Aristotelian tradition ever devised by the human mind. It is important to realise that logic is not the ‘same’ as sanity. It is possible to be both extremely logical and at the ‘same’ time ‘completely’ irrational or ‘mad’ (see also mad, bad and sad). In fact, the mental stability of Gödel was far from sound (see also Metalogic-A-supplement, under preparation). Gödel was clearly both extremely able and intelligent and his work most surely moved forward human understanding of logic. As J.E. Barrow said approximately, the work of Gödel showed mathematics as the only religion which had managed to prove itself unsound.[1]

  1. As a child, I learnt very little.
  2. In part, this was because a great deal of what adults babbled at me made no substantive sense to me.
  3. Much of what was taught as standard ‘education’ still makes no sense to me.
  4. Categories do not exist in the real world outside the head of an individual.
  5. Within the head of each individual, the categories differ by a greater or lesser amount.
  6. The attempt to stamp or condition categories into the minds of the young is part of any common culture; it may have large numbers of people thinking that they ‘agree’ with each other, it may give some fairly common culture; it also becomes a mind-trap, stopping people from thinking much outside the received categories.
  7. As a child, I was told that there was a difference between a category ‘verb’ and a category ‘noun’, I did not understand it then and I do not believe or accept it now. Yet this category/division has been transmitted for a thousand or more years, from generation to generation. Not understanding or accepting such a category/division, tends to label a person ‘uneducated’.
  8. ‘Educated’ and ‘uneducated’ are categories, therefore their usage varies from individual to individual.
  9. The world is continuous (see, for example, Section 3: words in Why Aristotelian logic does not work) but we act as if it is not.
  10. Jane is an individual. (See also universals and individuals.)
  11. Jane can be put into the categories, ‘all the people called Jane in the world’/women/females/people/things/animals.
    All these categories are different in each person’s head. For the purpose of this document I intend to set up some categories which I decide are useful to understanding how to communicate in the real world.
  12. The first of these categories is ‘things’. (See also categories.)

    Story: A child was asked by an adult whether they had been taking biscuits from a biscuit tin. At first, the child (for fear that it may be in trouble) denied responsibility. On being pressed by the adult, in the light of considerable evidence, the child admitted culpability. The child was then asked how many biscuits it had taken, to which it replied, “one”. On being pressed further, the child stated that it had, in fact, taken “lots of ones”. ‘One’ is another name for ‘thing’.

  13. The words ‘one’ and ‘thing’ are categories (Note: ‘philosophers’ often call categories, ‘universals’.) ‘Philosophers’ contrast universals or categories with individuals. Jane is an individual. A particular tree, or a particular pencil, is also called an individual.
  14. ‘Trees’ or ‘pencils’ are categories.
  15. Here is a list of words meaning category:return to the index
    category, universal, predicate, property.


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SECTION 2: Categorisation 1 – Boxes

AXIOM:

  1. To make sense every individual must refer to a definable real-world object.
  2. Every individual placed in a category must be a real-world object.
  1. A category, then, is a box into which are collected individuals.
  2. The box is always an individual.
  3. Other words for box are: ‘category’ and ‘set’. (See also Human classification systems.)
  4. A box or category may be made out of wood, be contained within a line of real chalk, be a mental pattern represented in the brain by an electro-chemical conformation; it may be a list written on real paper with real ink.
  5. To repeat, every category is an individual or real object, such as a box. See also Universals.

    AXIOM:

    No individual may sit in two boxes, or categories, at the same time.
    For an individual to sit in two boxes at the same time makes no REAL sense.
    Of course, one box may sit within another box.[7]

SECTION 3: Categorisation 2 – Contents of boxes

  1. Now for some examples of sane categorisation of some individual objects:
  1. I assume you will not have trouble understanding real ‘objects’ such as a tree or a pencil or a person.
  2. Green. Green is light, reflected off an object, to which you respond, “that is green”. Every time you see green, you see it (photons) coming off from an object. If I am to make a collection in a box, of green objects, I could include a green tree, a piece of green glass, a green car, and an elephant painted green.

  1. If I am now to set up the category ‘trees’, I will have to take the green tree out of the box of green objects and put it in the box of trees.

