Comparison:

versus

Dear Readers,

The foci of this post are two

The first of the two

The second of the two

-- as <<

Per the

-- whose space of "meta-numbers" is --

When a "meta-numeral" --

-- whereas, for a non-underscored [meta-]number, i.e., for a

For

http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

Unlike

Specifically, the two equations in question are the following --

-- and --

-- wherein

The general,

-- asks the question "What new axioms-systems follow,

The general,

-- asks the question "What new axioms-systems follow,

For

Exercising the

-- and --

The

-- and they abbreviate/assign ['

-- whose space is --

-- and --

-- whose space is --

The '

The

-- and they abbreviate/assign ['

-- whose space is --

-- and --

-- whose space is --

The '

Note that the second axioms-system -- the

This implies that the

Note that these two solutions, which

The third system, or

For example, the

If, in a given

If, in a given

The

They

If

If

Thus, per the

The

The two

http://www.dialectics.org/dialectics/Briefs_files/_Brief1-29JUL2008_OCR.pdf

*The*__Dialectic__*of***Algebraic Formal Logic**versus

TheThe

__Dialectic__*of*'''__W__**hole numbers**'''**Arithmetic**Dear Readers,

The foci of this post are two

**F**.**.**__E__**.**__D__*"*__dialectical__*equations**"*, both of which are expressed in the "ideo-graph-ical language" of the algebra of their__N____Q___*"**First*__Dialectical__*Arithmetic**"*.The first of the two

__dialectical__**expresses the***equations*__dialectical__**of the "Boolean" algebra of formal logic as <<***elaboration***>>.***arche'*The second of the two

__dialectical__**expresses the***equations*__dialectical__**of the "Peanoian" "natural" axiomatic arithmetic of***elaboration**"*__un__**ified**__qual__**ifiers", the "number-space" of the**__quant__**hole numbers --**__W__**W**...**...**__=__**{****0****,****1****,****2****,****3****, . . . }**-- as <<

**>>.***arche'*Per the

**F**.**.**__E__**. "solution" -- or "semantification" -- for the two post-<<**__D__**>> terms of both of these two equations, the categorial-progressions, or axioms-system progressions, that these equation generate both***arche'***and***converge***in an in-common second, "contra-system" term, which connotes the axioms-system of the***intersect***W**-based arithmetic of pure,__un__**ifiable ontological/categorial**__quant__**ifiers --**__qual____W____Q___-- whose space of "meta-numbers" is --

**W****...**__Q__**...**__=__**{q/****0****,**__q__/**1****,**__q__/**2****,**__q__/**3****, . . . }**.When a "meta-numeral" --

**rically, call it***gene***-- is**__x____"underscored__", or__underlined__, as are the**s in the**__q____q__**/n**above, except for**q/0**, then, per the**F**.**.**__E__**.**__D____Encyclopedia____Dialectica__**, this signifies that the "meta-number" thus notated exhibits the***notation**"*, i.e., that this "meta-number", as**contra**-**Boolean**"**characteristic**"*"*function", i.e., when it operates upon itself, e.g., when it is**-**__self____self__*-*multiplied, or "squared", produces, as its*"*product", a value which is**-**__self__**to its**__qual__itatively__un__equal*"*squared" value, viz. --__un____x__**^2**...**=**...**...**__xx__**=**...__x__**(**is__x__)**greater than**__not__**&**is**equal to**__not__**&**is**less than**__not____x__-- whereas, for a non-underscored [meta-]number, i.e., for a

*"*[meta-]number,**Boolean**"**rically denoted by***gene***x**--**x^2**...**=**...**xx**...**=**...**x(x)**is greater than**OR**is equal to**OR**is less than**x**.For

**F**.**.**__E__**.'s exposition of the centrality of this**__D__*"*or**-**__not__**-**__not__**" --**__not__*"*-- relation that characterizes their**-**__neti__**-**__neti__**"**__neti__**"meta-numbers" -- this relationship of**__dialectical__*"*, or of**-**__non__**itative**__quant__**equality"**__in__*"*-- and its ubiquity, and the relation-sign that they "coined" for it, and the revolution in mathematics implied by its transcendence of the mathematical**itative**__qual__**equality"**__in__*"*chotomy principle", and its supercession by their__tri____dialectical__*-*mathematical*"*chotomy principle", see --__tetra__http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

