This is where things get fuzzier. I think that by routing through logic and observing the empirical consequences of ordinal consistency axioms
I’m not sure I understand what you mean here.
Assume we have a formal system and we are using it to predict observations. In classical logic, if the formal system is inconsistent, then ex falso quodlibet, therefore all possible observations will be allowed by it.
I think the grandparent silently assumes we’ve solved logical uncertainty, and can use this to decide an axiom does not produce a contradiction by observing the lack of explosion. Though “routing through logic” seems like an incredibly bad way to say that.
I think the grandparent silently assumes we’ve solved logical uncertainty, and can use this to decide an axiom does not produce a contradiction by observing the lack of explosion.
I spoke imprecisely, or at least the OP argues that I did. It assumes we have a progressive solution to undecidability/ logical uncertainty that can take us pretty far, eventually, but does not allow self-knowledge at any time.
I think the grandparent silently assumes we’ve solved logical uncertainty, and can use this to decide an axiom does not produce a contradiction by observing the lack of explosion.
No. We don’t decide an axiom does not produce a contradiction by observing the lack of explosion. We just raise the probability of noncontradiction by observing the lack of explosion.
EDIT: Axiom, not acronym. Holy God I need more sleep.
Though “routing through logic” seems like an incredibly bad way to say that.
-_-. The post itself was a sketch, not a finished, formalized publication. Please refrain from being anal about phrasings or imagining that I’ve done far more math than I really have. If I want to be precise, I’ll just use symbols—that’s what they’re for.
I’m not sure I understand what you mean here.
Assume we have a formal system and we are using it to predict observations. In classical logic, if the formal system is inconsistent, then ex falso quodlibet, therefore all possible observations will be allowed by it.
I think the grandparent silently assumes we’ve solved logical uncertainty, and can use this to decide an axiom does not produce a contradiction by observing the lack of explosion. Though “routing through logic” seems like an incredibly bad way to say that.
Wouldn’t that imply having solved undecidability?
I spoke imprecisely, or at least the OP argues that I did. It assumes we have a progressive solution to undecidability/ logical uncertainty that can take us pretty far, eventually, but does not allow self-knowledge at any time.
No. We don’t decide an axiom does not produce a contradiction by observing the lack of explosion. We just raise the probability of noncontradiction by observing the lack of explosion.
EDIT: Axiom, not acronym. Holy God I need more sleep.
-_-. The post itself was a sketch, not a finished, formalized publication. Please refrain from being anal about phrasings or imagining that I’ve done far more math than I really have. If I want to be precise, I’ll just use symbols—that’s what they’re for.