I’m not sure if this is what you’re looking for, but Hofstadter gives a great analogy using record players which I find useful in terms of thinking about how changing the situation changes our results (which is paraphrased here).

A (hi-fi) record player that tries to playing every possible sound can’t actually play its own self-breaking sound, so it is incomplete by virtue of its strength.

A (low-fi) record player that refuses to play all sounds (in order to avoid destruction from its self-breaking sound) is incomplete by virtue of its weakness.

We may think of the hi-fi record player as a formal system like Peano Arithmetic: the incompleteness arises precisely because it is strong enough to be able to capture number theory. This is what allows us to use Gödel Numbering, which then allows PA to do meta-reasoning about itself.

The only way to fix it is to make a system that is weaker than PA, so that we cannot do Gödel Numbering. But then we have a system that isn’t even trying to express what we mean by number theory. This is the low-fi record player: as soon as we fix the one issue of self-reference, we fail to capture the thing we care about (number theory).

I think an example of a weaker formal system is Propositional Calculus. Here we do actually have completeness, but that is only because Propositional Calculus is too weak to be able to capture number theory.

What exactly is the aspect of natural numbers that makes them break math, as opposed to other types of values? Intuitively, it seems to be the fact that they can be arbitrarily large but not infinite.

Like, if you invent another data type that only has a finite number of values, it would not allow you to construct something equivalent to Gödel numbering. But if it allows infinite number of (finite) values, it would. (Not sure about an infinite number of/including infinite values, probably also would break math.)

It seems like you cannot precisely define natural numbers using first-order logic. Is that the reason of this all? Or is it a red herring? Would situation be somehow better with second-order logic?

(These are the kinds of questions that I assume would be obvious to me, if I grokked the situation. So the fact that they are not obvious, suggests that I do not see the larger picture.)

I’m not sure if this is what you’re looking for, but Hofstadter gives a great analogy using record players which I find useful in terms of thinking about how changing the situation changes our results (which is paraphrased here).

A (hi-fi) record player that tries to playing

every possible soundcan’t actually play its own self-breaking sound, so it is incomplete by virtue of its strength.A (low-fi) record player that refuses to play all sounds (in order to avoid destruction from its self-breaking sound) is incomplete by virtue of its weakness.

We may think of the hi-fi record player as a formal system like Peano Arithmetic: the incompleteness arises precisely because it is strong enough to be able to capture number theory. This is what allows us to use Gödel Numbering, which then allows PA to do meta-reasoning about itself.

The only way to fix it is to make a system that is weaker than PA, so that we cannot do Gödel Numbering. But then we have a system that isn’t even trying to express what we mean by number theory. This is the low-fi record player: as soon as we fix the one issue of self-reference, we fail to capture the thing we care about (number theory).

I think an example of a weaker formal system is Propositional Calculus. Here we do actually have completeness, but that is only because Propositional Calculus is too weak to be able to capture number theory.

What exactly is the aspect of natural numbers that makes them break math, as opposed to other types of values? Intuitively, it seems to be the fact that they can be

arbitrarily largebutnot infinite.Like, if you invent another data type that only has a finite number of values, it would not allow you to construct something equivalent to Gödel numbering. But if it allows infinite number of (finite) values, it would. (Not sure about an infinite number of/including infinite values, probably also would break math.)

It seems like you cannot precisely define natural numbers using first-order logic. Is that the reason of this all? Or is it a red herring? Would situation be somehow better with second-order logic?

(These are the kinds of questions that I assume would be

obviousto me, if I grokked the situation. So the fact that they are not obvious, suggests that I do not see the larger picture.)