# Is everything in math either an axiom or a theorem?

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• Feynstein100
In summary: I'm sorry, I misunderstood your previous statement. I thought you were asking for an example of something that is assumed to be true but cannot be proven. As for your question about the difference between definitions and axioms, definitions are used to clarify and define specific mathematical objects or concepts, while axioms are more general assumptions that serve as the foundation for a particular mathematical system.
Feynstein100
Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more?
I feel like we're venturing into Gödel incompleteness territory here but for this discussion, let's keep it simple. Is this duality of axiom/theorem all-encompassing or are there things that lie beyond?
It's also interesting to note that since theorems follow from axioms, it should be possible to write down all axioms and all their subsequent theorems, in a linear timelike/causality-like structure. Is this a coincidence?
And finally, where does the notion of mathematical objects fit into this? For example, you might find out all axioms and theorems related to numbers. But once you introduce the idea of vectors and tensors, which are different mathematical objects, the previous knowledge doesn't apply anymore. Because they are different objects, they will have different properties, meaning different axioms and theorems. Basically, any new mathematical object will have its own set of axioms and theorems. Which raises the question, will there always be some new mathematical object to discover i.e. there are an infinite number of them or will we eventually run out of them?

Feynstein100 said:
Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more?
I feel like we're venturing into Gödel incompleteness territory here but for this discussion, let's keep it simple. Is this duality of axiom/theorem all-encompassing or are there things that lie beyond?
It's also interesting to note that since theorems follow from axioms, it should be possible to write down all axioms and all their subsequent theorems, in a linear timelike/causality-like structure. Is this a coincidence?
And finally, where does the notion of mathematical objects fit into this? For example, you might find out all axioms and theorems related to numbers. But once you introduce the idea of vectors and tensors, which are different mathematical objects, the previous knowledge doesn't apply anymore. Because they are different objects, they will have different properties, meaning different axioms and theorems. Basically, any new mathematical object will have its own set of axioms and theorems. Which raises the question, will there always be some new mathematical object to discover i.e. there are an infinite number of them or will we eventually run out of them?
You also have definitions.

-Dan

vela
topsquark said:
You also have definitions.

-Dan
Eh...what? That was very cryptic

Feynstein100 said:
Eh...what? That was very cryptic
You have axioms: Statements that we take as given without proof.

You have theorems: Statements that we can prove from axioms.

You have definitions: Statements that define objects or features of objects.

For example, the ZF axioms can be used to show that an infinite set may be constructed. We make a definition that this set is (up to isomorphism, which is a later definition) the natural numbers. We also have a definition that this set is "countably infinite." etc. Definitions aren't axioms and can't be proven; they are statements that tell us what to call things.

-Dan

dextercioby, pinball1970, malawi_glenn and 1 other person
Feynstein100 said:
Or it's true but cannot be proven
I'm certainly not a mathematician, can you give an example of something known to be true but unproven? In my mind, axioms are "given" -- assumed to be true. But they may not "really" be true. Things like flat space.

FactChecker and topsquark
gmax137 said:
I'm certainly not a mathematician, can you give an example of something known to be true but unproven? In my mind, axioms are "given" -- assumed to be true. But they may not "really" be true. Things like flat space.
Good point. He should have said, "assumed true but can not be proven".
In fact, there is a lot of math where something is assumed true just to see what would logically follow.
There is the mathematics of flat space that is studied, and of curved space that is studied. The two have contradictory assumptions, but the mathematics of both exist.

A great deal of mathematics is done with assumptions that are not required to be true in any particular application. Once the logic of that situation is established, then we are free to use the entire set of logic where it applies. A good example is Abstract Algebra. If a particular subject of interest is determined to be an abstract algebra field, then the entire set of logic for a field can be used in that subject of interest. That gives us a "toolbox" of mathematics that can be used when appropriate.

topsquark said:
You have axioms: Statements that we take as given without proof.

You have theorems: Statements that we can prove from axioms.

You have definitions: Statements that define objects or features of objects.

For example, the ZF axioms can be used to show that an infinite set may be constructed. We make a definition that this set is (up to isomorphism, which is a later definition) the natural numbers. We also have a definition that this set is "countably infinite." etc. Definitions aren't axioms and can't be proven; they are statements that tell us what to call things.