Unlike

**F**.**.**__E__**., which uses the**__D____Dy__**to express both of these***adic Seldon Function equations***axioms-**__dialectical__**, I will use the***system*__s__progressions__Tri__**, for***adic Seldon Function equations***tep**__s__**s**..**=**..**1**, to express these**axioms-**__dialectical__*system*__s__**, in order to focus attention on the first three terms/***progressions***of each***systems**system*__s__progression**.**__only__Specifically, the two equations in question are the following --

**(**__W____E___**)^(****3****^****1****)**......**.....**__=____?__-- and --

**(**__W___**)^(****3****^****1****)**......**......**__=____?__-- wherein

__W__**connotes the "**__E___**lector" algebra of the "Boolean" axiomatic arithmetic system for algebraic formal logic, whose space is**__E__**W****E**...**...**__=__**{****0****,****1****}**, and wherein**connotes the "Peanoian" "**__W___**hole Number" axiomatic system of arithmetic, whose space is --**__W__**W**...**...**__=__**{****0****,****1****,****2****,****3****, . . . }**.The general,

**s**-as-variable form of the first of our two*"*__dialectical__*equations**"*--**(**__W____E___**)^(****3****^s)**......**.....**__=____?__-- asks the question "What new axioms-systems follow,

**, from the premise/<<**__dialectically__**>> of the "Boolean" axioms-system of***arche'**"***quasi**-__quant__*o**-***itative" algebraic formal logic?".**__qual__The general,

**s**-as-variable form of the second of our two*"*__dialectical__*equations**"*--**(**__W___**)^(****3****^s)**......**......**__=____?__-- asks the question "What new axioms-systems follow,

**, from the premise/<<**__dialectically__**>> of the "***arche'***hole Number" axioms-system of "purely-**__W__**itative", "Peanoian/Natural" arithmetic?".**__quant__For

**tep**__s__**s**..**=**..**1**,**F**.**.**__E__**.'s answers to these question/solutions to these equations -- their "semantification" of, or "meaning-discovery" for, the two new terms generated in each case -- is as follows.**__D__Exercising the

**ric***gene**"**self**-***combination***"***for the expansion of these***rules*__Tri__**, to obtain their still***adic Seldon Function equations***fically**__speci__**solved**__un__**ric expressions, we have --***gene***(**__W____E___**)^(****3****^****1****)**......__=____W__**..**__E___**+**..__W____q__**/****..**__EE___**+**..__W____q__**/**__EEE_______-- and --

**(**__W___**)^(****3****^****1****)**......**......**__=__

**..**__W___**+**..__q__**/****..**__WW___**+**..__q__**/****.**__WWW___The

**F**.**.**__E__**. solution, or "semantification", for the first equation's second and third terms is based upon the following overall intuition --**__D__**(**__W____hole-Numbers-based, unit-interval____Formal____-____Certainty Logic____System__**)^(****3****^****1****)**......

__=__**(**__W____hole-Numbers-based, unit-interval____Formal____-____Certainty Logic____System__**)**..**+**..**(**__W____hole-Numbers-based, unit-intervals____Ontological__*Possibility Logic*System**)**..**+**.**(**__W____hole-Numbers-based, unit-intervals____Ontological__*Actualization Logic*System**)**-- and they abbreviate/assign ['

**(---)**'] the two "ideo-ontologically" new terms/axioms-systems symbols thus generated essentially as follows --__W____q__**/**__EE__**......**_____**(---)**......__W____Q___-- whose space is --

**W****...**__Q__**...**__=__**{q/****0****,**__q__/**1****,**__q__/**2****,**__q__/**3****, . . . }**-- and --

__W____q__**/****......**__EEE___**(---)**......__W____q__**/**__Q____E_______-- whose space is --

**W**__q__**/****Q****E**...**...**__=__

{b{b

**.****/****0****, b****1****(t)**__b____.__**/****1****, b****2****(t)**__b____.__**/****2****, b****3****(t)**__b____.__**/****3****, ...}**.The '

**' "subscript" signifies unit-interval-restricted,**__.__*"*ifiablility" of the**quasi**-__quant__**ifier "meta-numeral" so "subscripted".**__qual__The