    AXIOM:

    Categories originate in your head, they do not exist in the world external to your head.
    In the limited case in this paragraph, we are dealing with one green tree. We are not dealing with two objects: one in the set of green objects and one in the set of tree objects.

  2. We could have taken one green leaf from the tree and put it in the ‘green’ box, that is, into our box of green objects. Then we could have put the rest of the tree, without the leaf we had taken, and put it into the box of trees. Thus we have divided one object into two objects. Likewise, we could now take the leaf out of the ‘green’ box and glue it back onto the tree, thus deciding that two objects are now one object.

    AXIOM:

How we divide the real world into objects is a matter of individual decision at a particular time. It is important that we do not confuse our times: we cannot call an object both ‘one object’ and ‘two objects’ at the same time. At any time we must decide, or choose, and then stick to that decision until we decide to change it.

  1. return to the index Putting an object into two lists (in two categories) does not conjure up two objects. It only produces two lists. Of course, the lists are ‘new’ objects.


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SECTION 4: Lists of Instructions

I now introduce another category. The category I choose is a list, but a particular kind of list: a list that gives instructions. See also Comparing predicates, relational strengths.

  1. As a human being, you can be squashed flat by a tree falling upon you, or your arm may be twisted up behind by Plod, and you may be frogmarched by Plod. But most of your life, your actions will be chosen by you (see choice, chance, cause). But your choices are also limited, if you do not eat for a sufficient period, you will run out of choices. You will die and, perhaps, become part of a tree or a worm.
  2. If you are sitting in a leafy glade, the trees do not make you get up and dance a Highland fling, though you may decide it to be a cool thing to do. The tree/s may or may not be impressed by the quality of your performance, or perhaps it/they will decide to walk over and shake your hand. You live in the same world as the trees and they live in the same world as you. Remember, reality is that you are all one and that you interact with all around you (see Feedback and crowding) . The tree shows you its green leaves and, apparently, you just did a dance for the entertainment of the tree/s. I expect it/they felt, through the earth, the vibrations made by you.
  3. A list of instructions is a strange and unusual object (at least, I think it is). It is only very recently that humans have developed lists of instructions in order to convey information to their fellows, or in order to control their actions. You might consider here why you obey lists of instructions, some of which instructions are called ‘laws’.
  4. Other names for instructions: laws, rules, algorithm.
  5. To play a game of chess, there exists a series of rules. Unless you are a serious chess player, you will be unaware of just how complicated and detailed these rules are. You will probably only be aware of the simple ones, such as: a board has 64 squares, arranged 8 by 8. A knight moves to the opposite corner of any block of 6 squares, arranged 2 by 3.
  6. Chess is ‘just’ a game, a mental gymnasium.

    Story: Every so often, some unusual happening causes a dispute over what should occur in a chess game, and the rules have to be updated. Chess is played with clocks to a very strict time limit. There is a little red ‘flag’ that is raised by the minute hand of the clock and is designed to drop when the minute hand passes the figure twelve. A few years back, in a critical game, the flag was twisted, thus shortening it. It, therefore, dropped 20 or 30 seconds early. There was a dispute. A new law had to be made. This process updating rules is fundamental to learning and to the advance of science (and to that of chess, for that matter).

    [30. continued]   Mathematics is often also treated as just a game, with a series of rules as meaningless as that for the knight move, but mathematics is also often used to model or represent the real world.
  7. It is important to remember that mathematics is a sub-set, or sub-category, of language. That is, everything in mathematics can be expressed in words. The symbols of mathematics are a mere shorthand.
  8. Some of the rules of mathematics model reality too crudely for my purposes. I cannot, therefore, use them in a manner which expresses my meaning clearly. Therefore, I have to drop such crudities from the language that I use. The rules/concepts of mathematics that I must drop are so widespread and habitual in Western communication that it is very difficult for most people to adapt quickly. Further, I do use these categories when communicating, especially when I am not concerned with ‘too much’ detail. (see iteration).
  9. Problems are also aggravated for humans because words are themselves treated as separated objects. A list of the major ‘nuisances’ inhibiting clear expression and communication can be found at Why Aristotelian logic does not work, from whence further detail may be found by following links.
  10. Mathematics contains large numbers of instructions. An example is the addition sign (+). It tells you how to handle the items on either side of itself. As with chess, the beginner will not at first be aware that there are many other complicated rules beyond the opening moves. Just as with chess these rules will be encountered as the novice increases in familiarity with the ‘subject’. These rules are, in general, reasonably clear, if you have them explained to you in a proper manner.
  11. Chess games can be written down move by move. The moves may then be reproduced by following what is written down. The written moves can then be used as a series of instructions. When this is done in mathematics, the instructions are called formulae and a series of them with a purpose is called a proof. In chess, a series of instructions can be called ‘a game’.