-Dan
Oh. I thought definitions were axioms too, since they can't be proven. What exactly is the difference between the two?

gmax137 said:
I'm certainly not a mathematician, can you give an example of something known to be true but unproven? In my mind, axioms are "given" -- assumed to be true. But they may not "really" be true. Things like flat space
I don't have a particular example in mind, but I'm most familiar with numbers so think of the fact that when we multiply two (natural) numbers, we get another number. I don't think it's possible to prove this. We just take it for granted. By contrast, when we mutliply vectors, we can get either a number(scalar), vector or even tensor.
What I'm trying to say is, multiplication is just a kind of interaction between mathematical objects. And yet for some reason, the same interaction between scalars behaves way differently than between vectors. Why is that so? How do we decide the rules of interaction? Well, we don't. They're just kind of there.
Idk about other mathematical objects but numbers at least are grounded in physical reality so the reason behind their interactions is simply that that's how we observe them in real life. 2*5=10 means that if you have 2 groups of 5 identical physical objects each or vice versa, you will have 10 objects in total. We know this to be true simply because we observe it to be true. Isn't that what an axiom is?

Feynstein100 said:
Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more?
I feel like we're venturing into Gödel incompleteness territory here but for this discussion, let's keep it simple. Is this duality of axiom/theorem all-encompassing or are there things that lie beyond?
Godel proved that either 1) there are theorems that cannot be proved within basic logic or 2) mathematics contains contradictions. No one believes (2).

The theorems in (1) can be proven using a higher system of logic but this systems also contains theorems that cannot be proven. And so forth.
Feynstein100 said:
It's also interesting to note that since theorems follow from axioms, it should be possible to write down all axioms and all their subsequent theorems, in a linear timelike/causality-like structure. Is this a coincidence?

It seems possible to construct a recipe for such a neverending process.

Feynstein100 said:
And finally, where does the notion of mathematical objects fit into this? For example, you might find out all axioms and theorems related to numbers. But once you introduce the idea of vectors and tensors, which are different mathematical objects, the previous knowledge doesn't apply anymore. Because they are different objects, they will have different properties, meaning different axioms and theorems. Basically, any new mathematical object will have its own set of axioms and theorems.
Hmm. It would be tautologically true if you define a new object as anything that requires a new axiom.
Feynstein100 said:
Which raises the question, will there always be some new mathematical object to discover i.e. there are an infinite number of them or will we eventually run out of them?
Well if you use that definition then I suppose once could make up new axioms forever if one felt like it.

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FactChecker said:
The two have contradictory assumptions, but the mathematics of both exist.
Ah but those aren't assumptions, those are properties. I don't see this as a contradiction, merely an indication that we're talking about different things. You wouldn't expect a solid to behave the same way as a liquid. And the existence of the liquid doesn't negate the existence of the solid, or vice versa. So why should you expect spherical geometry to behave the same way as flat geometry? They're different objects/domains with different properties.

It has already happened that there is math with axiom A and other math with axiom not A. It's OK as long as you don't mix the two.

A definition is just a name. You could use any other name and everything would still work just the same.

Hornbein said:
there are theorems that cannot be proved within basic logic
Isn't that just an axiom?
Hornbein said:
The theorems in (1) can be proved using a higher system of logic but this systems also contains theorems that cannot be proved. And so forth.
That's not really relevant for the original question. What I want to know is even if we add the higher system, would everything in it still be either an axiom or a theorem?
Hornbein said:
It seems possible to have such an infinite process.
Would you care to elaborate? I find it interesting because it means that even though math is supposed to be unbounded by physics, it still follows the same structure as our universe. That can't be a coincidence, can it?

Math100 and pinball1970
jedishrfu said:
There are also conjectures and hypotheses which are the names we give to unproven theorems
That's just a semantic issue. Logically, speaking, it's not a theorem until it has been proven. For our purposes, that doesn't matter. Either it can be proven, which means it's a theorem, or it can't, which means it's not. Wait, huh. If it's true but can't be proven then it'd be a new axiom now, wouldn't it? But if we can't prove it, how do we know it's true? I feel like I'm stuck in a loop here

dextercioby
How does Notion fit?

Feynstein100 said:
Ah but those aren't assumptions, those are properties. I don't see this as a contradiction, merely an indication that we're talking about different things. You wouldn't expect a solid to behave the same way as a liquid. And the existence of the liquid doesn't negate the existence of the solid, or vice versa. So why should you expect spherical geometry to behave the same way as flat geometry? They're different objects/domains with different properties.
Ok. I'll buy that. I admit that I don't know what official terminology is correct to use.

