Date: Aug 21, 2013 12:01 AM
Author: William Elliot
Subject: Re: Foundations of mathematics... the order of bootstrapping the<br> foundations
On Tue, 20 Aug 2013, Lax Clarke wrote:

> > > Please correct me if I'm wrong please:

> > > This is the order of bootstrapping the foundations of mathematics:

> > Boot strapping has nothing to do with mathematics.

> > It's part of computer science.

> Ok. I just want to know the order of learning I guess. Or say a really smart and "rigor-loving" alien landed in our backyard, in what order would we explain thing to him in (assuming the alien learns English first).

> > > 1) Naive logic (like the ones the Greeks played with).

> > > 2) Use 1) to talk about Naive Set theory (like Halmos' book).

> > > 3) Use 2) above to define Mathematical Logic / First-Order Logic

> > > 4) Use 3) above to define axiomatic set theory.

> > A metalanguage is used to describe a formal language.

> 1, 2 would be metalanguages? These would be used to build up formal

> languages 3,4?

The metalanguage is usually a simple native language with

simple logic which usually includes simple induction.

The metalanguage is used to describe a formal language and it's logic,

such as rules of inference.

It could be used to describe a FOL.

If axioms are added then it would be describing a theory.

The theory could be one of the set theories ML, ZF, ZFC or NBG.

Abbreviations are part of the metalanguage and within a

simple book, say ML, abbreviations can quickly represent hundreds

or thousands, perhaps millions, of primitive symbols.