SECTION 5: Freedom

  1. You do not have to follow a list of instructions, but you can. (For much more, see MetalogicB1 – Decision Processes.)
  2. A tree follows a list of chemically coded instructions when it grows. You might even wish to think that a rock falling down a mountain carries out a list of instructions:

    “If I hit another rock at a given angle, I bounce in that direction.
    If I hit it real hard, I shed a particular amount or piece of myself and leave it (the piece) to decide separately from myself what to do now that it is no longer tied to my apron strings.
    While I am on planet Earth, I will fall at an interesting speed.”

  3. Alan Turing says (On computable numbers with an application to the Entscheidungsproblem, p.249) “We cannot have an infinite number of symbols or we could not tell them apart .” This statement contains the hidden assumption that if we don’t have ‘too many’ symbols, we can tell them apart. It is clear that the number of symbols that a person can distinguish depends on such factors as training and acuity of the eye. No European can easily distinguish, reliably, one Chinese character from another. In fact, Turing’s system relies upon a very limited set of characters. It also relies upon an assumption that two characters written with different ink, on different parts of the page, at different times, with different movements of the hand are, in fact, ‘the same’.return to the index
  4. This amounts to categorising multiple copies of ‘particular’ symbols into a box.
  5. You can have a set of green trees!

SECTION 6: Internal and External Categories

  1. The purpose of communication is mutual cooperation between ‘individuals’. Such communication in general consists of working together to modify the environment for perceived advantage. Verbal communication uses categories. These categories, in the main, refer to the world external to the participating individuals. These categories include sets like trees and pencils.
  2. Internal categories include such terms as (I am) ‘happy’ or in ‘pain’. These categories are not usually accessible to outside others. They can only be related to by analogy with one’s own experience.

SECTION 7: Facts

  1. To say that a statement is ‘true’ is to state that the sentence points to a fact in the real world, e.g. “elephants have 4 legs”.
  2. To write, “this sentence is true”, does not point to a fact in the real world. A sentence is neither true nor false, only its external meaning can be true or false. The sentence, “there are 7 elephants in my room”, is currently false; for there are not any elephants presently in my room. The sentence, “unicorns exist”, is meaningless; for there is no real world object called a ‘unicorn’. Sentences are not true or false, it is to what they refer that can be true or false.

    ASYMMETRY OF NEGATIVES AND POSITIVES [for more detail see the asymmetry of ‘not’]

  3. Yes and no are not symmetrical. To state that I have a pair of pyjamas in my coat pocket tells you something rather specific, while to tell you that I have no pyjamas in my coat pocket tells you very little indeed.

    CONTEXT

  4. Statements without context are usually without much useful meaning. “Elephants exist” conveys far less content than “there is an elephant in my room on planet earth right now”.
  5. In the first case, you would not be given a location where an elephant or more were to be found but, given the local location of my room, you would at least know of a place where you could fruitfully look for an elephant, unless of course I’m lying.
  6. The statement, “There are no elephants, no horses and no aeroplanes in my room”, would not tend to enhance your life. Such a statement would not help you to locate a green parrot nor would it help you much if you required a horse.
  7. The statement, “There are no elephants painted green in my room”, tends to contain the suggestion that there are green elephants available somewhere on the planet. The statement that “There are no green-painted unicorns in my room” does not indicate that a search for such a being may well be successful.
  8. The statement, “this sentence is true”, suffers from both the problem raised in paragraph 44 (the truth of a sentence is not internal to the sentence) and the problems raised in paragraph 49 (the sentence refers to no outside, real object). The sentence contains a further problem:
  9. If being true were a meaningful statement about a sentence rather than about what a sentence indicated (which it is not!), it would be advisable to ask, “Which sentence is ‘true’?”
    Is it “this sentence” which is ‘true’; return to the index
    or is it
    this sentence is true”, which is ‘true’?

