Proposed new syntax

[Marko Rauhamaa]

Gregory Ewing <greg.ewing at 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