I may be the first person to notice that the great French philosopher and mathematician, René Descartes (1596-1650), used an indicator to create analytic geometry. That indicator is: <geometric point, number coordinate>. The "geometric point" is idealizartion of a spatial position, which is only vaguely perceived. Its representation by a number explicates it definitively. Please note that 1st and 2nd components, here, are from different languages, respectively, geometric language and arithmetic or algebraic language.
A critical translation problem between these languages wasknest discovered in the time of Pythagoras, 2500 years ago. In geometry, the diagonal of a unit square cannot be translated as a fraction or rational number. This led Platonists do declare that "arithmetic and geometry are incompatible". By declaring that "motion is geometry set to time", Platonists then reasoned that "motion cannot be described by arithmetic", resulting in a delay of 2000 years for The Mechanistic Industrial Revolution, prolonging slavery.
This was resolved by adjoining, to finite operations of arithmetic, the transfinite operation of LIMIT, allowing explication of "the real numbers" including "the square root of two" for the diagonal of a unit square. This problem is discussed in the literature but NOT AS A TRANSLATION PROBLEM BETWEEN TWO LANGUAGES.
I've introduced this to prepare for the novel approach of Kurt Gödel in his decidability proof. I, uniquely, note this proof also involves an indicator, of the form: <element of logical language, uniquely assigned number>. Again, two components from different languages.
Examples of "logical element": logical operators, such as and, or, conditional, biconditional; and operands of logical statements; and the universal, existential quantifiers.
Each "logical element" is uniquely assigned a prime integer. This induces (for a logical compound of ordered elements) the composite integer which is the product of the assigned primes of this logical ordering. The Fundamental Theorem of Arithmetic assures us that such a product is unique, assuring that "the second component" of the indicator, the assigned number is unique.
These numbers are known as "Gödel numbers". Then Gödel shows in detail that there exists a recursive procedure (comparable to that successor function which generators the counting numbers) which generates every possible Gödel number (comparable to every possible counting number). This implies that the above process can be reversed: every Gödel number can be interpreted as a simple or compound logical statement. Then "the trap closes": Gödel shows that a Gödel number must exist which says, "I am a valid statement of arithmetic and I cannot be proven". Since all this is conducted under the aegis of an axiomatic structure for predicate logic, the conclusion is that logically explicated arithmetic is incomplete.
We now consider language problems apparently (to date) overlooked.
We noted above the ancient problem of translating geometry into arithmetic, resolved only by introducing a transfinite operation.
Few know a comparable problem arises in the logical foundation of arithmetic. As M. Presburger proved (1929), decidability is possible within arithmetic lacking a multiplication operator. Why? Addition can be formalized in First Order Predicate Logic (FOP), requiring only sets. But SECOND ORDER PREDICATE LOGIC (SOP) needs sets of sets (2nd order sets) to provide for multiplication. (Addition involves a set of countables whereas multiplication subsumes a set of repeated addends : a set of sets.) And, in Second Order Predicate Logic, the Gödel problem arises.
My point? It is claimed that a logic adequate for arithmetic was used for Gödel's proof. If FOP failed for multiplication, does this imply failure for Gödel's proof about all of arithmetic? This point should be clarified. ("Just keeping you honest.")
Another overlooked problem arises in lattice structures common to the language of arithmetic and the language of predicate logic. Often, in the literature, (although not so explicitly stated) appears the lattice of a composite number (such as 30 = 2 · 3 · 5 ), isomorphic to the lattice of three simple statements (such as "It is raining"; "The streets are wet"; "The street lamps are lighted"). The partial ordering of factors maps into the partial ordering of inclusion; the arithmetic operations of least common multiple (LCM) and greatest common divisor (GCD) map, respectively, into the logical operations of disjnction and conjunction. But we've noted elsewhere (frintegers.htm) other omissions in the literature:
- Partial orderings such as factor and inclusion are not well-defined in
the way addition and multiplication are, that is, a + b = a + c
if, and only if, b = c. Similarly, for p
0, p · q = p · r
if, and only if, q = r. However, the LCM and GCD operators
are not well-defined. As counterexamples: LCM(2, 3) = 6
and LCM(2, 6) = 6, hence, LCM(2, 3) = LCM(2, 6)
, but, obviously, 3 6 . Similarly, GCD (30, 60) = 30 and GCD(30, 90) = 30, hence, GCD(30, 60) = GCD(30, 90), but 60 90
. - Being well-defined, both addition and multiplication have inverses
, respectively, subtraction, division. But, not being well-defined, LCM
and GCD (logical disjunction, conjnction) both lack an inverse.
