In this blog-entry, I am basing my account on the following

**F**.

__.__

**E**__. writings --__

**D**http://www.dialectics.org/dialectics/Primer.html

http://www.dialectics.org/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**__. "First Dialectical Arithmetic" as a "Non-Standard Model" of "__

**D****N**atural" 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

*-- in the "Standard Model", of "first order", of the "*

**immanence****N**atural" 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 "

**N**atural" Numbers,

**N = {1, 2, 3, . . .}**, a first order "axiomatization" makes assertions about only individual "

**N**atural" 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 "

**N**atural" Numbers would "quantify over", or make assertions about, the qualities shared by groups of "

**N**atural" 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 "

**N**atural" 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**"*

*"*

**N**

*atural number*

*s"*arithmetic:

"Most discussions of Gödel's proof [of his '''First Incompleteness Theorem''' -- M.D.] ... focus on itsquasi-nature.paradoxical

It is illuminating, however, to ignore the proof and ponder the implications of the theorems themselves.

It is particularly enlightening to consider together both theandcompletenesstheorems and to clarify the terminology, since the names of the two theorems might wrongly be taken to imply their incompatibility.incompleteness

The confusion arises from the two different senses in which the term "" is used within logic.complete

In thesense, "semantic" means "capable of proving whatever iscomplete", whereas in thevalidsense, it means "capable ofsyntacticorproving[i.e., of "refuting" -- M.D.]decidingof the theory".each sentence

Gödel'scompletenesstheoremstates that every (countable) [and ω-consistent -- M.D.]first-theory, whatever its non-logical axioms may be, isorderin thecompleteformer sense: Its theorems coincide with the statementsintrueof its axioms.all models

The, on the other hand, show that if formal number theory isincompletenesstheorems, it fails to beconsistentin thecompletesecond sense.

Theincompletenesstheoremshold also forhigher-formalizations of number theory [wherein the Godelorderno longer holds at all, neither semantically nor syntactically — M.D.].completeness theorem

If onlyfirst-formalizations are considered,orderthen the, andcompletenesstheorem applies as welltogether they yield not a contradiction, but an interesting conclusion.

Any sentence of arithmetic that ismust beundecidableoftrue in some models(lest it bePeano's axioms[as it would be were it true informally refutablemodels of the Peano axioms -- M.D.]) andno[false in]some(lest it beothers[as it would be were it true informally provablemodels of the Peano axioms -- M.D.]).all

In particular, there must be models offirst-.order Peano arithmetic whose elements do not "behave" the same as the natural numbers

Suchnon-werestandard modelsandunforeseenbut they cannot be ignored, for their existence implies thatunintendednofirst-orderaxiomatization of number theory can be adequate to the task of deriving as theorems["Standard" -- M.D.]exactlythose statements that are true of the."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**-*

**also, by itself, implies the inseparability of "Standard" and "Non-Standard" Models of the "first order" axioms of arithmetics:**

*Skolem theorem*"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**-*

**is this.**

*Skolem theory*Suppose one sets up axioms, logical and mathematical, for a branch of mathematics

*or for set theory as a foundation for*.

__all__of mathematicsThe most pertinent example is the set of axioms for the whole numbers.

One intends that these axioms should

**describe the**

*completely*

*positive whole***[i.e., the "**

*numbers***N**atural" numbers,

**N**-- M.D.] and

**the whole numbers. But,**

*only***.**

*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 «

**» 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**

*arché**aleph-sub-zero*, thence progressing to

*aleph-sub-one*, then to aleph-sub-two, etc. -- M.D.], there are

**that contain as many elements as the real numbers [ =**

*interpretations**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

**and one intends that these axioms should permit and indeed characterize non-denumerable**

*a theory of sets***.**

*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

**[from a '''finitist/constructivist''' point-of-view, a model**

*model***, but never "actually", involving**

*potentially**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

**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,**

*Gödel's incompleteness theorem*

*the Löwenheim-Skolem theorem**tells us that a set of axioms permits many more*['

**essentially different***unequal in a*

**qualitatively****different**', '**ideo**-**ontologically****different**',*sense -- M.D.]*

**non**-quantitative

*interpretations***.**

*than the one intended*The axioms do not limit the

**or**

*interpretations***[uniquely to the model intended -- M.D.]**

*models*Hence

*mathematical reality***.***

*cannot be unambiguously incorporated in axiomatic systems**Older texts did "prove" that the basic systems were

**; that is, all the**

*categorical***of any basic axiom system are isomorphic -- they are essentially the same but differ in terminology.**

*interpretations*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 othersNo set of axioms is categorical, despite "proofs" by Hilbert and others

One reason that

*unintended*

**are possible is that each axiomatic system contains**

*interpretations*

*undefined***.**

*terms*Formerly, it was thought that the axioms "defined" these terms

**.**

*implicitly*But the axioms do not suffice.

Hence the concept of

*undefined***must be altered in some as yet unforeseeable way.**

*terms***is as startling as**

The Löwenheim-Skolem theoremThe Löwenheim-Skolem theorem

**.**

*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 "

**N**atural" numbers can span a range of "models" which, at one extreme -- the "Standard" extreme -- describes the arithmetic of the

*"*of --

**pure**,**unqualified quantifiers**"**["**

N = {1, 2, 3, ...}

N = {1, 2, 3, ...}

**dialectical**

**THESIS**" system of arithmetic]

-- and, at another -- opposite -- extreme, describes the arithmetic of the

*"*of --

**pure**,**unquantifiable qualifiers**"**N**

__Q__=**{**["

__q__/1,__q__/2,__q__/3, ...}**dialectical**

**ANTI**-

**THESIS**" system of arithmetic].

*Terminological clarification, for present and future reference*:

In the sentence "I picked three oranges.",

**F**.

__.__

**E**__. terms the word "three" an "[ontological]__

**D****quantifier**

*"*, and the word "oranges" an "ontological category name" and an

*"*

**ontological***, or*

**qualifier**"*"*.

**kind**-of-being**qualifier**"In the sentence "I picked three pounds of oranges.",

**F**.

__.__

**E**__. terms the word "three" a "[metrical]__

**D**

**quantifier***"*, and the word "pounds" a "metrical unit(s) name", and a

*"*, and the word "oranges" an "ontological category name" and an

**metrical****qualifier**"*"*

**ontological***.*

**qualifier**"Regards,

Miguel

**F**.

__.__

**E**__. definitions of special terms utilized in the narrative above --__

**D**<<

*>>*

**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

difference

**ideo**-

**ontological**

no definition for this term is as yet available in Clarifications Archive

**,**

difference

difference

**qualitative**

no definition for this term is as yet available in Clarifications Archive

**,**

difference

difference

**quantitative**

no definition for this term is as yet available in Clarifications Archive

**no definition for this term is as yet available in Clarifications Archive**

ideogram

ideogram

**http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Immanent/Immanent.htm**

immanence

immanence

**,**

logic

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

**[of the "**

non-standard model

non-standard model

**N**atural" 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

**no definition for this term is as yet available in Clarifications Archive**

phonogram

phonogram

**no definition for this term is as yet available in Clarifications Archive**

pictogram

pictogram

**quantifier**

no definition for this term is as yet available in Clarifications Archive

**standard model**[of the "

**N**atural" Numbers]

no definition for this term is as yet available in Clarifications Archive

**no definition for this term is as yet available in Clarifications Archive**

unqualified quantifiers

unqualified quantifiers

**unquantifiable qualifier**

no definition for this term is as yet available in Clarifications Archive

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