Part 06: Seldon on ‘Dialectical Categorial Progressions’ Series. How nQ Axioms §7 & §9 “Conspire” Against Redundancy & Unwanted Quantifiability.

  

                   

Part 06: Seldon on

‘Dialectical Categorial Progressions’ Series.

 

 

 How

_NQ Axioms

§7 & §9

 

“Conspire”

Against

Redundancy

&

Unwanted

Quantifiability.

 

 

 

 

 

 

 

Dear Reader,

 

It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, Seldon’s commentaries on key Encyclopedia Dialectica concepts of Seldonian Theory.

 

 

This 6th text in this new such series is posted herewith, together with supporting text-images and diagrams [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].

 

 

 

 

 

 

 

 

 

 

 

Seldon –

 

“Tonight I want to show you how the interplay of axioms §7 & §9 of the _NQ “Non-Standard Model” of the Peano-Dedekind “Natural” Numbers – the set, or space, 

N = {1, 2, 3, …} 

– in the computation and formation of dialectical ontological categorial progressions in that _NQ language, blog both potential redundancies and potential ‘lack-cunae’ [E.D. editors: “lacunae”] in the ‘consecuum’ of those progressions, interpreted for a specific Domain of knowledge, and how they thereby – by blocking redundancies – preclude any, unwanted, quantifiability in that, “purely”-qualitative, “purely”-ontological-categorial, _NQ dialectical language.”

 

“If we take, for example, the categorial progression for our current model of ‘the dialectic of nature’ – for the specific Domain that is also the most general, most universal Domain of which we know, D = " – and expand it, using the ‘Dyadic Dialectical Function’, dyadically, up to its epoch _"t₂ = 3, we obtain the following ontic, categorial progression in its raw, axiomatically-unpolished form –

 

Epoch _"t₂ = 0:  q_x²⁰  =  q_x¹  =   q_x;

Epoch _"t₂ = 1:  q_x²¹  =  q_x²  =  

(q_x) Ä (q_x) =

 

q_x Å q_xx  |-=  q_x ~Å~ q_c; 

Epoch _"t₂ = 2:  q_x²²  =  q_x⁴  = 

(q_x ~Å~ q_c) Ä (q_x ~Å~ q_c) = 

(q_x Å q_xx) Å (q_c Å q_xc) Å 

(q_x Å q_cx) Å (q_c Å q_cc)  |-= 

q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r; 

But: we had two of category-symbol q_x and two of category-symbol q_c.  

How did we get rid of the redundant q_x and q_c?

How did we avoid incurring a ‘quantifiability’, unwanted in the, “purely”-qualitative, _NQ arithmetic – 

how did we avoid ‘2q_x’and ‘2q_c’?

Axiom  §7, the “additive idempotency” axiom: 

For every n in N and for every q_n in _NQ, 

q_n Å q_n  =  q_n.  

So, \, q_x Å q_x  =  q_x, and q_c Å q_c  =  q_c.  

Note also that, given the axiom §9 ‘double conservation aufheben evolute product rule’, we didn’t develop any breaks in the _NQ ‘consecuum’ undergirding this ontic categorial progression either: 

for every j, k in N 

and for every q_j and q_k in _NQ, 

q_k Ä q_j  =  q_j Å q_k+j.  

Thus, we didn’t get –

(q_x ~Å~ q_c) Ä (q_x ~Å~ q_c) = 

(q_xx) Å (q_xc) Å 

(q_cx) Å (q_cc)  |-= 

q_c ~Å~ q_cx ~Å~ q_r

– leaving out q_x, the “ever-present” arché-category, and thus replacing the ‘evoluteness’ of the aufheben product rule with a, partial, ‘convoluteness’.  

[Note: subscripts are ordinally commutative, so that we use q_cx to stand also for q_xc; q_xc to stand also for q_cx].

 

Likewise, for –

Epoch _"t₂ = 3:  q_x²³  =  q_x⁸  = 

(q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r) Ä

(q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r) = 

(q_xÅ q_xx)Å(q_c Å q_xc) Å 

(q_cx Å q_xcx)Å(q_r Å q_xr) Å  

(q_xÅ q_cx)Å(q_c Å q_cc) Å 

(q_cxÅ q_ccx)Å(q_r Å q_cr) Å  

(q_xÅ q_cxx)Å(q_c Å q_cxc) Å 

(q_cxÅ q_cxcx)Å(q_r Å q_cxr) Å  

(q_xÅ q_rx)Å(q_c Å q_rc) Å 

(q_cx Å q_rcx)Å(q_r Å q_rr)  |-= 

q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r ~Å~        

q_rx ~Å~ q_rc ~Å~ q_rcx ~Å~  q_a; 

 

“So now we have five of category-symbol q_cx or q_xc, and, again, seven of category-symbol q_r or q_cc or q_xcx, and so on.

