The reason I take second-order logic seriously is that it lets me pin down a single mathematical referent that I’m comparing to the realities of space and time.
I have my problems with the other two, but this is the only one I don’t understand. What do you mean?
it feels like talking about the collection of all collections—the supremum of an indefinitely extensible quality that shouldn’t have a supremum any more than I could talk about a mathematical object that is the supremum of all the models a first-order set theory can have
You seem to accept the notion that all finite numbers have a supremum. Why not just iterate whatever process accounts for that?
You seem to accept the notion that all finite numbers have a supremum. Why not just iterate whatever process accounts for that?
First of all, I’ve never seen an aleph-null, just one, two, three, etc. Accepting that the integers have a supremum is a whole different kettle of fish from accepting that the collection of finite integers seems to go on without bound. Second, taking a supremum once, using a clearly defined computable notation and a halting machine that can compare any two representations, is a whole different kettle of fish than talking about the supremum of all possible ways to define countable well-orderings to and beyond computable recursion.
you can’t talk about the integers or the reals in first-order logic.
It’s more accurate to say that you can’t talk about arbitrary subsets of the integers or the reals in first-order logic.
Accepting that the integers have a supremum is a whole different kettle of fish from accepting that the collection of finite integers seems to go on without bound.
I agree. This is the difference between completed and potential infinity. Nelson.
Second, taking a supremum once, using a clearly defined computable notation and a halting machine that can compare any two representations, is a whole different kettle of fish than talking about the supremum of all possible ways to define countable well-orderings to and beyond computable recursion.
I’m not so sure. Everything you’ve ever talked about, uncountable ordinals and all, you’ve talked about using computable notation. Computable, period is a whole different kettle of fish.
I have my problems with the other two, but this is the only one I don’t understand. What do you mean?
You seem to accept the notion that all finite numbers have a supremum. Why not just iterate whatever process accounts for that?
http://en.wikipedia.org/wiki/Second-order_logic#Expressive_power—you can’t talk about the integers or the reals in first-order logic. You can have first-order theories with the integers as a model, but they’ll have models of all other cardinalities too. http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem
First of all, I’ve never seen an aleph-null, just one, two, three, etc. Accepting that the integers have a supremum is a whole different kettle of fish from accepting that the collection of finite integers seems to go on without bound. Second, taking a supremum once, using a clearly defined computable notation and a halting machine that can compare any two representations, is a whole different kettle of fish than talking about the supremum of all possible ways to define countable well-orderings to and beyond computable recursion.
It’s more accurate to say that you can’t talk about arbitrary subsets of the integers or the reals in first-order logic.
I agree. This is the difference between completed and potential infinity. Nelson.
I’m not so sure. Everything you’ve ever talked about, uncountable ordinals and all, you’ve talked about using computable notation. Computable, period is a whole different kettle of fish.