Comparing predicates, relational strengths

More categories

Only by forming or imagining categories and ‘objects’, can one compare them.

Comparing ‘objects’ is a relativisation process. Such comparisons are useful for communication and, often, for thinking about how we will adjust reality to our wants. The world remains interactive and connected; our means of communication are a pragmatic convenience, not a series of intrinsic ‘rules of nature’ or reality.

1. Quality
E.g. apple/orange apple/elephant/tree green/blue apple/blue.
The mathematical comparator or axiom proposes ‘separation’ into ‘objects’.

2. Ordered
E.g. Places in a foot race, exits on a motorway (freeway), sizes of planets in the local solar system, human heights.
The mathematical comparator is greater than (>), or less than (<). The arithmetic expression is 1st, 2nd, 90th etc. Such ‘numbers’ cannot be reliably ‘added’.
Consider arranging several animals and a planet in order of ‘size’, then adding the second planet to three elephants (there are a lot of elephants around this place, even green-painted ones). In team races, often the positions of the first six competitors are added and the team with the lowest total wins. However, at the Olympics, they do not allow you to trade three second prizes for one first prize. As the Americans say, “Second is another word for loser”.

3. Even step ordered
E.g. Palings on a fence; the arithmetic expressions 1,2,3, … 90 … 247, etc.
This mathematical comparator is often regarded as having
‘equal’ (‘=’) steps. ‘Equality’ means, “I don’t care about differences at the moment”.

4. Continuous
E.g. Measuring stick, clock, no limit to subdivision, the measurement of continuous space and time, the arithmetic expressions 1.84, 1.333 recurring, 247, 247.838, 4701.367294, etc.
This is not the counting of supposedly separated objects, but is comparing one object with another, e.g. by using a length of stick with marks on called a ruler.
It is commonplace to confuse the numbers used to count separated ‘objects’ with the numbers used to indicate a place upon a ruler or clock. Note that the number 247, in Section 3.above, looks very like the number 247 in this section, but they are being used in very different ways. In Section 3., ‘the’ number is being used to count objects; in Section 4., to mark a position.

Every comparison is a personal choice:
Do you prefer cheese to chewing cabbage leaves? That is a greater than preference.
Will you swap three dollar bills for a ride on a donkey?
If you live in Rome with a good income, will you exchange a bottle of water for a gold watch? What if you are in the middle of a desert?return to the index
Are six elephants more useful to you than ham and eggs?
Would you prefer to be six feet tall or to come first in a race.


SECTION 8: Is a sentence well structured?

