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Topic: Foundations of mathematics... the order of bootstrapping the foundations
Replies: 13   Last Post: Aug 21, 2013 9:04 PM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Foundations of mathematics... the order of bootstrapping the

Posted: Aug 21, 2013 12:01 AM
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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.

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