Gödel’s confusions— METALOGIC AA1—Gödel and sound sets A4—The
Return of the Gödel 



The return of the Gödel 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. 

ON GÖDEL

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The
liar [1]

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Saying silly things

proveabilityGödel’s definition amounts to: Process: 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’. The actions to be carried out by the mathematician form a list, and the list is generated by following the predetermined rules of arithmetic. Humans call this type of list, a proof, and call each line of the proof, a formula.The rules can also be written on another list.(See discussion on lists in Gödel and sound sets.) Be very careful to note that these rules have been refined and preagreed 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 realworld 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. 

consistencyBy ‘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. 
completeBy ‘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. 

Gödel’s main confusionsCONFUSION 1: TRUTH AND proveability

A programmers approach

Gödel’s lists [addendum]

Reference material and bibliographyI have used Nagel and Newman as my main outline guide for Gödel’s results. The prime logic reference books that I use are: I have referred to many other works for minor details and cross checking, but they are of much lesser relevance or importance. 
Bibliography 

Barrow, John D.  Pi in the Sky – Counting, thinking and being  1993, Penguin Books, pbk, 0140231099: £7.19
1993, publisher unknown, 0316082597: $14.95 
Carroll, Lewis  Through the LookingGlass  [1st ed. 1872] 
Hilbert  
Hofstadter, Douglas R.  Gödel, Escher, Bach: an Eternal Golden Braid  1989 [1st ed. 1979] Vintage Books, NY, 0394756827 
Kahane, Howard and Tidman, Paul  Logic and Philosophy: A Modern Introduction (7th edition)  1995, Wadsworth Publishing Company, Belmont, 0534177603, $59.95 
Kleene, Stephen Cole  Introduction to MetaMathematics 
1991 [1st ed. 1952] WoltersNoordhoff Pub, Groningen 
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] RKP Ltd, 0710070780 
Regis (ed.)  Who got Einstein’s office? 
1987, Penguin Books, 0140149236 
endnotes

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