In this blog-entry, I am basing my account on the following F.E.D. writings --
http://www.dialectics.info/dialectics/Primer.html
http://www.dialectics.info/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf
[pages A-1 through A-35, especially page A-32].
In an earlier post to this thread, I identified the F.E.D. "First Dialectical Arithmetic" as a "Non-Standard Model" of "Natural" Numbers Arithmetic.
The purpose of this post is to sketch the remarkable story of how, while such "Non-Standard Models" have been relatively little explored in academic mathematics, their ineluctable inherence -- their immanence -- in the "Standard Model", of "first order", of the "Natural" Numbers Arithmetic was established long before any at all of such "Non-Standard Models" were known, by three of the deepest theorems ever to emerge, so far, in the modern inflorescence of mathematical formal logic, launched by Boole, Peano, Frege, Russell, Goedel, and others.
These three theorems are --
1. The Goedel Completeness Theorem
2. The Goedel First Incompleteness Theorem
3. The Lowenheim-Skolem Theorem
Key Background.
The meaning of "first order" as in "first order mathematical logic". "First order" mathematical logic is symbolic formal logic [or "ideographic" formal logic; formal logic expressed in "ideograms", as distinct from "phonograms" or "pictograms"] that "quantifies over" -- that makes assertions about "some" or "all" of -- only "logical individuals", i.e., about individual elements of the set of all elements, the universe of discourse, of a given logical [axiomatic] mathematical system.
In the context of the system of the "Natural" Numbers, N = {1, 2, 3, . . .}, a first order "axiomatization" makes assertions about only individual "Natural" numbers, not about the qualities of shared by and defining groups [non-empty, non-singleton "sub-sets"] of such numbers.
A "second order" axiomatization for the "Natural" Numbers would "quantify over", or make assertions about, the qualities shared by groups of "Natural" numbers, e.g., via ideographic formulas that would translate into English sentences like "all odd numbers are...", or "all even numbers are...".
And so on, to "third order" logic, and beyond.
A. The Goedel Completeness and First Incompleteness Theorems.
The inescapability of "Non-Standard Models" of the "first order" axioms of "Natural" arithmetic, given only the assertion of their "Standard Model", was "predicted" [rigorously implied] by the joint applicability of the Gödel Completeness and Incompleteness theorems to the "first order" axiomatic system of the Peano"Standard" "Natural numbers" arithmetic:
"Most discussions of Gödel's proof [of his '''First Incompleteness Theorem''' -- M.D.] ... focus on its quasi-paradoxical nature.
It is illuminating, however, to ignore the proof and ponder the implications of the theorems themselves.
It is particularly enlightening to consider together both the completeness and incompleteness theorems and to clarify the terminology, since the names of the two theorems might wrongly be taken to imply their incompatibility.
The confusion arises from the two different senses in which the term "complete" is used within logic.
In the semantic sense, "complete" means "capable of proving whatever is valid", whereas in the syntactic sense, it means "capable of proving or refuting [i.e., of "deciding" -- M.D.] each sentence of the theory".
Gödel's completeness theorem states that every (countable) [and ω-consistent -- M.D.] first-order theory, whatever its non-logical axioms may be, is complete in the former sense: Its theorems coincide with the statements true in all models of its axioms.
The incompleteness theorems, on the other hand, show that if formal number theory is consistent, it fails to be complete in the second sense.
The incompleteness theorems hold also for higher-order formalizations of number theory [wherein the Godel completeness theorem no longer holds at all, neither semantically nor syntactically — M.D.].
If only first-order formalizations are considered, then the completeness theorem applies as well, and together they yield not a contradiction, but an interesting conclusion.
Any sentence of arithmetic that is undecidable must be true in some models of Peano's axioms (lest it be formally refutable [as it would be were it true in no models of the Peano axioms -- M.D.]) and false in [some] others (lest it be formally provable [as it would be were it true in all models of the Peano axioms -- M.D.]).
In particular, there must be models of first-order Peano arithmetic whose elements do not "behave" the same as the natural numbers.
Such non-standard models were unforeseen and unintended but they cannot be ignored, for their existence implies that no first-order axiomatization of number theory can be adequate to the task of deriving as theorems exactly those statements that are true of the ["Standard" -- M.D.] natural numbers."
[John W. Dawson, Jr., Logical Dilemmas: The Life and Work of Kurt Godel, A. K. Peters [Wellesley, MA: 1997], pages 67-68, emphases added by M.D.].
B. The Lowenheim-Skolem Theorem.
The Löwenheim-Skolem theorem also, by itself, implies the inseparability of "Standard" and "Non-Standard" Models of the "first order" axioms of arithmetics:
"The research begun in 1915 by Leopold Löwenheim (1878-c. 1940), and simplified and completed by Thoralf Skolem (1887-1963) in a series of papers from 1920 to 1933, disclosed new flaws in the structure of mathematics.
The substance of what is now known as the Löwenheim-Skolem theory is this.
Suppose one sets up axioms, logical and mathematical, for a branch of mathematics or for set theory as a foundation for all of mathematics.
The most pertinent example is the set of axioms for the whole numbers.
One intends that these axioms should completely describe the positive whole numbers [i.e., the "Natural" numbers, N -- M.D.] and only the whole numbers. But, surprisingly, one discovers that one can find interpretations — models — that are drastically different and yet satisfy the axioms.
