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Gödel’s confusions— 
A1—Gödel and sound setsA2—Gödel
and sound numbers 



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. 
SECTION 1: IntroductionI 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 MetalogicAsupplement, 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]


SECTION 4: Lists of InstructionsI 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.

SECTION 5: Freedom

SECTION 7: Facts

Comparing predicates, relational strengthsOnly 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 2. Ordered 3. Even
step ordered 4. Continuous Every comparison
is a personal choice: 

Levels of languageThis 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’). Levels1. 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 mumbojumbo. 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 selfsame 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 illfounded, 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 metametalanguage and so on, but this is usually regarded as unnecessary or over the top. 

The error of ‘qualities’ or ‘properties’This box is for emphasis and focusFor 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 ‘interacting’ regardless. A tree is not ‘chopable’, a tree just stands
there doing its treeee 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) 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) 
Continue Gödel’s confusions with 
Endnotes

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 
email abelard at abelard.org © abelard, 2001, 25 january the address for this document is http://www.abelard.org/metalogic/metalogicA1.php 6800 words words 
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