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In a profound sense, science is the application of formal systems to the natural world. Within those systems "truth" absolutely does intend to mean "truth". >

In a profound sense, science is the application of formal systems to the natural world. Within those systems "truth" absolutely does intend to mean "truth". > the theorem "the previous axioms are inconsistent with each other" is also a valid axiom to add and results in a system of axioms that is consistent overall! Ehh, that's just another paradox. I don't think those are allowed in an overall-consistent system.

dwohnitmok 6y ago
Nope they're perfectly allowed! ZFC + ~Con(ZFC) is a perfectly valid axiom system that is consistent if and only if ZFC is consistent. In fact as a result ZFC + ~Con(ZFC) cannot prove any theorem that would contradict ZFC. Indeed basically all of modern mathematics looks the same under ZFC + ~Con(ZFC) as it does under ZFC. But how can that be? What happened to the Principle of Explosion? How is this even coherent? The trick is that ~Con(ZFC) being true doesn't mean that... well it's true (in the usual way that we talk about it). This is why I find the discussion of truth in Godel's incompleteness theorems to be distractions without a backdrop of model theory. Here it is important to delineate between an "internal" and "external" perspective. For a formal system S, the statement Con(S) is at the end of the day an arithmetic statement postulating the existence of a natural number satisfying certain constraints (this is why Godel's incompleteness theorems require the ability to perform arithmetic). This is the internal perspective, asking about a natural number. Externally, we know of the correspondence between the natural numbers and proofs and sentences, but that is external to our theory. ZFC cannot use the statement ~Con(ZFC), which we externally know encodes a proof that 0 = 1, to prove internally that 0 in fact is equal to 1. Thus we avoid the Principle of Explosion within ZFC. Indeed in this case, it turns out that what ZFC + ~Con(ZFC) thinks is a natural number is not unique! Different universes of sets that agree on ZFC might have different notions of natural numbers. Indeed the element that ZFC + ~Con(ZFC) thinks is a natural number satisfying ~Con(ZFC) might not be something another version of ZFC thinks is a natural number. And so when externally we go to try to prove that ~Con(ZFC) shows that ZFC is indeed inconsistent our mapping breaks down because we disagree that the witness to ~Con(ZFC) is a natural number externally. This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background.
- guscost 6y ago
  Thanks, I think this makes sense, and it's quite a trip! But I'm a little foggy about any differences between "externally inconsistent", "indeed inconsistent", and "inconsistent overall". I'm also guessing at a few pieces like what you mean by "our theory" - I assume that is "our system of ZFC + ~Con(ZFC)". Does this corollary have a name? > This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background. I don't agree at all, and you have not addressed my other point. Formal systems which are used for science aren't designed to be paradox-systems, and truth in those systems is always intended to correspond with reality.

dwohnitmok 6y ago

Nope they're perfectly allowed! ZFC + ~Con(ZFC) is a perfectly valid axiom system that is consistent if and only if ZFC is consistent. In fact as a result ZFC + ~Con(ZFC) cannot prove any theorem that would contradict ZFC. Indeed basically all of modern mathematics looks the same under ZFC + ~Con(ZFC) as it does under ZFC. But how can that be? What happened to the Principle of Explosion? How is this even coherent? The trick is that ~Con(ZFC) being true doesn't mean that... well it's true (in the usual way that we talk about it). This is why I find the discussion of truth in Godel's incompleteness theorems to be distractions without a backdrop of model theory. Here it is important to delineate between an "internal" and "external" perspective. For a formal system S, the statement Con(S) is at the end of the day an arithmetic statement postulating the existence of a natural number satisfying certain constraints (this is why Godel's incompleteness theorems require the ability to perform arithmetic). This is the internal perspective, asking about a natural number. Externally, we know of the correspondence between the natural numbers and proofs and sentences, but that is external to our theory. ZFC cannot use the statement ~Con(ZFC), which we externally know encodes a proof that 0 = 1, to prove internally that 0 in fact is equal to 1. Thus we avoid the Principle of Explosion within ZFC. Indeed in this case, it turns out that what ZFC + ~Con(ZFC) thinks is a natural number is not unique! Different universes of sets that agree on ZFC might have different notions of natural numbers. Indeed the element that ZFC + ~Con(ZFC) thinks is a natural number satisfying ~Con(ZFC) might not be something another version of ZFC thinks is a natural number. And so when externally we go to try to prove that ~Con(ZFC) shows that ZFC is indeed inconsistent our mapping breaks down because we disagree that the witness to ~Con(ZFC) is a natural number externally. This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background.

guscost 6y ago
Thanks, I think this makes sense, and it's quite a trip! But I'm a little foggy about any differences between "externally inconsistent", "indeed inconsistent", and "inconsistent overall". I'm also guessing at a few pieces like what you mean by "our theory" - I assume that is "our system of ZFC + ~Con(ZFC)". Does this corollary have a name? > This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background. I don't agree at all, and you have not addressed my other point. Formal systems which are used for science aren't designed to be paradox-systems, and truth in those systems is always intended to correspond with reality.