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Topic: The Invalidity of Godel's Incompleteness Work.
Replies: 87   Last Post: Oct 25, 2013 2:44 PM

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 fom Posts: 1,968 Registered: 12/4/12
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 non-trivial proper ideal. Conversely, every
non-trivial proper ideal contains an undecidable (closed)
element. These facts indicate that the Goedel incompleteness
theorem asserts the existence, in Peano algebras, of non-trivial
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 non-trivial
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 363-387

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?

Date Subject Author
10/4/13 namducnguyen
10/5/13 Peter Percival
10/6/13 LudovicoVan
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10/9/13 fom
10/18/13 Peter Percival
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10/19/13 Peter Percival
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10/19/13 Peter Percival
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10/19/13 Peter Percival
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10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
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10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
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10/19/13 Peter Percival
10/19/13 namducnguyen
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10/20/13 namducnguyen
10/20/13 namducnguyen
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10/20/13 namducnguyen
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10/24/13 namducnguyen
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10/24/13 namducnguyen
10/24/13 Peter Percival
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