On 2026-04-27, Alan Mackenzie <acm@muc.de> wrote:Olcott's Minimal Type TheoryIn sci.math Scott Hoge <nospam@nospam.com> wrote:I own a copy of Gödel's /On Formally Undecidable Propositions ofI learned what directed graphs were in high school.>
>
It seems our views are somewhat in agreement, but my directed
graph looks like this:
>
(D1)
· ─→ · ─→ · ─→ · ─→ ...
I strongly urge you to read and understand an actual proof of
Gödel's incompleteness theorem[*]. There are no looping or
endless directed graphs in these. Such notions result from
misunderstandings by those lacking formal training in
mathematics.
Principia Mathematica and Related Systems/ and have read the
original proof.
You're correct that the proof does not refer to directed graphs.
What I want to argue, rather, is that such graphs can be used to
/visualize the meaning/ of the Gödel sentence. That meaning
exists for the first unencoded sentence, which (as Gödel showed)
is true. But for what I'm calling the "formally abstracted
sentence" -- that is, the concept of being any sentence
represented by some node in the graph -- perhaps there is no such
meaning.
-- Scott Hoge
G ↔ ¬Prov_PA(⌜G⌝)
Directed Graph of evaluation sequence
00 ↔ 01 02
01 G
02 ¬ 03
03 Prov_PA 04
04 Gödel_Number_of 01 // cycle
BEGIN:(Gödel 1931:39-41)
...We are therefore confronted with a proposition which
asserts its own unprovability. 15
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.
This required establishing a new foundation
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