How many axioms do you need?

[Xeranox]

In mathematics you can derive almost everything provided that you know some fundamental laws or axioms. I was wondering what the least number of such axioms you need to be able to derive the 'entire mathematics', and if it's even possible to do. I would also like to know if there is somewhere I can find the 'entire mathematics' derives in this way from the very beginning.

[ugly_bob]

er..
Mathematics is not complete yet.
And a lot of it is based on conjecture, rather than axioms

[Xeranox]

That's why I said 'entire mathematics'. I actually found something called ZFC Set Theory, from which you can prove pretty much everything. Some pages say 7, some say 9 and some say that you need more axioms. And apparently there are some flaws, but nevertheless. Check it out: http://us.metamath.org/index.html

[jake-the-peg]

It sounds as if you're looking for principia mathematica by Russell and Whitehead

http://plato.stanford.edu/entries/principia-ma thematica/

It's famous for two things, firstly for taking a thousand pages before arriving at the conclusion that 1+1=2

And secondly for being totally undermined by Godel's Incompleteness theorum.

This also rather contradicts your first sentence
see here:
http://www.ncsu.edu/felder-public/kenny/papers /godel.html

[fo3nix]

There's an excellent book called "Godel's Proof", which you may want to read (referencing jake's answer).

[Peter Pedant]

erm sounds as tho you may be one of the people who know the answer before they post it.

No answers on the back of a post-card,
BUT Boolos and Jeffrey 3 rd edn - Computablity and Logic,
and or (eek!) Cutland Computablity
are helpful in this
and I think the OU still run Computability and Logic - M483 ?

The only problem for you is that you have to wade through an awful lot of other stuff (Turing, Church, Oh God and all sort of bo==lox) before you get to the good bit which in your case is towards the end of the books and is
der daaah axiomatisation of arithmetic

I think the two books I have mentioned go after peano arithmetic (and axiomatise it)
and the problem with that is that the principle of induction is difficult to shoe-horn into the scheme of things, and turns out to be rather weak.

and having done that, all the books go onto the Biggie in this area, and that is Godel's first and second theorems whic is basically - there are some arithmetic truths which cant be proven even though they are true

OK and so make a better go at axiomatising the maths to take in the defect, and you find there are still some truths which can't be proven......

Godel Escher and Bach go into this in some detail -

whcih leads to the Real Biggie - "there is no axiomatisable extension of arithemtic wh is complete."

and so dear reader, it now doestn matter if there are five seven or eleven axioms, there will still be bits left out.

Oh and somewhere else in the book you will find a statement about Zermelo Franklin set theory whcih says that group theory is not axiomatisable competely either.


This completely (hahahhaa) answers your question, but most Abers will read a bit of this and pass on......

[Xeranox]

Very interesting. Thanks guys!

If I understood this correctly:

A complete, consistent system describing arithmetics can not prove it's own consistency; that is, there might be true statements in the system that cannot be proven.

However, this does not mean that all the mathematics WE KNOW cannot be proven. All known mathematical theorems can be proven from a set of axioms (such as the ZFC set theory), and they are indeed true! But there might be some theorems we cannot prove to be true even if they are.


The last point is what is interesting to me. If all mathematics we know (or even all mathematics I know) is provable from a set of axioms based on first-order logic, then it's good enough for me. Even though it's not enough to prove the consistency of the theory itself.

[mibn2cweus]

Certain fundamental philosophies (Objectivism for example) are also built on axioms; concepts which can not be proven but upon which all proof must rest.
Among these is the concept of existence. Existence is self-evident and must be accepted, even in any attempt to disprove it. Identity and consciousness are two axioms as well that must be accepted and understood before any further examination of reality can lead to understanding. The very act of claiming that any of these do not exists provides proof that they do and is a contradiction and therefore proof of an error in logic, (the art of non-contradictory identification).

[Peter Pedant]

X
I think your third paragraph is Tarnow's theorem

but you are still scratchhing at Godel's theorem which is that there is no complete axiomaatisable extension of Q

and theis means that there ARE mathematical truths which cannot be proven

I dunno why you are asking us: the only people I found who could follow computability were the others on the course.