Reflection On Recursion • 2

Turning to the form of a simple recursive function the clause we used to define it earns the title of “syntactic recursion” due to the way the function name occurring in the defined phrase re‑occurs in the defining phrase

It needs to be clear there is no circle in the definition — each instance of the type is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat.  Instead of a circle then we have two gyres, the gyre down via the predecessor function and the gyre up via the modifier function

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2) (3)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Do we have any Gödel experts in the house?

I'm trying to understand why Gödel used this encoding in the original Gödel numbering. To be clear, these are supposed to be the exponents in the unique factorisation 2ᵃ3ᵇ5ᶜ⋯, not the bases which are also coincidentally prime.

Why did Gödel pick prime exponents too?

And why did is the 0 assigned to the exponent 1 and not 2? Why is that first one not prime?

At first I thought this might be a typo in the inset figure in Wikipedia, but upon consulting the cited reference I saw that the same encoding is used in the original paper. https://en.wikipedia.org/wiki/G%C3%B6del_numbering#G%C3%B6del's_encoding

#GödelNumbers #MathematicalLogic

Higher Order Sign Relations • Discussion 1
• https://inquiryintoinquiry.com/2025/03/27/higher-order-sign-relations-discussion-1-a/

Re: FB | Charles S. Peirce Society • John Corcoran
• https://www.facebook.com/groups/peircesociety/posts/1768975423238442/

Questions about the proper treatment of use and mention from the standpoint of Peirce’s theory of signs came up recently in discussions on Facebook. In pragmatic semiotics the trade‑off between “signs-of-objects” and “signs-as-objects” opens up the wider space of Higher Order Sign Relations. In previous work on Inquiry Driven Systems I introduced the subject in the following way.

When interpreters reflect on their use of signs they require an appropriate technical language in which to pursue their reflections. They need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations. The orders of signs developing as reflection evolves can be organized under the heading of “higher order signs” and the reflective sign relations involving them can be referred to as “higher order sign relations”.

References —

John Corcoran
• https://johncorcoran.academia.edu/

Schemata : The Concept of Schema in the History of Logic
• https://www.academia.edu/12691868/SCHEMATA_THE_CONCEPT_OF_SCHEMA_IN_THE_HISTORY_OF_LOGIC

Use And Mention, Use Without Mention, Mention Without Use
• https://www.academia.edu/s/ea64a3484e/schemata#comment_525151

Resources —

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2024/02/28/survey-of-inquiry-driven-systems-6/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
#InquiryDrivenSystems #ReflectiveInterpretiveFrameworks
#Arithmetization #GödelNumbers #Quotation #UseAndMention