Do science and religion have rights to some "explanatory space"?

I think that your terminology is a bit unclear.It is probably the case that most scientific theories are false in the sense that they are only very close to the truth. But we can't even know that for sure.But what is really confusing is your "unproven true" category.It appears that you cannot mean that the statements in this category actually are true because you also include their negations in the same category. And as a consequence of that it follows that no statement in this category can be proven true or false. Perhaps it would be better to label the category "unknowable" ?

I think that you are misunderstanding your philosophy of science. We do not accept a statement as true simply because it has not been proven false - but because it could be proven false and has not been. I do not know of many people who label an unfalsifiable statement as true simply because it is unfalsifiable. I certainly would not.I'll also ass that your understanding of Gdel's theorem is a bit dodgy, too. For instance it only applies to systems capable of handling arithmetic. I very much doubt that Special Relativity, for instance is useful for doing arithmetic.

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Only theorems that cannot be contradicted by (future) observations can be true.

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This is a necessary condition for truth, but not - as is quite clear from your posts - a sufficient one.

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All other theorems are necessarily false, because every falsifiable statement will eventually be falsified.

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That is simply not true. There certainly can be falsifiable statements that will not be falsified. Indeed, if a statement and it's negation are both falsifiable it is necessarily the case that only one will be falsified.

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So, how do you label a statement that can impossibly be contradicted by reality?

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I would label it unfalsifiable. If it's negation was also unfalsifiable I would label it unknowable. I certainly wouldn't call it true because there would be no basis for doing so.

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.. but it will be. So, it is not true. It is false. We simply don't know how and why.

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But it might not be.

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That includes every system that has -- or could have -- a digital representation.

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No, it does not. Systems that lack the richness required to construct the representations needed for the proof are unaffected by the Theorem.

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Ok. We have an issue here:

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(1) Every statement that cannot be proven to be true, is false
(2) Every statement that cannot be proven to be false, is true

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What is the correct default?

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Both are wrong, so the conclusion is that we should not choose a default.

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The scientific method is necessarily (2).

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No. The scientific method says that unfalsifiable statements should be binned.

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In order to be useful and predict events in the future, the scientific method must accept these statements "to be true", even though they are correctly suspected to be false.

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No, only statements shown to be very close to the truth (i.e. we have grounds for thinking that they will be true in almost every case, even if there may be unforseen cases where they are not) are taken as true. And - to make useful predictions - a statement MUST be falsifiable. That is why unfalsifiable statements are binned.

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Consequently, we have to label statements that cannot be proven to be false: true.

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That doesn't follow. "Scientifically useless" would be a better label.As I said, the scientific ideal is falsifiable, but unfalsified - in the face of serious investigation.

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Such systems are too simple to be useful. Do you know of any scientific theorem which is so simple that it would be unaffected?

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I don't know of even one that WOULD definitely be affected. Can you give me an example and show how Gdel's proof applies ?

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It says that unfalsifiable statements are not part of science. And I agree with that.

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So we are agreed that science does NOT accept that every statement that cannot be proven false is true.

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These statements are not "very close to the truth". They will remain unproven and false, until they are finally proven false [Popper].

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A false statement may be "very close to the truth". Science aims for continuous improvement, getting closer and closer to the truth. That indeed is the only reason for replacing a theory - it is closer to the truth.

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Anyway, it is not because a statement is hard to disprove, that the statement is true. It is still false.

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You seem to be confusing our positions. You state that statements that cannot be falsified must be accepted as true. I don't demand that falsification should be difficult in principle - just the opposite. The difficulty in falsifying the statement should come from it's being true or close to the truth. I take a view close to Popper's - that we should accept theories that survive serious attempts at falsification. Odd that you seem to disagree, given that you invoke his name so often.

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For Gdel's proof to apply, it must be possible to represent the statements in the theory with natural numbers

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That is inaccurate. What is needed is a way for a statement in the system to represent itself, so that it is possible to create a version of the "liar paradox". Gdel used natural numbers because he was working with mathematics - that was the system he was interested in.In fact your criterion cannot possible be correct because the poorer the system is, the easier it is to represent in other forms !

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Now, instead of using the axiomatic reduction of the statement as proof, we define its collection of all possible observations as proof. Each potential observation can be represented as a number too. The Gdel number for the proof is then the number representing its collection of possible observations (Y) for statement (X).

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We must therefore show that for any such given statement X, the number Y (its proof) does not exist. That would prove that all such statements X are false.

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In other words you are vaguely presenting something that looks a bit like Gdel's proof - but leaves out the essential element, among other fatal flaws.

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This is what Popper says; and what Gdel implies, even though the latter uses axiomatic reduction as proof instead of collections of observations

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I very much doubt that Popper said any such thing.

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Not true. As long as you can represent anything as natural numbers, the theory applies. Gdel represents statements as natural numbers, because then he can apply number theory to theorems.

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Complete and utter rubbish.A trivial axiomatic system with a single axiom (1) and identity as the only allowable derivation rule is quite easily represented by natural numbers, (1 and 1 = 1). However, Gdel's theorem does NOT apply because the system is not sufficiently rich.

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What's more, this is an issue of definition. Assuming the following definitions of true and false:

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A theory is false, if at least one observation contradicts it.
A theory is true, if all possible observations concur with it.

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That definition of "true" is in fact the definition of "unfalsifiable". Since it would be incorrect to accept a theory as true because it is unfalsifiable the definition is obviously faulty.

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A fact by itself is also true, since the observation obviously cannot contradict itself.

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But since observing the contrary is also a "possible observation", it fails to fit your definition of "true".A definition of "true" which includes falsehoods while excluding true statements is no good at all.