 

How do we escape 5q_cx and 7q_r? 

Answer: Again, Axiom §7.

 

How do we avert losing interpreted/applied ontic category-symbols q_x, q_c, q_cx, and q_r, which corresponding, in the generic/-unapplied/minimally-interpreted _NQ dialectical arithmetic to its first four ‘meta-numerals’ – 

q₁, q₂, q₃, and q₄? 

 

Answer: Again, Axiom §9.”

 

“Finally, how do we get from the four rows and thirty-two category-symbols of the “raw” product for the third-epoch, with q_x raised to the power 8?”

“The key is to be consistent about our solution [‘|-=’] for each repeat subscript category-symbol, and for the ordinal number corresponding to the subscript(s) of each distinct category-symbol.” 

 

“For example, the ordinal number values corresponding to the subscripts of category-symbols q_x [|x| = ordinal 1];

q_xx |-= q_c [|c| = ordinal 2];

q_cx [|cx| = |c| + |x| = ordinal 3], and;

q_cc |-= q_r [|cc|= |c| + |c| = ordinal 4]

– are –

1^st;

2^nd = 1^st plus 1^st;

3^rd = 2^nd plus 1^st, and;

4^th = 2^nd plus 2^nd,

respectively.”

 

“Per these considerations, the progression of subscript commutations and consistent double-subscript designations/solutions that gets us to the eight, mutually opposing [‘~’] – as well as gapless, via Axiom §9, and ‘de-redundantized’, via Axiom §7 – category symbols –

q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r ~Å~      

q_rx ~Å~ q_rc ~Å~ q_rcx ~Å~  q_a;

– are the following –

(q_xÅq_xx)Å(q_cÅq_xc)Å(q_cxÅq_xcx)Å(q_rÅq_xr) 

Å  

(q_xÅq_cx)Å(q_cÅq_cc)Å(q_cxÅq_ccx)Å(q_rÅq_cr) 

Å  

(q_xÅq_cxx)Å(q_cÅq_cxc)Å(q_cxÅ q_cxcx) Å

(q_rÅ q_cxr) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_rr) 

– transforms to –

(q_xÅq_xx)Å(q_cÅq_cx)Å(q_cxÅq_cxx)Å(q_rÅq_rx) 

Å  

(q_xÅq_cx)Å(q_cÅq_cc)Å(q_cxÅq_ccx)Å(q_rÅq_cr) 

Å  

(q_xÅq_cxx)Å(q_cÅq_xcc)Å(q_cxÅq_ccxx) Å

(q_rÅq_rcx) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_a)

– which next transforms to –

(q_xÅq_c)Å(q_cÅq_cx)Å(q_cxÅq_cc)Å(q_rÅq_rx) 

Å  

(q_xÅq_cx)Å(q_cÅq_cc)Å(q_cxÅq_ccx)Å(q_rÅq_cr) 

Å  

(q_xÅq_cc)Å(q_cÅq_xr)Å(q_cxÅq_ccc) Å

(q_rÅq_rcx) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_a)

– which next transforms to, with now-reveled redundant category-symbols terms crossed-out –

(q_xÅq_c)Å(q_cÅq_cx)Å(q_cxÅq_r)Å(q_rÅq_rx) 

Å  

(q_xÅq_cx)Å(q_cÅq_r)Å(q_cxÅq_rx)Å(q_rÅq_rc) 

Å  

(q_xÅq_r)Å(q_cÅq_rx)Å(q_cxÅq_rc) Å

(q_rÅq_rcx) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_a)

– which arrives at –

q_x~Å~q_c~Å~q_cx~Å~q_r~Å~q_rx~Å~  

q_rc~Å~  

q_rcx~Å~  

q_a.”

 

 

 

 

 

 

 

 

 

For more information regarding these Seldonian insights, and to read and/or download, free of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:

 

www.dialectics.info

 

and

 

https://independent.academia.edu/KarlSeldon

 

 

 

 

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights – specimens of ‘dialectical art’ – as well as dialectically-illustrated books published by the F.E.D. Press, see:

 

https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

 

 

 

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

 

 

 

 

Regards,

 

Miguel Detonacciones,

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

 

 

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