Is there a rule system on which all rule systems can be underpinned?

Question

The question came to me when wondering how to begin defining addition.

On that question I came across this:

I noticed they reasonably begin by defining the natural numbers.

However I also noticed they resort to other already established systems from where to draw those definitions, be it set theory or the lambda calculus.

So is there a more foundational theory, within which all such systems, eg. addition, the natural numbers, the lambda calculus and set-theory, can be defined?

You can see this post with many useful links to other posts. — Mauro ALLEGRANZA, Jul 03 '20 at 13:03
@Mauro ALLEGRANZA: I probably should have worded that better. In fact even as I posted it I though the question is clearer without that final sentence. The real question is: Are there a more foundational theories, within which all such systems, eg. addition, the natural numbers, the lambda calculus and set-theory, can be defined? — Georgina Davenport, Jul 03 '20 at 20:43

score 2 · Answer 1 · answered Jul 03 '20 at 12:43

Unique book, which I know starting from "nothing" is Nicolas Bourbaki "Theory of sets", English translation 1968. May be it is little old and not everybody loves it, but, correct me somebody, please, with constructive examples, it introduced formal meta-language for creation mathematics.

Is there a rule system on which all rule systems can be underpinned?

1 Answers1