Gregory Ewing <greg.ew...@canterbury.ac.nz>: > Marko Rauhamaa wrote: >> You will always have to step outside your formal system and resort to >> hand-waving in a natural language. > > If the hand-waving is rigorous, this amounts to expanding your formal > system by adding new axioms and/or rules to it.
Ultimately, it's not rigorous. You can add rigor to any number of meta-levels, but on top of it all, you will need an informal "observer" level. It's turtles all the way down. For example, you can't have a set of all sets. The NBG set theory solves the problem by introducing the concept of classes. You can have a class of all sets. But you can't have a class of all classes. > If the hand-waving is not rigorous, then you haven't really proved > anything. The mathematicians have stopped caring. In fact, even if metamathematics were a closed, formal system, the best it could achieve would be circular reasoning. That would still be satisfactory and "convincing." However, no interesting system can prove its own consistency (but it *can* prove it can't prove its own consistency). Recommended reading: <URL: https://www.amazon.com/Unprovability-Consistency-Essay-Moda l-Logic/dp/0521092973> Marko -- https://mail.python.org/mailman/listinfo/python-list