Feynstein100 said:
If it's true but can't be proven then it'd be a new axiom now, wouldn't it?
How would you know some statement is true, if you can't prove it?

Feynstein100 said:
Either it can be proven, which means it's a theorem, or it can't, which means it's not. Wait, huh. If it's true but can't be proven then it'd be a new axiom now, wouldn't it? But if we can't prove it, how do we know it's true?
There are statements that are not provable (and their negation is not provable either) within a system but can be included consistently with the system. The continuum hypothesis is an example of such a statement. Neither it nor its negation can be proven from within the standard Zermelo Fraenkel set theory, but one can include either it or its negation (but obviously not both!) as an axiom without creating an inconsistency.

Theorems are specifically provable from axioms, and axioms are taken as true at the outset. But the continuum hypothesis is an example of a well-formed statement that is neither a theorem nor an axiom of ZFC.

PeroK, Hornbein, FactChecker and 1 other person
Feynstein100 said:
I don't have a particular example in mind, but I'm most familiar with numbers so think of the fact that when we multiply two (natural) numbers, we get another number.
Most people (including mathematicians) think about math in such an informal manner. They conduct their internal mental dialog as if some mathematical "things" exist and have certain properties independently of anything humans write down about them. ( In the early history of mathematics, it was customary to call a self evident mathematical fact an "Axiom". Other statements assumed to be true but somehow not so self evident were called "Postulates".) But this approach is fallible and depends on culture. For example, it took time for a consensus about concepts such as zero, negative numbers, irrational numbers, and complex numbers to develop in human history. In the present day, you can observe online disagreements (mostly among non-mathematicians) about concepts involving the word "infinity".

So the modern approach to pure mathematics is legalistic. From this point of view, mathematical concepts and their properties must be explicitly defined. A definition is just a handy way to abbreviate a collection of assumptions. For example if I write "Let k be a natural number" then I have assumed that k is a thing that satisfies the properties that are assumed for natural numbers.

The modern (or "formal") approach to mathematics is not how mathematics education proceeds! Elementary education encourages children to think intuitively about mathematical ideas. It encourages the approach that examples of applications of mathematics show us what is true in mathematics.

It's tempting to think that we can prove certain mathematical statements are true in the same sense that scientific facts are proven. So if we are asked to prove that 3+2=5, we might formulate an argument like: "If I put 3 apples on a table and then put 2 more apples on the table then...." etc. However, in pure mathematics , a proof of a general statement can't be done only by showing examples where the statement is true. By contrast, a large collection of examples where a statement is true and lack of any where the statement is false is scientific proof of the statement - but scientific statements accepted as true are always subject to revision in the light of new discoveries. In pure mathematics we'd like to avoid such unrelenting revision; we'd like innovation to take form of new theorems about old concepts or new concepts - not the redefinition of old concepts. So we do not regard empirical examples as general proofs.

Almost everyone likes to reason about mathematical concepts in terms of their favorite application of those mathematical concepts. People who like physics can effectively reason about the concepts of vectors and scalars in terms of physical examples. Yet in pure mathematics, the fact (in the scientific sense) that a statement is true in some applications of mathematics does not prove the statement is true (in the mathematical sense) in general. So a proof about vectors and scalars must be based on the assumptions that define concepts like "vector space". The definition of "vector space" can be written using other definitions of mathematical ideas such as "abelian group". These other definitions abbreviate further assumptions.

David Lewis, PeroK, symbolipoint and 2 others
Stephen Tashi said:
In the early history of mathematics, it was customary to call a self evident mathematical fact an "Axiom". Other statements assumed to be true but somehow not so self evident were called "Postulates".
Thanks, I was wondering about the distinction between the two.

Stephen Tashi said:
They conduct their internal mental dialog as if some mathematical "things" exist and have certain properties independently of anything humans write down about them.
Yes, that's called Platonism. What exactly is wrong with this line of thinking? You talk about building mathematical "intuition" while somehow failing to realize that intuition is built from experience. It arises from pattern recognition i.e. recognizing something that's already there, not arbitrarily creating it yourself.
Don't mistake the representation of a thing as the thing itself. Every culture might have a different word for water but that doesn't mean water itself is an artificial construct that you can manipulate simply by tinkering with the word that represents it.
Math follows an if........then structure that's independent of subjectivity. No matter who discovers it in whatever way, the structure remains the same. "If you draw a triangle in a flat plane, its angles will add up to 180 degrees." This statement is true no matter how you you look at it.

