fom
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Re: The Invalidity of Godel's Incompleteness Work. (Halmos quote)
Posted:
Oct 9, 2013 10:24 PM


On 10/4/2013 10:47 PM, Nam Nguyen wrote: > > On the Incompleteness, since the requirement that T be _informally_ > adequate enough to describe the concept of the natural numbers is _not_ > a syntactical notion [as that of a T's consistency], it's logically > invalid to assume that T always be syntactically consistent, simply > because we _informally assume_ T adequately describe the concept of > the natural numbers. QED. >
If you would learn to be civil and use terms correctly, others might be able to figure out what you are trying to say.
This is what you are looking for:
"The word 'incompleteness' in 'incompleteness theorem' refers to syntactic incompleteness. The result here is not so general as the completeness theorem; The algebras covered by the theorem are usually described by saying that they are adequate for elementary arithmetic. Because a precise explanation of what this means would involve a rather long and technical detour, no such explanation will be presented here, and, consequently, the incomleteness theorem will be described rather than stated. The striking qualities of the theorem are sufficiently great to remain visible even under such cavalier treatment.
"Even if the definition of the class of algebras under consideration is not made explicit it is convenient to have a short phrase in which to refer to them; in what follows 'Peano algebra' will be used instead of 'polyadic algebra that is adequate for arithmetic'. The Goedel incompleteness theorem asserts the existence of 'undecidable propositions'. Since in the passage from logics to algebras the statement that an element p is refutable (or provable) was identified with the statement p=0 (or p=1), and since 'undecidable' means 'neither refutable nor provable', the assertion reduces to the existence of a (closed) element different from both 0 and 1. The ideal generated by such an element is a nontrivial proper ideal. Conversely, every nontrivial proper ideal contains an undecidable (closed) element. These facts indicate that the Goedel incompleteness theorem asserts the existence, in Peano algebras, of nontrivial proper ideals; the definition of syntactic completeness shows now that the theorem does indeed assert that something is (syntactically) incomplete. The usual, logical, formulation explicitly makes the assumption that the underlying logic is consistent; the treatment of algebras instead of logics makes it unnecessary to mention such an assumption here.
"The Goedel theorem does not assert that every Peano algebra is syntactically incomplete. It asserts, instead, that the definition of Peano algebras is not a faithful algebraic transcription of all intuitive facts about elmentary arithmetic. In algebraic terms this means that while some Peano algebras may be syntactically complete, there definitely exist others that are not. The situation is analogous to the one in the theory of groups. A class of polyadic algebras could be defined that is adequate for a discussion of elementary group theory. For this class the assertion that every two elements of a group commute would be undecidable, in the obvious sense that the assertion is true for some groups (and, therefore, corresponds to the unit element of some of the polyadic algebras under consideration) and false for others.
"What has been said so far makes the Goedel incompleteness theorem take the following form: not every Peano algebra is syntactically complete. In view of the algebraic characterization of syntactic completeness, this can be rephrased thus: not every Peano algebra is a simple polyadic algebra. This is the description that was promised above. What follows is another rephrasing of the description; the rephrasing, possibly of some mnemonic value makes its point by making a pun. Consider one of the systems of axiomatic set theory that is commonly accepted as a foundation for all extant mathematics. There is no difficulty in constructing polyadic algebras with sufficiently rich structure to mirror that axiomatic system in all detail. Since set theory is, in particular, an adequate foundation for elementary arithmetic, each such algebra is a Peano algebra. The elements of such a Peano algebra correspond in a natural way to the propositions considered in mathematics; it is stretching a point, but not very far, to identify such an algebra with mathematics itself. Some of these 'mathematics' may turn out to possess no nontrivial proper ideals, i.e., to be syntactically complete; the Goedel theorem implies that some of them will certainly be syntactically incomplete. The conclusion is that the crowning glory of modern logic (algebraic or not) is the assertion: mathematics is not necessarily simple."
Paul Halmos "The basic concepts of algebraic logic" American Mathematical Monthly, vol 53 1956 pp 363387
reprinted in
Algebraic Logic Chelsea Publishing Co New York, NY 1962
Too bad you do not know enough mathematics to understand what he said.
But, here is a question:
What makes it everyone else's job to do your research while you do nothing more than repeat flawed statements, insulting remarks, and expletives?

