What do you mean by omniscience and potential omnipotence? — Posty McPostface

The ability to understand and maybe manipulate the world without limitation.

I take it from a Platonic POV and assume that math is the reality, therefore what does that imply according to Godel's Incompleteness Theorems? — Posty McPostface

I do not understand. There is math and there is... stones. How are stones math?

It (the G-sentence) is not true in the formal system as it cannot be deduced.
If it was true in the system then the system would be self-contradictory.
Either incomplete or self-contradictory. It was the mathematicians taking a look at Gödel's proof who thought the sentence was true.


Sorry my original message mysteriously disappeared after I edited out a comma and not having saved it I replaced it with a simpler one. Contains the essentials though.

I am not sure of the source, but I am sure I read that Kurt Gödel himself believed the G-sentence to be true outside the system. It didn't matter for his incompleteness theorems as they only depend on "undecidability" which is obviously a weaker proposition.

Modern mathematics holds the view that the G-sentence is true in some models (notably the standard model) and false in others. I view it as an information problem. Since some essential "reflective" information was not passed on from the model to the formal system, it couldn't decide the outcome. Then again, formal systems actually lack the tool required to receive that information and we would have to conclude that Gödels concept simply cannot be completely coded into the language of the formal system. Which means that mathematical logic halted around the age of 4 for a human child. Therefore I am not overly impressed when a mathematician speaks of "undefinable truth".

Modern mathematics holds the view that the G-sentence is true in some models (notably the standard model) and false in others. — Arisktotle

Who? Which model?

Since some essential "reflective" information was not passed on from the model to the formal system, it couldn't decide the outcome. — Arisktotle

Again: Which model?

Then again, formal systems actually lack the tool required to receive that information and we would have to conclude that Gödels concept simply cannot be completely coded into the language of the formal system. — Arisktotle

No, no... The sentence could be coded into the all-system. It just blew up then.

Therefore I am not overly impressed when a mathematician speaks of "undefinable truth". — Arisktotle

Would a definition be true in respect to itself?

Modern mathematics holds the view that the G-sentence is true in some models (notably the standard model) and false in others.
— Arisktotle
Who? Which model?


From Wikipedia, Gödels incompleteness theorems:

Although the Gödel sentence of a consistent theory is true as a statement about the intended interpretation of arithmetic, the Gödel sentence will be false in some nonstandard models of arithmetic, as a consequence of Gödel's completeness theorem (Franzén 2005, p. 135). That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false. As described earlier, the Gödel sentence of a system F is an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the system F, and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number (Raatikainen 2015, Franzén 2005, p. 135).

Since some essential "reflective" information was not passed on from the model to the formal system, it couldn't decide the outcome.
— Arisktotle
Again: Which model?
Then again, formal systems actually lack the tool required to receive that information and we would have to conclude that Gödels concept simply cannot be completely coded into the language of the formal system.
— Arisktotle
No, no... The sentence could be coded into the all-system. It just blew up then.


Note that the latter citations constitute my view as indicated in the comment. This is not standard mathematical stuff but something I have been working on for a while. For instance, I doubt the all-system will be capable of handling new definitions for function domains and co-domains necessary to be compatible with mature human logic.

Okay... as I read this those non-standard models are those where contradictions would be allowed without leading to arbitrary conclusions. ( like https://en.wikipedia.org/wiki/Paraconsistent_logic ).
Those are indeed interesting as the G-sentence could be put onto an island for it's own without any connection to the rest of the world, like: "There is everything that is the case. And then there is the G-sentence."

Thanks for the exchange of views. Discussions on Gödel tend to be as incomplete as his theorem domains and are guaranteed to haunt us another day. I recently stumbled upon a message from an author who was in the process of writing a book to introduce Gödels work to his students and concluded that he now needed to write an intro book to that intro book as its size had grown to hundreds of pages. A deceptively simple subject (if ye know what I mean)!

Meanwhile some visitors may choose to continue this thread in accordance with its original intention to link up Quantum Theory and Incompleteness from which we got a bit sidetracked.