TeethWhitener said:
There are statements that are not provable (and their negation is not provable either) within a system but can be included consistently with the system. The continuum hypothesis is an example of such a statement. Neither it nor its negation can be proven from within the standard Zermelo Fraenkel set theory, but one can include either it or its negation (but obviously not both!) as an axiom without creating an inconsistency.

Theorems are specifically provable from axioms, and axioms are taken as true at the outset. But the continuum hypothesis is an example of a well-formed statement that is neither a theorem nor an axiom of ZFC.
Yeah but that doesn't answer the question, how would you know it's true if you can't prove it?

Feynstein100 said:
Yeah but that doesn't answer the question, how would you know it's true if you can't prove it?
We don't know that mathematical axioms are true and we don't prove them. Axiom is just another name for assumption. The truth of a mathematical axiom can be discussed in the context of whether it is true in a particular application of mathematics. That's a discussion of science, not pure math.

Feynstein100 said:
Yes, that's called Platonism. What exactly is wrong with this line of thinking?

Do you mean what's wrong with it in the practical sense of solving real world problems? - or do you mean what's wrong with it in task of developing pure mathematics?

Nothing is wrong with mathematical Platonism In the practical sense. In thinking about practical situations, anything goes that is empirically effective. However, in developing mathematics consistently, intuition doesn't always work. When things get technical, one person's idea of what must be true about a mathematical concept may differ from another person's. The output of one person's pattern recognition from experiences may differ from the output of a different person's pattern recognition.

Feynstein100 said:
Don't mistake the representation of a thing as the thing itself.

The frustrating situation is that everything is a representation. If you discuss a topic, you use words that represent it. Even the phrase "the thing itself" is a representation. In formal mathematics we acknowledge that we must use undefined concepts in discussing mathematics. As history progressed, concepts originally accepted as undefined ( like "number") were defined in terms of concepts that are regarded as simpler.

In elementary mathematics, childen are taught that the commutative "law" of addition is a "property" of real numbers - as if the real numbers exist and have objective properties in the same sense that your favorite chair exists. In more advanced mathematics, such "laws" are presented either as theorems or assumptions.

In very advanced mathematics, people developed ways to prove the properties of real numbers beginning only with assumptions about simpler concepts. I think they use concepts from set theory, but I haven't studied this myself. Mathematician's call this type of investigation "foundations". Most mathematicians I have known are not interested in foundations.

PeroK and FactChecker
Stephen Tashi said:
In very advanced mathematics, people developed ways to prove the properties of real numbers beginning only with assumptions about simpler concepts. I think they use concepts from set theory, but I haven't studied this myself. Mathematician's call this type of investigation "foundations". Most mathematicians I have known are not interested in foundations.
IMO, this is in the domain of philosophers (who may also be great mathematicians). Bertrand Russel and such.

dextercioby and symbolipoint
Stephen Tashi said:
We don't know that mathematical axioms are true and we don't prove them. Axiom is just another name for assumption. The truth of a mathematical axiom can be discussed in the context of whether it is true in a particular application of mathematics. That's a discussion of science, not pure math.
Do you mean what's wrong with it in the practical sense of solving real world problems? - or do you mean what's wrong with it in task of developing pure mathematics?

Nothing is wrong with mathematical Platonism In the practical sense. In thinking about practical situations, anything goes that is empirically effective. However, in developing mathematics consistently, intuition doesn't always work. When things get technical, one person's idea of what must be true about a mathematical concept may differ from another person's. The output of one person's pattern recognition from experiences may differ from the output of a different person's pattern recognition.
The frustrating situation is that everything is a representation. If you discuss a topic, you use words that represent it. Even the phrase "the thing itself" is a representation. In formal mathematics we acknowledge that we must use undefined concepts in discussing mathematics. As history progressed, concepts originally accepted as undefined ( like "number") were defined in terms of concepts that are regarded as simpler.

In elementary mathematics, childen are taught that the commutative "law" of addition is a "property" of real numbers - as if the real numbers exist and have objective properties in the same sense that your favorite chair exists. In more advanced mathematics, such "laws" are presented either as theorems or assumptions.

