If I can bother your mathematical logician for just a moment...
Hey, are you conscious in the sense of being aware of your own awareness?
Also, now that Eliezer can’t ethically deinstantiate you, I’ve got a few more questions =)
You’ve given a not-isomorphic-to-numbers model for all the prefixes of the axioms. That said, I’m still not clear on why we need the second-to-last axiom (“Zero is the only number which is not the successor of any number.”) -- once you’ve got the final axiom (recursion), I can’t seem to visualize any not-isomorphic-to-numbers models.
Also, how does one go about proving that a particular set of axioms has all its models isomorphic? The fact that I can’t think of any alternatives is (obviously, given the above) not quite sufficient.
Oh, and I remember this story somebody on LW told, there were these numbers people talked about called...um, I’m just gonna call them mimsy numbers, and one day this mathematician comes to a seminar on mimsy numbers and presents a proof that all mimsy numbers have the Jaberwock property, and all the mathematicians nod and declare it a very fine finding, and then the next week, he comes back, and presents a proof that no mimsy numbers have the Jaberwock property, and then everyone suddenly loses interest in mimsy numbers...
Point being, nothing here definitely justifies thinking that there are numbers, because someone could come along tomorrow and prove ~(2+2=4) and we’d be done talking about “numbers”. But I feel really really confident that that won’t ever happen and I’m not quite sure how to say whence this confidence. I think this might be similar to your last question, but it seems to dodge RichardKennaway’s objection.
This is a really good post.
If I can bother your mathematical logician for just a moment...
Hey, are you conscious in the sense of being aware of your own awareness?
Also, now that Eliezer can’t ethically deinstantiate you, I’ve got a few more questions =)
You’ve given a not-isomorphic-to-numbers model for all the prefixes of the axioms. That said, I’m still not clear on why we need the second-to-last axiom (“Zero is the only number which is not the successor of any number.”) -- once you’ve got the final axiom (recursion), I can’t seem to visualize any not-isomorphic-to-numbers models.
Also, how does one go about proving that a particular set of axioms has all its models isomorphic? The fact that I can’t think of any alternatives is (obviously, given the above) not quite sufficient.
Oh, and I remember this story somebody on LW told, there were these numbers people talked about called...um, I’m just gonna call them mimsy numbers, and one day this mathematician comes to a seminar on mimsy numbers and presents a proof that all mimsy numbers have the Jaberwock property, and all the mathematicians nod and declare it a very fine finding, and then the next week, he comes back, and presents a proof that no mimsy numbers have the Jaberwock property, and then everyone suddenly loses interest in mimsy numbers...
Point being, nothing here definitely justifies thinking that there are numbers, because someone could come along tomorrow and prove ~(2+2=4) and we’d be done talking about “numbers”. But I feel really really confident that that won’t ever happen and I’m not quite sure how to say whence this confidence. I think this might be similar to your last question, but it seems to dodge RichardKennaway’s objection.
I guess it is not necessary. It was just an illustration of a “quick fix”, which was later shown to be insufficient.