Thus, whereas the set of whole numbers is countable, or, in Cantor's notation, there are only aleph-subscript-0 [spoken as either aleph-sub[script]-zero, aleph-null, or aleph-nought -- M.D.] of them [i.e., there are only the minimal infinite number of them — only an «arché» infinity of them — according to Cantor's theory of an endlessly-escalating progression of "actual" infinities, starting with the infinity that he denoted by the ideogram for aleph-sub-zero, thence progressing to aleph-sub-one, then to aleph-sub-two, etc. -- M.D.], there are interpretations that contain as many elements as the real numbers [ = aleph-sub-one elements, per the "Cantor Continuum Hypothesis" -- M.D.], and even sets larger in the transfinite sense.
The converse phenomenon also occurs.
That is, suppose one adopts a system of axioms for a theory of sets and one intends that these axioms should permit and indeed characterize non-denumerable collections of sets.
One can, nevertheless, find a countable (denumerable) collection of sets that satisfies the system of axioms and other transfinite interpretations quite apart from the one intended.
In fact, every consistent set of ['''first-order''' -- M.D.] axioms has a countable model [from a '''finitist/constructivist''' point-of-view, a model potentially, but never "actually", involving aleph-sub-zero "logical individuals" in its '''universe''' [of discourse], but no more than that -- M.D.]
.... In other words, axiom systems that are designed to characterize a unique class of mathematical objects do not do so.
Whereas Gödel's incompleteness theorem tells us that a set of axioms is not adequate to prove all the theorems belonging to the branch of mathematics that the axioms are intended to cover, the Löwenheim-Skolem theorem tells us that a set of axioms permits many more essentially different ['qualitatively different', 'ideo-ontologically different', unequal in a non-quantitative sense -- M.D.] interpretations than the one intended.
The axioms do not limit the interpretations or models [uniquely to the model intended -- M.D.]
Hence mathematical reality cannot be unambiguously incorporated in axiomatic systems.*
*Older texts did "prove" that the basic systems were categorical; that is, all the interpretations of any basic axiom system are isomorphic -- they are essentially the same but differ in terminology.
But the "proofs" were loose in that logical principles were used that are not allowed in Hilbert's metamathematics and the axiomatic bases were not as carefully formulated then as now.
No set of axioms is categorical, despite "proofs" by Hilbert and others....
One reason that unintended interpretations are possible is that each axiomatic system contains undefined terms.
Formerly, it was thought that the axioms "defined" these terms implicitly.
But the axioms do not suffice.
Hence the concept of undefined terms must be altered in some as yet unforeseeable way.
The Löwenheim-Skolem theorem is as startling as Gödel's incompleteness theorem.
It is another blow to the axiomatic method which from 1900 even to recent times seemed to be the only sound approach, and is still the one employed by logicists, formalists, and set-theorists."
[Morris Kline, Mathematics: The Loss of Certainty, Oxford University Press [NY: 1980], pages 271-272, emphases added by M.D.].
Thus, the first-order Peano axioms for the "Natural" numbers can span a range of "models" which, at one extreme -- the "Standard" extreme -- describes the arithmetic of the "pure, unqualified quantifiers" of --
N = {1, 2, 3, ...} [core of "dialectical THESIS" category/variant-systems units of arithmetics]
-- and, at another -- opposite -- extreme, describes the arithmetic of the "pure, unquantifiable qualifiers" of --
NQ = {q/1, q/2, q/3, ...} [core of "dialectical ANTI-THESIS" category/variant-systems units of arithmetics].
Terminological clarification, for present and future reference:
In the sentence "I picked three oranges.", F.E.D. terms the word "three" an "[ontological] quantifier", and the word "oranges" an "ontological category name" and an "ontological qualifier", or "kind-of-being qualifier".
In the sentence "I picked three pounds of oranges.", F.E.D. terms the word "three" a "[metrical] quantifier", and the word "pounds" a "metrical unit(s) name", and a "metrical qualifier", and the word "oranges" an "ontological category name" and an "ontological qualifier".
Regards,
Miguel
F.E.D. definitions of special terms utilized in the narrative above --
<<arche'>>
http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arche/Arche.htm
dialectical antithesis
no definition for this term is as yet available in Clarifications Archive
dialectical thesis
no definition for this term is as yet available in Clarifications Archive
difference, ideo-ontological
no definition for this term is as yet available in Clarifications Archive
difference, qualitative
no definition for this term is as yet available in Clarifications Archive
difference, quantitative
no definition for this term is as yet available in Clarifications Archive
ideogram
no definition for this term is as yet available in Clarifications Archive
immanence
http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Immanent/Immanent.htm
logic, first order; second order; third order . . .
no definition for this term is as yet available in Clarifications Archive
metrical qualifier
no definition for this term is as yet available in Clarifications Archive
non-standard model [of the "Natural" Numbers]
no definition for this term is as yet available in Clarifications Archive
ontological qualifier
no definition for this term is as yet available in Clarifications Archive
phonogram
no definition for this term is as yet available in Clarifications Archive
pictogram
no definition for this term is as yet available in Clarifications Archive
quantifier
no definition for this term is as yet available in Clarifications Archive
standard model [of the "Natural" Numbers]
no definition for this term is as yet available in Clarifications Archive
unqualified quantifiers
no definition for this term is as yet available in Clarifications Archive
unquantifiable qualifier
no definition for this term is as yet available in Clarifications Archive
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