  1. Received ‘logic’ speaks of whether a sentence is ‘valid’ or not. By this is meant, does the sentence conform to certain rules. This is usually confused with some sort of assumption that the rules are ‘meaningful’ or realistic. It is my intention to show that these accepted ‘rules’ are widely dodgy and often in direct contravention of empirical evidence (see Why Aristotelian logic does not work for details)
  2. It is widely imagined that sentences conforming to received ‘logic’ guard against the making of invalid arguments or statements, but that ain’t necessarily so. As has been argued elsewhere, much of the standard ‘logic’ does not hold in the real world.
  3. There is no excluded middle, only excluded ends, thus the concept of ‘contradiction’ is unsafe (see the excluded middle). The notion of consistency depends upon the notion that contradiction will show up ‘inconsistency’.
  4. The idea of completeness depends upon the (silly) idea that ‘every’ statement that can be made in a language must either be ‘completely’ true or ‘completely’ false. If a statement can be made within a language within the rules laid down for that language, then for completeness that statement must be decidable. That is, one must be able to decide whether the statement is ‘true’ or ‘false’.
  5. Decidable can be a synonym for ‘provable’.
  6. Writing down formulae assumes a set or rules or instructions. See commentary above on instructions.
  7. To repeat, Alan Turing correctly states on p. 249 of On computable numbers with an application to the Entscheidungsproblem, “I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent”.
  8. Landauer’s work concentrates on the physical limitations placed upon computational machinery by reality. Gregory Chaitin has done much on the predictability and the reliability of computing machinery [2].
  9. The outcome of these streams of logic and fact is that computation, though highly reliable, is neither predictable nor ‘certain’. There is no essential difference between modelling a complex system like the weather and modelling a more limited system like symbolic representation. The act of simplifying does not lead to some safe predictable nirvana.
  10. Mathematics is a relatively safe game, which can be used to model some limited simpler problems in the real world with considerable facility and usefulness. But the models are never 'completely’ trustworthy, nor do the real models ever magically ‘become’ the elements of the real world that are being modelled.
  11. Gödel introduces another either/or. Either the system is incomplete or it is inconsistent. As has been noted, such an appeal to the ‘law of the excluded middle’ will not hold up in empiric reality (see also the excluded middle).
  12. Further, I have stated above that negatives are not somehow naturally symmetrical with positives (see also the error called ‘zero’or ‘not’ or ‘negative’ and paragraph 45 above) and suggested that the sentence upon which Gödel’s edifice is constructed is not unambiguously formed (see paragraph 51 above).
  13. When forming a category or predicate, that category or predicate is an internal construct or algorithm in an individual mind.
  14. Even though similar sounding words be used by two human entities, the internal mental mapping[3] cannot be the ‘same’ in the two ‘minds’/brains; for the ‘two’ brains are at ‘different’ space/time ‘positions’.
  15. Having formed categories, there is no external guarantee that each person will assign each item to be classified to the supposedly imagined ‘same’ boxes.
  16. Thus, a rather large number of relaxations (ignoring/avoidance) of empiric rigour are required, before we even start to play our individual understanding of return to the indexGödel’s game through our own private mental worlds.

Levels of language

This is another useful manner of categorising, though one must constantly remain mindful that ‘all’ categories are arbitrary and somewhat foggy.

Language is used to point at chosen ‘objects’ or ‘parts’ of the real world (see also the error of the verb ‘to be’).

Levels

1. The real interactive and continuous world, of which we consider ourselves a somewhat autonomous, interactive part (see feedback and crowding). This level is often rather awkwardly called a model of number 2. This terminology is used, despite mathematical schemas also being widely referred to as models.

2. The mathematical or logical language frequently used to describe 1. This language is often claimed by mathematicians to be contentless, a system of marks on paper combined according to prescribed rules (for instance, Hilbert was keen on this approach). I dispute this claim, as did Gödel [4] , though possibly for differing reasons. Remember that we cannot know what is in the mind of another.

Others claim that mathematics is somehow ‘abstract’. Again, I dispute this formulation as being meaningless mumbo-jumbo. Every expression of mathematics is performed by marks on paper, sound vibration in the air activated by the human voice, or some such other real form. Otherwise put, mathematics is an intrinsic part of the real world every bit as much as is a house, or a stick used to point at some other ‘object’ or ‘part’ of the world. Thus mathematics is intrinsically level 1, whatever claims of detachment may be asserted for mathematics.

Mathematics is subject to the self-same reality as the rest of our environment, it is not somehow ‘special’ and thus above physical ‘law’ or empiric investigation. Level 1 is the legitimate test of mathematical consistency (see paragraph 54), not some arbitrary human construct such as the logical axioms of Peano or Fraenkel and Zermelo, or any other such system.

The concept of an 'independent’ variable or axiom is ill-founded, for all of reality is interactive. All that is possible is degrees of independence relative to other potential variables or axioms.

Operators in mathematics, such as addition (+), give instructions for the manner in which other symbols in a page of mathematics are to be manipulated. In this sense, then, the operators are ‘external’ to the mathematics. They are thus, to some degree, metalanguage, i.e. level 3.

3. A metalanguage in which number 2 is discussed. It is also possible to discuss the metalanguage in a meta-metalanguage and so on, but return to the indexthis is usually regarded as unnecessary or over the top.


