Comment by burrows
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.
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’?
Arithmetic is complete in ZFC. Well, in each model of ZFC sits a model of arithmetic (first order Peano axioms) and that model is complete. The incompleteness theorem doesn’t apply in this case because the incompleteness theorem is a statement about the first order Peano axioms and not about the situation in which they are residing in a larger theory (which in our case is ZFC). The Peano axioms are not able to prove their own completeness but if they reside in a larger theory then that larger theory may be able to prove their completeness.
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