cbostrum (02/07/83)
I wasnt ignoring the first order nature of LST so much as accepting them, as Quine would for example, of indicative of what we would really call a logic. There are a number of nasty things involved in going to second order. The worst is that we lose a handle on things since we dont have any form of completeness theorem for second and higher order unless we lobotomise the structures into what are called (i believe) general structures. (Where you get to pick the relation universe as well as the constant universes but as a result, you are left with just a sugared version of FOL). Another related item is the ontological commitment. Amazing. Properties free for the asking! How many thing do the numbers 2 and 3 have in common? An uncountable number!! This seems like so much mysticism. On the other hand, sticking to first order means that you cant even uniquely characterise common objects. Forget complete ordered fields, you cant even characterise the natural numbers! Oh gosh, what a dilemma! It seems we have to accept one of two thing: 1) there are some truths that we can fully understand yet will never ever (even in principle) be able to discover 2) the natural numbers are not really things that that are well defined at all. so neither are turing machines. so it doesnt make sense to say a TM halts or it doesnt. Either of these is a very upsetting thing to accept. I think Id have to go with 1, however. I guess this is devastating to logical positivists. What should we do???