On 4/27/2026 5:01 PM, Scott Hoge wrote:There is no cycle in any Gödel number. Gödel numbers are natural numbersOn 2026-04-27, Alan Mackenzie <acm@muc.de> wrote:Olcott's Minimal Type TheoryIn sci.math Scott Hoge <nospam@nospam.com> wrote:>I 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.
I own a copy of Gödel's /On Formally Undecidable Propositions of
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.
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
and there are not any cycles in any natural number.
--
Mikko
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