Is there is minimum number of axioms on which one can build an axiomatic system, and is there a proof of it? Thanks for any answers.

Axioms are the buildingblocks of reasoning , you would at least need 1 assumption , otherwise you would have no material to construct anything.

Is one assumption enough?

How about the axiom: "I think therefor I exist"

I think all the other axioms and constructs can be derived from the assumption that you are a thinking being....

i suppose you can make an axiomatic system with any number of axioms you choose. so the minimum is probably 1. but you probably couldn t make a very interesting system that way.

as to proof? there are many axiomatic systems that are proven to be complete and consistent. in other words, it is proven that they have the correct number of axioms. for finite systems, this isn t hard to prove.

i think there are very few infinite systems for which this is also true. gödel proved that any system interesting enough to contain the natural numbers must be incomplete (in other words, there are true statements which do not follow from the axioms), and that it is impossible to prove whether the system is consistent (consistent means that the axioms do not lead to contradictions).

but to answer your question, i think that the minimum number of axioms in your system depends on the system you want to talk about. but for interesting systems, it is known that you would need effectively an infinite number of axioms to really axiomatize the system, and you can never be guaranteed that the system is consistent.

Thanks - I think I phrased my question badly.

What meant was slightly different. Perhaps I should have said what is the minimum number of terms and/or theorums required to constitute the foundation of a mathematical system? I was checking if I was right to think that it's three, as in 'I think - therefore - I am', although on a lower level this breaks down into many more than that.

I hope this doesn't seem daft. It seems like a curiously interesting question to me.

Originally posted by Canute
Thanks - I think I phrased my question badly.

What meant was slightly different. Perhaps I should have said what is the minimum number of terms and/or theorums required to constitute the foundation of a mathematical system? I was checking if I was right to think that it's three, as in 'I think - therefore - I am', although on a lower level this breaks down into many more than that.

I hope this doesn't seem daft. It seems like a curiously interesting question to me.

ok, i guess i m not really sure what you re asking. i still think the answer is going to be: there is no minimum.

Originally posted by lethe
ok, i guess i m not really sure what you re asking. i still think the answer is going to be: there is no minimum.
There can't be NO minimum, unless you're going to get very metaphysical.

To be honest I can't decide myself if it's an interesting question or a trivial one. It seems to me that you can't say anthing in any linguistic system without at least two subjects and an operator. (postulates if you like). That suggests that three may be some sort of minimum. But Euclid had five and couldn't seem to get away with less.

Feel free to leave this one. I begin to see it's a ambiguous question with a number of different answers.

Canute,

Perhaps I should have said what is the minimum number of terms and/or theorums required to constitute the foundation of a mathematical system?

For a mathematical system I would say one : 1+1=2

Originally posted by Canute
There can't be NO minimum, unless you're going to get very metaphysical.

you can make a logical system with as few axioms, and as few fundamental terms as you want.

whether you want to call a system with zero axioms a system at all, is a semantic question.

Originally posted by Prosoothus
Canute, For a mathematical system I would say one : 1+1=2
Yes that's what I meant. Me too. What I'm exploring is whether this is true for ALL systems. In other words must all non-trivial systems (mathematical, linguistic, metaphysical, explanatory etc.) start by making at least three assumptions? What do you think?

Is the whole of mathematics (the schoolbook kind) derivable from 1+1=2? It seems to be.

Originally posted by Canute
In other words must all non-trivial systems start by making at least three assumptions?

No.


Is the whole of mathematics (the schoolbook kind) derivable from 1+1=2?

no.

Originally posted by lethe
you can make a logical system with as few axioms, and as few fundamental terms as you want. Whether you want to call a system with zero axioms a system at all, is a semantic question.
That's what I meant by metaphysical. I suspect that any system with zero axioms does not exist, which seems more an ontological issue than a semantical one. Do you have an example of a system with just one term?

Originally posted by lethe
No.
Any idea why?
no. [/B]
Can you explain why not?

Originally posted by Canute
That's what I meant by metaphysical. I suspect that any system with zero axioms does not exist, which seems more an ontological issue than a semantical one. Do you have an example of a system with just one term?

this is not ontological. this is a purely semantic question. by that i mean it s just a question of what you want to call a system. it s like the question of whether P=>Q is true when P is false. it s a matter of convention, or language.

want an example?

how about this: let my fundamental term be a set. let my single axiom be this: there exists one set, the empty set.

ok? not a very interesting system, but it has fewer than 3 axioms.

that axiom, by the way, is one of the ZFC axioms that are used to axiomatize modern mathematics. there are 10 of them.

One mathematical assumption that led to a long list of axioms in trigonometry, is that the diameter of a circle is relative to its circumference. There is no exactly defined relationship of the diameter of a circle to its circumference, but we developed a number, pi, to define it nonetheless.

May not be a number for axioms, but it requires:
1 variable to count. 1 2 3 4 ...
2 variables to do arithmetic. A + B
3 variables to do algebra. C = A + B

Originally posted by lethe
how about this: let my fundamental term be a set. let my single axiom be this: there exists one set, the empty set.

ok? not a very interesting system, but it has fewer than 3 axioms.

that axiom, by the way, is one of the ZFC axioms that are used to axiomatize modern mathematics. there are 10 of them.
Yes, but is there not another way of looking at it? Your axiom hides more than 'fewer than three' assumptions in it, viz. the set, its emptiness, and its existence. In this way it is able to be a statement. Thus the system in which the statement is made has these three terms as a base.

At the next level (the level at which this axiom is taken to be just one thing), there will be other axioms to accompany it, and so on to the next level ad finitum. However complex the system becomes with its tiers of recursive axioms it rests on these three primary assumptions, these links with the outside world.

?

Originally posted by hlreed
May not be a number for axioms, but it requires:
1 variable to count. 1 2 3 4 ...
2 variables to do arithmetic. A + B
3 variables to do algebra. C = A + B
I think you have disguised some others. Thus:

3 'givens' to count (1+1 is 2)
3 'givens' to do arithmetic (A + B is C)
3 'givens' to do algebra (C is A+B)

In each case there are two terms predefined and one operator. Then we just give the result a name (which is not an assumption, just a name).

I am beginning to think I asked a stupid question. My apologies.

Originally posted by Canute

I am beginning to think I asked a stupid question. My apologies.

agreed. apology accepted.

Fluidity:

<i>There is no exactly defined relationship of the diameter of a circle to its circumference, but we developed a number, pi, to define it nonetheless.</i>

What about the relationship C=pi *times; D?

pi is an exactly defined number, and C and D are exactly defined terms, so where's the problem?