the set of all formal systems is an ontology: you can have formal systems that talk about formal systems, and you run into the same problems you embarked to avoid
you do not run into completeness issues, but if you want your system to have any utility you have to assert the consistency of the systems you’re using
the set of all formal system? How do you define that, or check it’s consistent? Without resorting to higher order theory?
Not quite: what I have here is game formalism with the additional claim that the games are ‘real’ in a Platonic sense (and nothing else is).
you can have formal systems that talk about formal systems
But you cannot prove, formally, that formal system A talks about formal system B, without appealing to another formal system C (and then how do you know that C talks about A?). I already made this point in the OP.
if you want your system to have any utility you have to assert the consistency of the systems you’re using
Yes, but you only have to make that assertion as a condition of the application; in order to claim that two pebbles plus two pebbles makes four pebbles I have to assert that PA applies to pebbles, which assertion involves asserting that PA is consistent—but if it turns out that PA is inconsistent, that only refutes the assertion that PA applies to pebbles; it does not ‘refute’ PA, since PA makes no claims of consistency (indeed, PA does not have a notion of “truth”, so even Q=PA+”Q is consistent” makes no claims of consistency). Thus, while paradoxes may conceivably destroy the utility of mathematics, they still could not destroy mathematics itself.
But you cannot prove, formally, that formal system A talks about formal system B, without appealing to another formal system C (and then how do you know that C talks about A?). I already made this point in the OP.
If in formal system C you can proove that formal system A prooves statements about formal system B, and since
Not quite: what I have here is game formalism with the additional claim that the games are ‘real’ in a Platonic sense (and nothing else is).
then it means that system C is a real meta-theory for system A and B, and since B is real, then A is making ‘true’ statements.
Thus, while paradoxes may conceivably destroy the utility of mathematics, they still could not destroy mathematics itself.
It’s the same thing. What distinguishes mathematics from fantasy narrative is the strict adherence to a set of rules: if some system is inconsistent, then it’s equivalent to have no rules (that is, all inconsistent systems have the same proving power).
Good point, that should have been ‘class’. Fixed.
The distinction between set and class is only meaningful in a formal system… and I don’t think you want a theory able to talk about the quantity of the totality of the real formal systems...
Four observations:
you’re basically restating formalism
the set of all formal systems is an ontology: you can have formal systems that talk about formal systems, and you run into the same problems you embarked to avoid
you do not run into completeness issues, but if you want your system to have any utility you have to assert the consistency of the systems you’re using
the set of all formal system? How do you define that, or check it’s consistent? Without resorting to higher order theory?
Not quite: what I have here is game formalism with the additional claim that the games are ‘real’ in a Platonic sense (and nothing else is).
But you cannot prove, formally, that formal system A talks about formal system B, without appealing to another formal system C (and then how do you know that C talks about A?). I already made this point in the OP.
Yes, but you only have to make that assertion as a condition of the application; in order to claim that two pebbles plus two pebbles makes four pebbles I have to assert that PA applies to pebbles, which assertion involves asserting that PA is consistent—but if it turns out that PA is inconsistent, that only refutes the assertion that PA applies to pebbles; it does not ‘refute’ PA, since PA makes no claims of consistency (indeed, PA does not have a notion of “truth”, so even Q=PA+”Q is consistent” makes no claims of consistency). Thus, while paradoxes may conceivably destroy the utility of mathematics, they still could not destroy mathematics itself.
Good point, that should have been ‘class’. Fixed.
If in formal system C you can proove that formal system A prooves statements about formal system B, and since
then it means that system C is a real meta-theory for system A and B, and since B is real, then A is making ‘true’ statements.
It’s the same thing. What distinguishes mathematics from fantasy narrative is the strict adherence to a set of rules: if some system is inconsistent, then it’s equivalent to have no rules (that is, all inconsistent systems have the same proving power).
The distinction between set and class is only meaningful in a formal system… and I don’t think you want a theory able to talk about the quantity of the totality of the real formal systems...