User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-27513631-20180611232757


 * notes that you can work internal to a model and just call it the universe*


 * notes that in normal mathematics the axioms (for, say ZFC) are all implicit and that in our context we usually mean "additional axioms"*


 * notes that axioms are just a special case of rules (the ones with no requirements) so technically it's possible (easy) to do mathematics without axioms*


 * notes that a definition is valid if \(\exists !x \phi(x)\) is true, even if not provably so, if you also consider the case where it is false separately*


 * notes that for pretty much all high-end recursive and non-recursive googology to always define actual naturals, as opposed to model-relative ones, it's usual to restrict attention to models with the actual naturals*


 * recommends you explicitly state languages separate from their theories and don't restrict attention to set theories only when speaking in generalities*