Chopability, rideability and proveability

CHOPABILITY

  1. A person stood in a forest glade before a stately tree, in their hand they held a device called an axe. An idea was mulling in the person’s head; “Is this tree chopable?”
  2. Consider a planet with no persons and no axes; would the tree still be ‘chopable’. What makes a tree ‘chopable’?
  3. Chopability demands both a tree and a person with the means to chop it. The tree is not chopable outside a context.
  4. Is a steel I-beam chopable? What about a germ?
  5. For the question, “Is a tree chopable?”, there must be a tree, an axe and a means of wielding the axe. The term ‘chopable’ has no meaning without the total context. To ask whether a tree is chopable without a context is a misuse of logic or language. Out of context, such questions are sometimes referred to as ‘meaningless’.
  6. The human mind often has an inclination to accept meaningless sentences without close examination. It is quite easy to fool a defenceless and untrained mind with words that have no meaning in the real world. Such is done to young children when they are taught of Santa Claus and unicorns. Plato clearly recognised this in objecting to the teaching of fiction. It is not the greatest of wisdom to present fiction to the young as if it were fact. Such practise confuses and leaves the adult ever unsure of how to distinguish reality from meaningless words.

    RIDEABILITY

  7. Is a bicycle rideable? Is there a person to ride it? Can you ride on an aeroplane? Just what does rideable ‘mean’? Every word has a context; every word is real. A word is made of ink on paper, or excited phosphor on a computer screen, or vibrating air masses. A word is just as real as a tree or an axe or a person or a bicycle.
  8. So, just what does rideable mean? If I suggest that it means mounting a machine and waggling your legs about, what of riding a horse that does the leg-waggling for you? What of riding in an aeroplane, no legs are being waggled; in fact, you just sit there and let the ‘plane take the strain’. Whatever you ride, you may decide that the critical issue is that you start in one place and end up in another. But what of riding an exercise bicycle, or even a girlfriend? All these are acts in the real world. Wittgenstein has a similar discussion of the word game. However, the word game is often falsely claimed to be ‘abstract’, whereas none of my discussion of bicycles involves any ‘abstraction’.
  9. The word abstract, I assert has no clear meaning. I claim , that every useful meaning of ‘game’ can be defined in terms of a group of real matter and likewise for any other useful term. The word game is merely yet another pointer like unto a stick, as with ‘rideable’ or ‘chopable’. To repeat yet again, a word is also a proposed ‘section’ of reality, it is not somehow ‘abstract’.
  10. We use the sections of reality that we call words to communicate about ‘other’ sections of reality, as we discuss how to manipulate our earthly home.
  11. The words and symbols that we use in mathematics have no essential qualitative difference from the words we use in writing a novel. The terms of mathematics have no ‘special’ status. They are neither more or less ‘abstract’. They are just more sticks we use to point at other ‘bits’ of reality.
  12. To state that some set of words is ‘meaningless’ does not make the words or terms any less real. It just ‘means’ that the words are not referring to anything else real. There is no tree to be chopable or bicycle to be rideable. There is an axe, but no tree to chop. There is a rider, but there is no bicycle to ride. There is no unicorn to be unicorned.
  13. Sometimes we wish to discuss the real words themselves. To discuss the words, we use words. This often confuses the humans doing the discussing.

