Gödel’s confusions— METALOGIC AA1—Gödel and sound sets A3—Gödel and the ‘paradoxes’ 



Gödel and the paradoxes is the fourth and last 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. 

Introduction

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Cantor’s diagonalisation


Richard’s ‘paradox’

Russell’s ‘paradox’
The Cretan liar ‘paradox’

The Return of the Cretan Liar

The asymmetry of ‘not’Any statement properly pointing at reasonably clearly defined elements of the real world can be imagined to be factual or not. A ‘lie’
is a statement regarded as factually not
so in the real world. Types of negative sentence Type 1 Type 2 ‘There are no horses in this room’ is not isometric with ‘there are no horses’ ‘Liar’ means, ‘what you say is not so’ or it is ‘untrue’. (See here for discussion of psychological ‘lie’states.) The relative interpretation of not notWhere is the horseConsider the statement, “there is not not a horse in this room”. This can be interpreted relative to the horse or relative to the room. Mathematicians habitually interpret ‘not not a horse’ to mean that there is a horse: this is an unsafe and potentially illdefined practice. Consider ‘not (not a horse)’. Given that from Type 3 above, when ‘not a horse’ means there are ‘no horses at all’, then not not a horse cannot mean there is a horse; for there was no horse to be notted in the first place (as with the unicorn in Type 1). So in the case of no horse or three unicorns, not not would tend to mean not at all at all. [14] As Brouwer might have said, “An absurdity of an absurdity is still an absurdity.” [15] Unfortunately, many a mathematician seems to think that absurdity of an absurdity is a ‘proof’. It will be seen that when deciding just what is not, it is sensible to first decide where the item is; or where it might have been or where it is now, if anywhere! If the horse was not in the room and was last seen in
the paddock, it may turn out to be reasonable to interpret
‘the horse is not not in the room’ as ‘the
horse is However, if the room was the focus of discussion, then concluding that there were no horse in the room would perhaps suggest the room was a suitable place to work. Interest in the absent horse could well then become an issue to be ignored unless, of course, there were now an elephant in the room. In the absence of any zoo in the room, attention would very likely be upon whether there was a table, some pencils, paper and a bottle of whisky in the room. Where is the room?Consider the empty room—think about what it is empty of, is it empty of horses or empty of cockroaches or empty of air? Consider, ‘not an empty room’—do we have a full room? If so, how full? Or does ‘not an empty room’ mean no room at all at all to be ‘empty’? In set terms; does ‘the’ set ‘exist’? What is supposedly ‘in’ the set? Or even ‘not in the set’? Where ‘is’ that which is currently ‘not in the set’? Is it ‘anywhere’? Or have you been lured into discussing unicorns? Nots can tie you in knots if your attention wanders from reality. One must always know just what it is that one intends to get notted. First catch your room, then find your horse; if your communications are to maintain contact with reality and, thus, with sanity. 
Grelling’s ‘paradox’

Humans describe things; words do not

Nagel and Newman

Berry ‘paradox’

More on ordering [24]

More on lists [27]

The word factory

INSTRUCTIONS
Next week: (Continue Gödel’s confusions with A4—The Return of the Gödel) 
Endnotes

Bibliography 

Barrow, John D.  Pi in the Sky – Counting, thinking and being  1993, Penguin Books, 0140231099: £7.19
1993, publisher unknown, 0316082597: $14.95 
Hilbert  For background reading: Hilbert by Constance Reid 
1996, Copernicus (SpringerVerlag) 0387946748 
From Brouwer to Hilbert, the debate on the foundations
of mathematics in the 1920s by Paolo Mancosu 
1998, Oxford University Press 0195096312 hbk 0195096320 pbk 

Hofstadter, Douglas R.  Gödel, Escher, Bach: an Eternal Golden Braid  1989 [1st ed. 1979] Vintage Books, New York, 0394756827 
Kahane, Howard and Tidman, Paul  Logic and Philosophy: A Modern Introduction (7th edition)  1995, Wadsworth Publishing Company, Belmont, 0534177603 
Kleene, Stephen Cole  Introduction to MetaMathematics  1991 [1st ed. 1952] WoltersNoordhoff Publishing, Groningen, NorthHolland Publishing Co., Amsterdam 0720421039 / New York 04444100881 
Kneale, William and Kneale, Martha  The Development of Logic  1994 [1st ed. 1962] Clarendon Press/Oxford University Press 
Nagel and Newman  Gödel’s Proof  1976[1st ed. 1959] Routledge & Kegan Paul Ltd, 0710070780 

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