**F**.**.**__E__**. solution, or "semantification", for the second equation's second and third terms is based upon the following overall intuition --**__D__**(**__W____hole-Numbers,____Pure, Un____qual____ified____Quant____ifiers____Arithmetic System__**)^(****3****^****1****)**......__=__**(**__W____hole-Numbers,____Pure, Un____qual____ified____Quant____ifiers____Arithmetic System__**)**..**+**..**(**__W____hole-Numbers-based,____Pure, Un____quant____ifiable____Qual____ifiers____Arithmetic System__**)**..**+**.**(**__W____hole-Numbers-based,____Quant____ifiable____Qual____ifiers____Arithmetic System__**)**-- and they abbreviate/assign ['

**(---)**'] the two "ideo-ontologically" new terms/axioms-systems symbols thus generated essentially as follows --__q__**/**__WW__**......**_____**(---)**......__W____Q___-- whose space is --

**W****...**__Q__**...**__=__**{q/****0****,**__q__/**1****,**__q__/**2****,**__q__/**3****, . . . }**-- and --

__q__**/**__WWW__**......**_____**(---)**......__W____q__**/**__Q____W__**......**_____**(---)**......__W____U___-- whose space is --

__W__**...**__U___**...**__=__

**{****u****o****/****0****,****u****1****(t)**__u____o__**/****1****,****u****2****(t)**__u____o__**/****2****,****u****3****(t)**__u____o__**/****3****, . . . }**.The '

**' "subscript" signifies "full-multiplicity**__o__**ifiablility" of the**__quant__**ifier "meta-numeral" so "subscripted" [reflecting the "syncopated" notation for the**__qual__**ric <<***gene***nad>>, or**__Mo__**ric***gene**"*itative__qual__**unit**", in Diophantus of Alexandria's*circa***250****C**.**E**. proto-ideographic-algebraical manuscript,__The____<<__].**>>***Arithmetike'*Note that the second axioms-system -- the

*"*-- in each of the two progressions per the above "semantification"/solution is the same: the**contra**-**system**"__W__**axioms-system of arithmetic.**__Q___This implies that the

__W__**axioms-system is both a system of**__Q___*"*and a system of**"**__dialectical____ARITHMETIC__*"*[**]***arithmetic for an algebra of*__dialectical__*.***"**__LOGIC__Note that these two solutions, which

**verge and**__con__**in the***intersect*__W__**system for their second terms,**__Q___**verge again thereafter, for the third terms of each.**__di__The third system, or

*"*, in each case, is**uni**-**system**"**itatively different from --**__qual__**itatively**__qual__**equal to -- the third system of the other case --**__un____W____q__**/**__Q____E__**......is**_____**itatively, "ideo-ontologically"**__qual__**equal to......**__un____W__**.**__U___For example, the

**b****n****(t)***"*ifiers" of the**quasi**-__quant____W____q__**/**__Q____E__**system's "ontological**_____**ifier"**__qual__**symbols,***ontological category*__b____.__**/n**, are restricted to the "Boolean" unit-interval. That is, they are*"formal-***logical****ifiers" -- "**__quant__**all****(1)**, or**none****(0)**"__quasi__*-*ifiers -- whose only admissible values are confined to the "Boolean", "unit-interval end-points" set__quant__**W****E**...**...**__=__**{****0****,****1****}**, inherited from the__W__**component of this third,**__E___*"*axioms-system as a**-**__uni__**system**"*"*. The**complex unity**"__b____.__**/n***"quasi-*ifiable ontological__quant__**ifiers" still connote whole**__qual__**.***ontological categories*If, in a given

**being modeled using the**__dialectical__progression__W____q__**/**__Q____E__**system, the**_____**assigned to the ontological***ontological category***ifier**__qual____b____.__**/k****during s**__actualizes__**age**__t__**t**, then the "quasi-**ifier" function for that**__quant__**ifier,**__qual__**b****k****(t)**, in a well-fitting "meta-model", should generate the value**1**for s**age**__t__**t**, so that the combined value of**b****k****(t)**and/"times"__b____.__**/k**will be**1**__b____.__**/k**, or, equivalently, just__b____.__**/k**, indicating that the**assigned to category***ontology*__b____.__**/k****extant/manifest in s**__is__**age**__t__**t**.If, in a given