- Wherefore? I note elsewhere -- relations.htm -- a
binary relation, if many-one or one-one is a binary function, which, in
turn, can specialize (if "input" includes "output") as an operator. Since this is rarely,
if ever, discussed in the literature, it is perhaps not noted that only the one-one function
can specialize as a well-defined operation with inverse, say, in arithmetic. A
(noncyclic) many-one function can only specialize as a nonwell-defined operation
without inverse , as with LCM and GCD.
- Discussion of similar implications for logic lattices seem missing! ("Just ....")
- The indicator reappears to induce a property involving the creation of
Gödel numbers. (This indicator procedure belongs to a combinatoric proof
theory replacing axiom-theoretic proofing and model-theoretic proofing.) Given a
binary predicate, such as "is member of, is not member of" or "is true or is false", etc.,
and a binary number (0, 1), the indicator is: < binary predicate, binary number > . (In statement logic, this is "a
truth table". with homologues in set theory and probability theory. I may be
the only person to apply it in lattice theory.) Thus, a true/false statement, S, is assigned the indicator table 0, 1. Two
such statements induce the binary numbers, 00, 01, 10, 11
; etc. (I created a measure, "bank", homologue of a casino bank, which
adds digits in all positions (listed as row or column) of a table . More on this below.)
The logical operator of disjunction is mapped as LCM of the binary
numbers; and conjunction, as GCD; etc. (This relates to explications in the
first three items, above.)
- Rarely, if ever, discussed is the nonstandard nature of free lattice elements.
Applied to integers, the result (not to be confused with pseudoprimes) is
comparable to what happens when rational numbers arise between "next" integers.
As in set-theoretic topology, applying only LCM (as join) -- no GCD (as meet)
-- to basis set, {1, 2, 3, 5}, generates the standard
complemented distributive lattice, D30 on 30, i.e.
{1, 2, 3, 5} = D30
. Lattice ranking distributes (please see
Fig1.htm) according to a binomial table row: 1, 3, 3, 1, yet lacks banks for 2, 3, 5.
Applying the "free" operator GCD (meet) can generate the free lattice on
30, i.e. 
(D30) = F
30, whose ranking distributes (please see
Fig2.htm) as 1, 3, 3, 4, 3, 3, 1
and has the full range of banks: 1, 2, 3, 4, 5, 6, 7. Thus, one
element in F30 not present in L30
is 6 10 15
(using lattice "meet" to represent arithmetic GCD), induced with bank of 4 (independent of the bank 4 assigned to each
of the prime factors or lattice atoms, 2, 3, 5). - Implication for the present enquiry? That THE FUNDAMENTAL THEOREM OF ARITHMETIC (with
WELL-DEFINED PRIMARY OPERATORS) BREAKS DOWN FOR THE FREE LATTICE (with NONWELL-DEFINED PRIMARY
OPERATORS)! (For example, in D6, the distributive lattice on
6, we have 6 = 2 · 3 =
2 3, while in D30,
6 = (2 * 3
2 3)
(2 3) (2 3 5)
(6 10) (6 15)
(6 10 15), because each of
these has a different bank. And, by adjoining prime 7 for
D210, its element 6 has still more
frinteger factors, so that, in general, it has an infinity of frinteger factors!)
