Arguably, “basic logical principles” are those that are true in natural language. Otherwise nothing stops us from considering absurd logical systems where “true and true” is false, or the like. Likewise, “one plus one is two” seems to be a “basic mathematical principle” in natural language. Any axiomatization which produces “one plus one is three” can be dismissed on grounds of contradicting the meanings of terms like “one” or “plus” in natural language.
The trouble with set theory is that, unlike logic or arithmetic, it often doesn’t involve strong intuitions from natural language. Sets are a fairly artificial concept compared to natural language collections (empty sets, for example, can produce arbitrary nestings), especially when it comes to infinite sets.
Arguably, “basic logical principles” are those that are true in natural language.
That’s where the problem starts, not where it stops. Natural language supports a bunch of assumptions that are hard to formally reconcile: if you want your strict PNC, you have to give up on something else. The whole 2500 yeah history of logic has been a history of trying to come up with formal systems that fulfil various desiderata. It is now formally proven that you can’t have all of them at once, and it’s not obvious what to keep and what to ditch. (Godelian problems can be avoided with lower power systems, but that’s another tradeoff, since high power is desirable).
Formalists are happy to pick a system that’s appropriate for a practical domain, and to explore the theoretical properties of different systems in parallel.
Platonists believe that only one axiom system has truth in addition to usefulness, but can’t agree which one it is, so it makes no difference in practice
I’m not seeing a specific problem with sets—you can avoid some of the problems of naive self theory by adding limitations, but that’s tradeoffs again.
Otherwise nothing stops us from considering absurd logical systems where “true and true” is false, or the like.
“You can’t have all the intuitive principles in full strength in one system”
doesn’t imply
“adopt unintuitive axioms”.
Even formalists don’t believe all axiomisations are equally useful.
Likewise, “one plus one is two” seems to be a “basic mathematical principle” in natural language.
What’s 12+1?
Any axiomatization which produces “one plus one is three” can be dismissed on grounds of contradicting the meanings of terms like “one” or “plus” in natural language.
They’re ambiguous in natural language, hence the need for formalisation.
The trouble with set theory is that, unlike logic or arithmetic, it often doesn’t involve strong intuitions from natural language.
It involves some intuitions . It works like clubs. Being a senator is being a member of a set, not exemplifying a universal.
Sets are a fairly artificial concept compared to natural language collections (empty sets, for example, can produce arbitrary nestings), especially when it comes to infinite sets.
If you want finitism, you need a principled way to select a largest finite number.
Arguably, “basic logical principles” are those that are true in natural language. Otherwise nothing stops us from considering absurd logical systems where “true and true” is false, or the like. Likewise, “one plus one is two” seems to be a “basic mathematical principle” in natural language. Any axiomatization which produces “one plus one is three” can be dismissed on grounds of contradicting the meanings of terms like “one” or “plus” in natural language.
The trouble with set theory is that, unlike logic or arithmetic, it often doesn’t involve strong intuitions from natural language. Sets are a fairly artificial concept compared to natural language collections (empty sets, for example, can produce arbitrary nestings), especially when it comes to infinite sets.
That’s where the problem starts, not where it stops. Natural language supports a bunch of assumptions that are hard to formally reconcile: if you want your strict PNC, you have to give up on something else. The whole 2500 yeah history of logic has been a history of trying to come up with formal systems that fulfil various desiderata. It is now formally proven that you can’t have all of them at once, and it’s not obvious what to keep and what to ditch. (Godelian problems can be avoided with lower power systems, but that’s another tradeoff, since high power is desirable).
Formalists are happy to pick a system that’s appropriate for a practical domain, and to explore the theoretical properties of different systems in parallel.
Platonists believe that only one axiom system has truth in addition to usefulness, but can’t agree which one it is, so it makes no difference in practice
I’m not seeing a specific problem with sets—you can avoid some of the problems of naive self theory by adding limitations, but that’s tradeoffs again.
“You can’t have all the intuitive principles in full strength in one system”
doesn’t imply
“adopt unintuitive axioms”.
Even formalists don’t believe all axiomisations are equally useful.
What’s 12+1?
They’re ambiguous in natural language, hence the need for formalisation.
It involves some intuitions . It works like clubs. Being a senator is being a member of a set, not exemplifying a universal.
If you want finitism, you need a principled way to select a largest finite number.