**being modeled using the**__dialectical__progression__W____q__**/**__Q____E__**system, the**_____**assigned to the ontological***ontological category***ifier**__qual____b____.__**/k**__does____not__**during s**__actualize__**age**__t__**t**, then the*"quasi-*ifier" function for that__quant__**ifier,**__qual__**b****k****(t)**, in a well-fitting "meta-model", should generate the value**0**for s**age**__t__**t**, so that the combined value of**b****k****(t)**and/"times"__b____.__**/k**will be**0**__b____.__**/k**, or, equivalently, just**b****.****/****0**, a*"*value, indicating that the**-**__quant__o**itative zero/absence"**__qual__**assigned to***ontology**category*__b____.__**/k**is**manifest throughout s**__un__**age**__t__**t**.**, the***On the contrary***u****n****(t)**full-multiplicity**ifiers [in the**__quant__**hole-numbers sense] of the**__W____W__**system's**__U___**ifiers,**__qual____u____o__**/n**, range over the entire**W**space.The

__u____o__**/n***"*ifiable__quant__**ifiers" do**__qual__**make sense as connoting**__not__**whole****, as do the***ontological categories*__q__**/n**, and the__b____.__**/n**.They

**make sense as connoting the**__do__**, or <<***units***>>, of the given***monads***, or <<***ontological category***>>.***arithmos*If

__u____o__**/k**denotes a single**, or <<**__u__*nit***>> of the***monad***, or <<***ontological category***>>,***arithmos*__q__**/k**, then the**u****k****(t)***"*ifier" function for that__quant____u____o__**/k**<<**>>***monad**-***ic****ifier, in a well-fitting "meta-model" of a given**__qual____dialectical__**, should generate the "population census count" -- the "number" of <<***ontological progression***>> of***monads***, or, e.g., of <<**__k__ind**>>,***genos*__q__**/k**-- that were/will-be extant as of stage**t**, e.g., the**number of type***average*__u____o__**/k****/ <<**__u__*nits***>> /***monads**logical**-*that were/will-be in existence during epoch/stage**individuals****t**.If

**u****k****(t)**...**=**...**0**, then**u****k****(t)**__u____o__**/k**...**=**...**0**__u____o__**/k**...**=**...**u****o****/****0**, which signifies either the extinction of**<<**__all__**>> of the***monads*__q__**/k**<<**>>, or the***arithmos**not-yet-*manifest,*not-yet-*irrupted state of the__q__**/k**<<**>>/***arithmos***.***ontology*Thus, per the

**F**.**.**__E__**. solution, the**__D____W____q__**/**__Q____E_______*"*"inherits" the "unit-interval-endpoints" restrictedness that characterize its**uni**-**system**"*"quasi-*ifier functions", the__quant__**b****n****(t)**, from its "<<**>>***arche'**-*, the "Boolean" system of a formal-logic calculus, as simulated by an arithmetic restricted to the endpoints of the unit interval, i.e., restricted to**system**"**0**and**1**, namely, the__W__**"<<**__E___**>>***arche'**-*.**system**"The

__W____U___*"*, on the contrary, "descends" from a "full-multiplicity" "<<**uni**-**system**"**>>***arche'**-*, the**system**"__W__**system of "**_____**hole numbers" arithmetic, and, consequently, shows that "inheritance" in its "full-**__W__**W**-multiplicity**ifier functions", the**__quant__**u****n****(t)**.The two

__dialectical__*system*__s__*-*considered in this post, per**progressions****F**.**.**__E__**., coincide neither in their first, "<<**__D__**>>***arche'**-*, nor in their third,**system**"*"*, but only in their second,**uni**-**system**"*"*:**contra**-**systems**"__w____Q__**in both cases.**_____**F**.**.**__E__**. source materials for this post --**__D__http://www.dialectics.org/dialectics/Briefs_files/_Brief1-29JUL2008_OCR.pdf

[pages

http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

**12**through**19 &**pages**3**-**1**through**3**-**8**].http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

[page

Regards,

Miguel

**A**-**42**].Regards,

Miguel

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