    PROVEABILITY

  14. Returning to our lists and to instructions, as discussed above in Section 4. A sentence or a formula is not 'provable’ out of the context of a human to decide that it is provable, or of a human to go through a process that the human decides to call ‘proving’ the given sentence or formula.
  15. Proving a formula is a process in a context, just as is riding a bicycle or chopping a tree. There is no chopping without a chopper and there is no proving without a prover. The tree does not chop itself nor the formula prove itself. For a formula to ‘say of itself’, “I am provable”, is as nonsensical as a tree to say of itself, “I am chopable”. It is humans who decide what sentences say.
  16. What humans tend to mean by ‘provable’ is that a formula is provable if certain rules or instructions are followed. A proof tends to mean a list of formulae following those rules, ending in an entry called the ‘provable’[5] formula.
  17. The instructions for forming formula one from another are pre-determined ‘rules’. The proof is not itself a series of instructions, but a list of stages on the road to the ‘provable statement’. Do not confuse the list, which is called a proof, with the end provable statement.
  18. Keep remembering, words can be formed to ‘say’ anything, sensible or ridiculous. You may say that a bicycle is provable or a formula is rideable or chopable. Are you alert? I can write a formula on a piece of paper and I can chop it with an axe. Can you think of a way of proving a bicycle or a tree? Is the tree following instructions laid down in its DNA?
  19. Gödel’s statement relies upon being able to present a comment about ‘proveability’ in a meta-language. That is, he ‘speaks’ about the ‘sentences’ or formulae of arithmetic as one would discuss whether a tree were chopable.
  20. The problems addressed by Gödel were particularly referred to a ‘logicised’ arithmetic. This means that the arithmetic was expressed through a different but assumed similar symbolism. He attempted to embed a sentence that reads ‘I am not provable’ in the logicised arithmetic, which he then suggested somehow, proves that arithmetic can make statements that are ‘unprovable’ within the arithmetic! (see also paragraph 89) Don’t get too worried smiley at abelard.org, I have studied this in ginormous detail and it still does not make much sense to me either.

Story: A young logician (perhaps that should be magician or priest) named Paul Cohen wanted to become famous, so he asked which problem in logic he should solve. He was told that the independence of the axiom of choice (don’t lose sleep over it!) was a good bet. So he went away and studied the problem, coming back a year later with his answer. The story continues, local mathematicians knew, “there was only one way to know whether the proof was correct”. The spell was duly submitted to the high priest, Gödel! He pronounced it kosher, and Cohen indeed became famous. I suggest that the king has no seriously substantial clothes.[6]

  1. So now Gödel developed a sentence in the meta-language commenting upon ‘proveability’, and had managed to copy a version of the sentence into the logicised symbols of arithmetic. (We will meet this in more detail in Metalogic A4—The Return of the Gödel.)
  2. I first define all sentences shorter than 25 letters as short. Now for my next sentence: “this is not a short sentence”. Is this sentence true or false? Is this sentence short or long? Well, I have defined it as a short sentence because it only has 23 letters; looking at it from my aeroplane from above (from the meta-language), I can see that the sentence is ‘short’. So the sentence is surely ‘lying about itself’. But, of course, a sentence cannot lie, it cannot tell lies or truth or anything else. The sentence does not talk. We read it!
  3. Does this awkward sentence follow, more or less, the generally accepted ‘rules’ of English? Clearly it does: it ‘says’ something that has some sort of meaning, a meaning which we can define. Therefore, we now have a sentence written in English that some people appear to claim ‘says of itself’ that it is ‘not short’, despite the fact that we have decided to define it as ‘short’ (again, refer to the excluded middle). So, we have shown that we can form a legitimately structured sentence that makes a false statement.
  4. But that statement is not ‘about itself’, for sentences do not speak. It is only our interpretation that gives the sentence ‘truth’ or ‘falsity’. We ask in our heads, “Is the sentence short?”, and we ask it about the sentence on the page.
  5. It is essential not to confuse what we say about sentences, or any other phenomenon, with the phenomenon at which the sentences are constructed to point. Sentences do not point, we do. Sentences just sit there like the tree in the glade; they do nothing ‘of themselves’. They ‘say’ nothing ‘of themselves’. Even sentences which ‘say’ things like, “I am a very long sentence”, in fact say no-thing. Sentences are more like bicycles or trees than they are like bicycle riders or tree choppers. Sentences do not ‘say’, they do not speak.
  6. Nor is a sentence any more ‘provable’ than a tree is chopable, just so long as no prover or chopper approaches it.
  7. Just what was in Gödel’s head when he formed his conundrums I cannot know and never will; for I, like the rest of the species, am entirely incapable of reading the mind of another. Very great amounts of ink and paper and great cudgelling of brains have been expended upon the puzzles of Gödel and his intellectual predecessors. Each puzzler had a different space/time brain, each understood as best they could the meaning, if any, of the puzzles. Each one of them understood differently.
  8. There is no ‘Gödel’s theory’ sitting somewhere ‘out there’. There are just various reactions to a problem in human communication. My interest in studying Gödel has been to understand the nature of human communication, far more than any interest in the puzzles ‘of themselves’. In the pursuit of that objective, I have come close to squeezing dry the puzzle and the various human commentaries upon the puzzles.return to the index
  9. I hope that my efforts will go someway to clarify the problems and widespread confusions of human communication in your own separate mind.