In very advanced mathematics, people developed ways to prove the properties of real numbers beginning only with assumptions about simpler concepts. I think they use concepts from set theory, but I haven't studied this myself. Mathematician's call this type of investigation "foundations". Most mathematicians I have known are not interested in foundations.
Conjectures? Do they fall in-between? They could be true or false but may or not be provable?

Feynstein100 said:
Yeah but that doesn't answer the question, how would you know it's true if you can't prove it?
It answers your question from the OP as to whether all statements were either axioms or theorems. In the case of the continuum hypothesis from within Zermelo Fraenkel set theory (ZF), you can’t prove it’s true, but you also can’t prove it’s false, so it’s not a theorem. It’s also not an axiom of the theory, and adding either it or its negation doesn’t render the theory inconsistent. Therefore, you could expand ZF by adding the continuum hypothesis as an axiom, but you could equally validly expand ZF by adding the negation of the continuum hypothesis as an axiom. We say that, because of this, the continuum hypothesis is independent of ZF, even though it is expressible in the language of set theory.

To answer your new question (very non-rigorously), Gödel proved that there are logical systems—including standard arithmetic—which contain sentences that are “true but not provable.” What he did was construct a sentence (call it ##G##) that basically says “this sentence is not provable,” and then he proved that if the logical system is consistent (i.e.,doesn’t prove a false statement), then ##G## is true. The details are way above a B-level explanation, and include semantic notions like “interpretations” of logical systems, but the upshot is this: even in fairly straightforward theories, either there are true and unprovable sentences, or the system is inconsistent.

dextercioby and Stephen Tashi
pinball1970 said:
Conjectures? Do they fall in-between? They could be true or false but may or not be provable?

Yes. Mathematical conjectures are speculations. When they are made they are unproven. They may eventually be proven true or proven false. In rarer cases they may be proven "undecidable". The notion of undecidable (un-provable) does not refer to a limitation of human capabilities or endurance. In other words, it doesn't mean that people just gave up trying to prove or disprove the conjecture.

It is important to remember that the provability or non-provability of a statement is relative to the assumptions ( axioms) of the mathematical system in which the statement is made. What's unprovable in one system may be provable in a different system.

An attempt at analogy is the following. Suppose there is a certain type of puzzle and a known method for whether determining whether any given set of proposed answers is a valid solution. A person might write a computer program using certain algorithms to solve this type of puzzle. There might be examples of puzzles that the program cannot solve, even though the program can input a proposed solution to such a puzzle and determine whether the solution is valid. It would fair to call such puzzles unsolvable (with respect to the algorithms in the program) even though the puzzles have solutions. That's probably a better analogy for "uncomputable" than "undecidable", but it's what comes to mind at the moment.

Feynstein100, jbriggs444, jedishrfu and 1 other person
Stephen Tashi said:
We don't know that mathematical axioms are true and we don't prove them
That's true but somehow I feel like it's not a bad thing. Like I said before, I like to think of math as an "If....then....." structure. Consider the statement, "If unicorns exist, they will have horns". In this case, the axiom is unicorns existing and isn't true, however, the statement that follows it is kind of true. Because if unicorns did exist, they would have horns. Same with numbers. You can say, if I have 3 apples and I eat 2 of them, I will have 1 apple remaining, basically saying that 3-2=1. Whether or not you actually have 3 apples is irrelevant. The statement is still true, in a sense that it could be true, should you choose to fulfil the premise i.e. the axiom. Huh. It also reminds me of the scientific method. Let's say you've never seen microbes with your own eyes before. Does that mean they don't exist? Well, no. Because you can see them if you fulfil the criterion/axiom, which in this case is getting a microscope and a sample.
Stephen Tashi said:
The output of one person's pattern recognition from experiences may differ from the output of a different person's pattern recognition.
You gave me an interesting thought. Now we're entering the realm of subjectivity vs objectivity. I'm arguing that mathematical Platonism is useful for pure math because it frees us from subjectivity. And just like objective truths in reality, we have objective truths in math. Again, provided we follow the axioms. That's what theorems and proofs are for, right? Basically seeing how different axioms interact with each other in logically consistent ways.
I have to admit, my viewpoint on this is basically taken from how in physics everything can be reduced to objects and interactions. This duality makes the real world very easy to understand. I've simply applied the same pattern to math, basically reducing all of math to mathematical objects and their interactions. I find this viewpoint, again, like with physics, very helpful. The fact that this viewpoint aligns somewhat with mathematical platonism is kind of coincidental. Although, perhaps there's a reason behind it too.
Stephen Tashi said:
In more advanced mathematics, such "laws" are presented either as theorems or assumptions.
That's kind of my point. We get that from real life too. Assumptions are the same as properties. Why is the electron negatively charged? There's no answer for that. It just is. Of course, in this case "negative" is just an arbitrary label we put on the electron's charge. We could've just as easily called it positive. In fact, didn't we do that for a large period of history? Anyway, the deeper truth here is that the electron is different from say, the proton. Why? Again, no answer. That's just how it is. Similarly, we can apply the same thinking to mathematical "objects" and them having certain properties. Numbers are a kind of mathematical object that have certain properties. Why do they have these properties? There's no answer to that. Vectors are a different mathematical object with different properties than numbers. Whether or not they "exist", is once again, like with our unicorn example, irrelevant. What's important is that if you follow the axioms/properties, you will always get the same results. Which, in a sense, kind of feels like objective reality since reproducibility is how we define science. Hmm I feel like this is more the realm of formal logic than math per se.

