first-order logic is complete, so giving a unique definition is sufficient to tell you the relevant properties
That inference seems questionable, though I’m not sure what you mean with “irrelevant properties”. (Actually, in first-order logic many concepts can’t be defined, e.g. “natural number”, because we can’t express ”… and nothing else is a natural number.” Another example is “power set”. Yudkowsky has written about this.)
Forgot to say, for first-order logic it doesn’t matter what properties are considered relevant because Gödel’s completeness theorem tells you that it allows you to infer all the true properties.
The properties that hold in all models of the theory.
That is, in logic, propositions are usually interpreted to be about some object, called the model. To pin down a model, you take some known facts about that model as axioms.
Logic then allows you to derive additional propositions which are true of all the objects satisfying the initial axioms, and first-order logic is complete in the sense that if some proposition is true for all models of the axioms then it is provable in the logic.
I’m still not sure what you want to say. It’s a necessary property of natural numbers that they can be reached from iterating the successor function. That condition can’t be expressed in first-order logic, so it can’t be proved and it holds in some models and in others it doesn’t. It’s like trying to define “cat” by stating that it’s an animal. This is not a sufficient definition.
You’re the one who brought up the natural numbers, I’m just saying they’re not relevant to the discussion because they don’t satisfy the uniqueness thing that OP was talking about.
In these examples, the issue is that you can’t get a computable set of axioms which uniquely pin down what you mean by natural numbers/power set, rather than permitting multiple inequivalent objects.
That inference seems questionable, though I’m not sure what you mean with “irrelevant properties”. (Actually, in first-order logic many concepts can’t be defined, e.g. “natural number”, because we can’t express ”… and nothing else is a natural number.” Another example is “power set”. Yudkowsky has written about this.)
Forgot to say, for first-order logic it doesn’t matter what properties are considered relevant because Gödel’s completeness theorem tells you that it allows you to infer all the true properties.
What do you mean with “all the true properties”?
The properties that hold in all models of the theory.
That is, in logic, propositions are usually interpreted to be about some object, called the model. To pin down a model, you take some known facts about that model as axioms.
Logic then allows you to derive additional propositions which are true of all the objects satisfying the initial axioms, and first-order logic is complete in the sense that if some proposition is true for all models of the axioms then it is provable in the logic.
I’m still not sure what you want to say. It’s a necessary property of natural numbers that they can be reached from iterating the successor function. That condition can’t be expressed in first-order logic, so it can’t be proved and it holds in some models and in others it doesn’t. It’s like trying to define “cat” by stating that it’s an animal. This is not a sufficient definition.
You’re the one who brought up the natural numbers, I’m just saying they’re not relevant to the discussion because they don’t satisfy the uniqueness thing that OP was talking about.
In these examples, the issue is that you can’t get a computable set of axioms which uniquely pin down what you mean by natural numbers/power set, rather than permitting multiple inequivalent objects.