Comment by syki
4 years ago
How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent. The second order Peano axioms are categorical so I assume you mean only the first order theory.
At any rate, what does any of this have to do with information capacity in the universe? Is the information capacity of the universe related to the consistency/completeness of arithmetic?
> How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent.
Aren’t you just begging the question by assuming ZFC is consistent to demonstrate that arithmetic is consistent?
I don’t think so. The point is, saying “arithmetic is inconsistent” doesn’t mean anything without talking about where this theory resides. The larger point is that this has no relationship to whether or not the universe is infinite so it shouldn’t be talked about at all within that context.
Suppose for a moment that it makes sense to say arithmetic is part of the fabric of the universe (whatever that is supposed to mean). How would one know if arithmetic is consistent or complete within the context of being part of the universe? Suppose ZFC is part of the universe. Then arithmetic (the model of it as being part of the universe) is complete and consistent. Now what? I claim nothing of consequence follows from this in relation to whether or not space is continuous.
> How would one know if arithmetic is consistent or complete within the context of being part of the universe?
The same way I know that it’s incomplete when formulated in ZFC (assuming ZFC is consistent), from the first incompleteness theorem.
> Suppose ZFC is part of the universe. Then arithmetic (the model of it as being part of the universe) is complete and consistent.
Why does this conclusion follow?
Are you raising the question of whether mathematical propositions can be justified a posteriori?
Or are you arguing that the incompleteness theorems don’t necessarily apply to theories formulated ‘as part of the universe’?
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