Great sentences! No discussion can flourish without good examples and metaphores though they make easier targets for criticism than dull theory. Or may be precisely because of that!

So far it seems to me that defining the "nothing" and "everything" quantifiers in a meaningful manner is considerable easier than performing that feat for "nothingness". Of course, we can generate the standard expansion for "-nesses" and define nothingness as "the quality of being nothing" but that is where trouble begins. One needs something to have that quality and a something which is nothing appears a trivial contradiction (as MindForged suggested). Unless of course, the something is a nothing itself - as a thing (noun). A nothing possesses the quality of being nothing and therefore possesses nothingness. That's not a genuine property but a description of the fixed relationship.of these concepts.

The point I try to make here is that, in order to access nothingness without contradiction one needs a nothing, not as a quantifier but as an object. I am not sure meaningful examples of a nothing object exist but if they do, and they are not unbounded, they also provide a peek at bounded mental manifestations of nothingness. Wonder what that looks like.

Not that any of this promises individual significance for the nothingnesses. They are only addressable through nothing objects and that is probably where the buck stops wih regard to any functional uses.

Nothingness is the dual concept of everything (or would it be "everythingness"?) "Everything" would be the mereological sum of all objects. So nothingness is the mereological sum of no objects.

I suspect if you get down to it nothingness I probably a contradictory concept.



This is true if nothingness and everythingness always apply to unrestricted object and property categories/domains. However, in practical cases a property filter is applied before nothingness is declared, e.g.:

A physical object commonly has a color but occasionally not because it is transparent. Its color then is described as "no-color" or "nothing". Its color attribute becomes a nothingness, another way of stating that this particular object has no color attribute. The color-domain restriction assures that the other mereological attributes - especially the objectness of the object - need not be erased.

Is this a valid mereological argument?

You may very well be right here as my knowledge of mereology is minimal. From the title of the thread and its opening post I had not deduced we were dealing with a specialized subject and treated it with the looseness commonly associated with human language.

Note that I was aware that zero is not the same as nothing or nothingness. I tried to show that many sentences with the word "nothing" can be easily converted to similar sentences with the word "zero" with the same meaning. That the sentence as a whole has the same meaning does not imply that "zero" and "nothing" have the same meaning but it could imply that the understanding of one can be derived - via a detour - from the understanding of the other.

My view on the emptyness is that - because it refers to an entity which must be empty - it is equivalent to a container with a nothingness (property). For its lack of property the nothingness itself cannot be tied to the container, but there exists no reason why the container would have no information on its emptyness.

I see your points about nouns and quantifiers.

"Universal nothingness" is pretty uninteresting as it is a pertinent lie. And consequently, its complement of "incidental somethingness" (there is something somewhere, some time) is true. Descartes "cogito, ergo sum" covers about the same point by stating that the something "I" exists.

Therefore it is more realistic to look at "nothing(-ness)" in terms of the specific context and attribute it is used for. For instance:

Mum: "The porselain is broken. What did you do, Jacob?" Jacob: "I did nothing, Mum!"

Clearly, this neeeds a little editing before feeding it to the household robot to decide upon proper punishment for Jacob. like: Mum: "During the time I was shopping this afternoon, the porselain in the kitchen was broken. Did you do do anything which caused that porselain to be broken, Jacob?" Jacob: "I did zero things to cause the broken state of the kitchen porselain this afternoon, Mum!".

This example shows that "nothing" is often convertible to "zero" when using a different phrasing. One might try to debate the usefulness of the concept of "nothingness" but no one will deny the perks of having the "zero". Lacking it kept Roman calculus backward for centuries.

Math makes perfect use of nothingness by means of the "empty set" - a set which contains nothing - for the construction of number-objects in ZFC Set-Theory, probably the most important theory in all of mathematics. All sets based on natural numbers are derived from the empty set. It tells us that "nothing" in itself may not be much but "nothing in a container" can be a great tool. Another example is the assocation of "nothing" with "non-separation" in number theory. What's between 5 and 10? Well, 6 and 7 and ... But what is between 5 and 6? Pecisely, nothing! Tells you these natural numbers are contiguous, a non-existent property for most other number types.

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.

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.

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".

The (G-)sentence cannot be deduced and hence does not exist in the system.

This is incorrect since the sentence exists in the syntax of the system. It is true though that it does not have a semantic value and therefore does not exist as a theorem. The whole point of incompleteness is to show that the syntax of many formal systems is "bigger" than their semantics.

The point of my original comment is however metaphorical. It describes what a human being living inside a formal system would need to make sense of the syntactically present G-sentence. Could he read into the model associated with the system - as a contingent truth not as a necessary one - he could perceive its semantic value in the same way he could for a quantum state in our physical reality by making an observation.

The (G-)sentence cannot be deduced and hence does not exist in the system.

That is incorrect since it is covered by the syntax of the system. It is true though that it has no semantic value and therefore does not exist as a theorem. The whole point of incompleteness is to show that the syntax of many formal systems is "bigger" than their semantics. The Gödel-sentence would never have been debated had it been excluded a priori by the syntax-checker of the active system.

That is one of the interesting points of the chess rules (notably their extensions in the Codex conventions). Illegal positions - not axiomatically derivable from the game starting position - are not permitted to be associated with a semantic value for "winnability" though this is often possible. The chess community decided to declare these positions syntactically illegal even where a lot of analysis is required to prove this point. Note that chess is more complicated than common mathematical examples as it contains a separate geometric syntax layer for the placing of chess units on the board.

Your comment would hold in chess where illegal positions would be syntactically placed outside the evaluation system for winnability.

I pondered on the relationship between QM and Gödelian incompleteness a while ago and found some interesting parallels especially to the G-sentence. What comes to mind straight away is the similarity of quantum state uncertainty and undecidability of the G-sentence which is probably what made you look for a deep connection in the first place.

One way of perceiving quantum uncertainty is as an incomplete process of physical manifestation. The laws of physics only go this far in deciding the fate of a particular quantum state and will just conclude it upon an observation. In this context, the laws of physics may be considered a priori and the observation a posteriori.

In Math, the a priori axiomatic system is only capable of generating certain truth (and certain falsity by negation) leaving the undecidable G-sentence out in the cold. However, by picking an appropriate model to shell the formal system, the semantic value of the G-sentence becomes accessible (though not by formal proof)..From a viewpoint inside the formal system the model is the a posteriori observation required to determine the state of the G-sentence.

I am very comfortable with this image as it connects perfectly with scenarios I developed earlier for certain chess problems in relation to Gödelian incompleteness on one side and to Quantum entanglement on another. And it emphasizes the non-existence of undefinable truth other than by random choice of model.