Lowenheim-Skolem is going to give you trouble, unless “coherently-thinkable” is meant of as a subtantive restriction. You might be able to enumerate finitely-axiomatisable models, up to isomorphism, up to aleph-w, if you limit yourself to k-categorical theories, for k < aleph-w, though. Then you could use Will’s strategy and enumerate axioms.
Edit: I realised I’m being pointlessly obscure.
The Upwards Lowenheim-Skolem means that, for every set of axioms in your list, you’ll have multiple (non-isomorphic) models.
You might avoid this if “coherantly thinkable” was taken to mean “of small cardinality”.
If you didn’t enjoy this restriction, you could, for any given set of axioms, enumerate the k-categorical models of that set of axioms—or at least enumerate the models of whose cardinality can be expressed as 2^2^...2^w, for some finite number of 2′s. This is because k-categoriciticy means you’ll only have one model of each cardinality, up to isomorphism.
So then you just enumerate all the possible countable combinations of axioms, and you have an enumeration of all countably axiomatisable, k-categorical, models.
I don’t think it’s unfair to put some restrictions on the universes you want to describe. Sure, reality could be arbitrarily weird—but if the universe cannot even be approximated within a number of bits much larger than the number of neurons (or even atoms, quarks, whatever), “rationality” has lost anyway.
(The obvious counterexample is that previous generations would have considered different classes of universes unthinkable in this fashion.)
Sure, reality could be arbitrarily weird—but if the universe cannot even be approximated within a number of bits much larger than the number of neurons (or even atoms, quarks, whatever), “rationality” has lost anyway.
Why? If the universe has features that our current computers can’t approximate, maybe we could use those features to build better computers.
A start is to choose some language for writing down axiom lists for formal systems, and a measure on strings in that language.
Lowenheim-Skolem is going to give you trouble, unless “coherently-thinkable” is meant of as a subtantive restriction. You might be able to enumerate finitely-axiomatisable models, up to isomorphism, up to aleph-w, if you limit yourself to k-categorical theories, for k < aleph-w, though. Then you could use Will’s strategy and enumerate axioms.
Edit: I realised I’m being pointlessly obscure.
The Upwards Lowenheim-Skolem means that, for every set of axioms in your list, you’ll have multiple (non-isomorphic) models.
You might avoid this if “coherantly thinkable” was taken to mean “of small cardinality”.
If you didn’t enjoy this restriction, you could, for any given set of axioms, enumerate the k-categorical models of that set of axioms—or at least enumerate the models of whose cardinality can be expressed as 2^2^...2^w, for some finite number of 2′s. This is because k-categoriciticy means you’ll only have one model of each cardinality, up to isomorphism.
So then you just enumerate all the possible countable combinations of axioms, and you have an enumeration of all countably axiomatisable, k-categorical, models.
I don’t think it’s unfair to put some restrictions on the universes you want to describe. Sure, reality could be arbitrarily weird—but if the universe cannot even be approximated within a number of bits much larger than the number of neurons (or even atoms, quarks, whatever), “rationality” has lost anyway.
(The obvious counterexample is that previous generations would have considered different classes of universes unthinkable in this fashion.)
Why? If the universe has features that our current computers can’t approximate, maybe we could use those features to build better computers.