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. 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.