TeethWhitener said:
Gödel proved that there are logical systems—including standard arithmetic—which contain sentences that are “true but not provable.”
I've been meaning to ask about that. It seems to me that what Gödel actually discovered is that self-reference is different kind of thing that isn't compatible with normal everyday logic i.e. the pattern of true/false doesn't necessarily apply to self-referential statements. I have yet to see an example of it in a non-self-referential context i.e. a statement that isn't self-referential and not true/false. It seems that as long as you stay away from self-reference, you should be fine.

jbriggs444 and PeroK
Feynstein100 said:
I've been meaning to ask about that. It seems to me that what Gödel actually discovered is that self-reference is different kind of thing that isn't compatible with normal everyday logic i.e. the pattern of true/false doesn't necessarily apply to self-referential statements. I have yet to see an example of it in a non-self-referential context i.e. a statement that isn't self-referential and not true/false. It seems that as long as you stay away from self-reference, you should be fine.
The so-called axiom of choice cannot be proved to be true or false, but is generally adopted as an additional axiom.

PeroK said:
The so-called axiom of choice cannot be proved to be true or false, but is generally adopted as an additional axiom.
I don't get it. Would you mind elaborating?

The other very important aspect of mathematics is the objects we choose to study. From the natural numbers to groups to differential equations to topology etc.

There's also the way we think about mathematics. Which is outside the formal system of axioms and theorems. Question's like how do I go about proving this? Is it true? Can I find a counterexample? Etc.

Feynstein100 said:
I don't get it. Would you mind elaborating?
Try an Internet search.

The premise of your posts seems to be that we should forget Goedel and pretend Incompleteness isn't a thing.

Feynstein100 said:
I've been meaning to ask about that. It seems to me that what Gödel actually discovered is that self-reference is different kind of thing that isn't compatible with normal everyday logic i.e. the pattern of true/false doesn't necessarily apply to self-referential statements. I have yet to see an example of it in a non-self-referential context i.e. a statement that isn't self-referential and not true/false. It seems that as long as you stay away from self-reference, you should be fine.
There are a few different things going on here. First is that Gödel doesn’t specifically say that the Gödel sentence (the technical version of “this is not provable”) is unconditionally true. He proved that if it is false, then the logical system is inconsistent (NB—I’m skipping over a lot of model theory technicalities here). Or, from the contrapositive, if the system is consistent, then the Gödel sentence is true.

Second is the necessity of self-reference for unprovability. Note that Gödel uses self-reference to prove his incompleteness theorems, but unprovable statements don’t require self-reference. Here’s a list of unprovable statements (called “independent”) on Wikipedia:
https://en.m.wikipedia.org/wiki/List_of_statements_independent_of_ZFC
There are subtleties here that I’m eliding because they go well beyond a B level thread.

Third is the relationship between self-reference and the bivalence (two-valuedness) of standard logic. This is actually pretty well-trodden: it can be shown, for instance, that for certain semantic instantiations of the liar paradox (i.e., “this sentence is false”), if you model the sentence with a logic that has more truth values than just “true” and “false,” the liar sentence will take one of those non-Boolean values.

But these are three separate ideas that are all kind of scrambled together in your post, when really they should be treated separately.

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