Agent Smith wrote: ↑Mon May 01, 2023 9:42 am
I'm still in the dark about Gödel numbering; I hear it's critical to his proofs of incompleteness.
Anyway,
PeteOlcott seems to have homed in on a flaw in Gödel's argument and it kinda sorta aligns with my own suspicions in re Gödel's work. The penny ultimately ... hasta drop, oui?
I can somewhat sense the urgency of the matter i.e. there's at least one very good reason why Gödel should keep us awake at night, but that's a topic for another discussion. Oui?
The key urgency of the matter is to divide the opposing ideas:
(a) Putin invaded Ukraine as a land grab to bring back that Russian empire.
(b) Putin invaded Ukraine to legitimately prevent hostile powers from getting too close to its borders
in a way similar to the Cuban Missile Crisis.
Whether or not there was substantial voter fraud that changed the outcome of the
2020 presidential election.
Whether or not climate changed is actually caused by humans and what are the reasonably
plausible range of best case to worst case scenarios of simply ignoring it.
My ultimate goal is to make True(L,x) computable.
Here is a brief overview of my greatest impediment to that goal:
The idea of the proof: If there were such a formula of the language of F, an easy application
of the Diagonalization Lemma to its negation would result in the paradoxical sentence
L (for “Liar”; see the Liar paradox)), such that: F ⊢ ¬Tr(┌L┐) ↔ L,
which, together with the T-equivalences, which were assumed to be derivable, would quickly
give an explicit contradiction, thus contradicting the assumption that F is consistent.
https://plato.stanford.edu/entries/goed ... rTheUndTru
In other words Tarski could not prove that the self-contradictory Liar Paradox is true
so he concluded that True(F,x) cannot be correctly defined in F.
Here is how Tarski "proved" that the Liar Paradox is true in his metatheory:
This sentence is not true: "This sentence is not true"
Here is his original proof:
https://liarparadox.org/Tarski_275_276.pdf
The Liar Paradox is very obviously not a truth bearer
https://plato.stanford.edu/entries/truthmakers/
My unique contribution to this field is that there are a set of finite strings of language L
that are stipulated to have the semantic property of Boolean True these are the truth makers of L.
For formal languages Haskell Curry called these the
elementary theorems of::t
https://www.liarparadox.org/Haskell_Curry_45.pdf
I extend this notion to natural languages such that {cats are animals} is stipulated
to be true in the model of the current world of all possible worlds.
Thus True(L, x) merely means that x is derived from the axioms of formal or natural language L.
What I mean by derived requires a separate discussion.
It is essentially a formal syntactic proof from the axioms of L to x using only semantically
truth preserving operations. At a minimum this requires relevance logic.
https://plato.stanford.edu/entries/logic-relevance/