The error of ‘qualities’ or ‘properties’

This box is for emphasis and focus

For some groundwork, see also ontology and essence.

To discuss objects in terms of ‘their’ ‘qualities’ can very easily lead to confusion and to unrealistic modes of ‘thinking’.

It is we who decide to view the world in terms of ‘separated’ ‘objects’ and thereby, to develop ideas of quantity and countability. The real world is an interacting maelstrom (see also Feedback and crowding). We mentally break it into ‘parts’, the better to manage our interactions with the world. The ‘parts’, however, go right on ‘inter-acting’ regardless.

A tree is not ‘chopable’, a tree just stands there doing its tree-ee business. A bicycle is not ‘rideable’, a bicycle is an object put together by human ingenuity. These ‘things’ are ‘objects’, they do not possess qualities in some such analogy as you ‘possess’ a pen or, for that matter, a bicycle. (for wider discussion see Chopability, rideability and proveability)

‘Owning’ is an arrangement between humans designed, perhaps, to lessen friction, or to enhance power. It is not written in the stars. Ownership of ‘qualities’, or ownership by a tree, is really rather pushing clarity.

A tree (or its leaves) does not ‘have’ or ‘possess’ the ‘quality’ of being ‘green’, it does not ‘have’ the ‘property’ of being green; the tree merely reflects a wavelength of real light which we perceive as ‘green’. This confusion also links to problems with the verb to be.

Neither do ‘two’ ‘objects’, which we have decided to regard as separate, ‘possess’ some etheric ‘quality called ‘twoness'!! Nor even does a formula ‘possess’ ‘proveability’.

To attempt to handle our communications in terms of more than ‘one’ imagined ‘quality’ ‘simultaneously’ is a design bound to flirt easily with confusion. The very greatest of care must be exerted and each circumstance tested rigorously against reality. (see Categorisation 1 - Boxes)return to the index


  1. To any of those who have studied this issue and wish to ‘contend’ or seek to clarify any of this document; constructive discussion with you will be welcome.

Continue Gödel’s confusions with
A2—Gödel and sound numbers
A3—Gödel and the ‘paradoxes’
A4—The Return of the Gödel


Endnotes

  1. J.D. Barrow says, “One would normally define a ‘religion’as system of ideas that contained statements that cannot be logically or observationally demonstrated ... Gödel’s theorem, not only demonstrates that mathematics is a religion, but shows that mathematics is the only religion that proves itself to be one.” [J.D. Barrow, The World within the World, 0192861085]
    I’m not entirely happy with the tautness of this statement, but the idea’s the thing.
  2. For more of Chaitin’s work, see http://www.cs.auckland.ac.nz/CDMTCS/chaitin/.
  3. ‘Mapping’ is a word used by mathematicians to mean linking various items together. For more details see Gödel and sound numbers.
  4. e.g. see p.124 of Pi in the Sky
  5. Later you will see that this is sloppy usage.
  6. Pi in the Sky, p.215.
  7. Thus an individual item in the inner box may be said to be inside two boxes. The item remains in the inner box and, for clarity, this should not be forgotten. The nose of the horse is on the face of the horse, the horse is in the shop, the shop is in the town, the town is on the planet, etc. The horse’s nose is, therefore, on the planet. Because of the nature of reality, I assert that you may attend to only one of these relationships at any ‘one’ time.

Bibliography

Barrow, John D. Pi in the Sky – Counting, thinking and being 1993, publisher unknown, 0316082597, $14.95
1993, Penguin Books, 0140231099, £7.19
Barrow, John D The World within the World

1988, Clarendon Press (OUP), hbk, 0198519796, $35.00
1990, Oxford Paperbacks, 0192861085, £10.39

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