And that challenges the uniqueness of Gödel numbers, which depends upon THE
FUNDAMENTAL THEOREM, for just one way of factoring. Is it a langabuse to say that
final arguments are based upon the partial ordering of logic, but the Gödel numbers
backing these arguments are restricted to the total ordering of standard arithmetic? - The results of free lattices also mean that an "infinity" of cases may be ignored in applying probability measure, and for relational datbase querying (as in Oracle).
An important way in which langabuse may enter into Gödel's proof is the following.
The distributive lattice was created by Richard Dedekind (1831-1916) to explicate arithmetical factor theory of (positive) integers, since the complemented distributive lattice cannot. For, as with standard set theory, it recognizes only TYPE, not ORDER. Thus, 60 = 2 · 2 · 3 · 5 must be treated the same as 30 = 2 · 3 · 5, multiple tokens of type must be ignored. (I taught that in 1957 in a National Science Foundation Institute. Since NSF was planning "The New Mathematics" Program, based upon stndard set theory, their leaders did not want some one saying that "The New Math does not support The Old Math", so I became persona non grata to NSF.) Relevance here? The distributive lattice can be entirely generated from an atomic basis -- in factory theory, these are primes -- using only lattice meet and intersection (factoring LCM and GCD). Numbers are complemented iff containing each prime maximally. But that presents problems when the lattice structure in arithmetic is confronted with the lattice structure in logic when confronted by tne need for negation in logic.
Of many forms of negation, two stand out: exception and exclusion. The complement in a distributive lattice is isomorphic to exception in logic, since, for example, to say that 4 an 15 are mutually complementary in the lattice on factors of 60, is to say that each contains 60 except for the other. But the really useful logical negation is the exclusionary one.
Enter indicator again. The indicator factor tables show that operational factor change is monotonic: you cannot operationally "lose" a "1" (i.e. change it to "0"). That can only happen in the antilattice. Furthermore, you cannot operationally generate the homologues of logical "Truth" and "Falsity", respectively, with an indicator of all ones, all zeros. To resolve such problems leads to axiomatics translateong transfinitarily in generatics. ("Just ....")
Perhaps this and other "goodies" can be achieved without langabuse in this MIX-MATCH-BATCH. But, so far, one must wonder about what might have been "swept under the rug"!
Polish-American mathematician, Alfred Tarski (1902-1983), in, "The concept of truth in formalized languages" (1936), (cited in http://www.bun.kyoto-u.ac.jp/phisci/Gallery/tarski_note.html) "tried to give 'a materially adequate and formally correct definition of the term 'a true sentence', and obtained an important result closely related with Gödel's incompleteness theorem. That is, under very general assumptions, the set of sentences which are true in a model is not definable in that model. If you replace the concept 'truth' by that of 'provability', this result easily turns into Gödel's theorem."
As a second opinion (http://plato.standford.edu/entries/tarski-truth), "Tarski's own conclusion was that a truth defiition for a language L has to be given in a metalanguage which is essentially stronger than L."
I have my own version of Tarski's "truth theorem":
Given a game G with one or more players P and requires a
referee/umpire R/U to decide the legality of any play of the game. Then R/U
cannot be a member of P.
Dig? In any game requiring a referee or umpire, that referee or umpire should not be a player.
Profound? Resembles saying that police cannot judge arrested persons as "guilty" or "not guilty" of the alleged crimes or misdemeanors. Saying that an examined student cannot proctor the exam. Saying physicians testing a medical procedure or medicine cannot be the physicians judging the effects. Kenneth J. Arrow (1972 Economics Nobelist) proved that intuitively reasonable members of a set of rules (axioms) for electing representatives were incompatible, requiring decidability by a dictator (outside). My semiotic version (semiotic.htm): mathematics is primarily syntactic, lacking semantics (e.g., "truth", "legality") as well as pragmatics (for value-assignment).
We are told, in the literature, that "truth" can be decided in Pressburger's arithmetic (without multiplication). Perhaps langabuse arises when so much comment follows decidability failure in full arithmetic and no enquiry when decidability does not fail