How many axioms are there in a theory of natural numbers?

3 hours ago, Genady said:

How many axioms are there in a theory of natural numbers?

do you mean (something like) "arithmetic units" by "arithmetic(?). (some potential keywords in the contexts that I predict might be are : algebraic numbers, transcendent numbers etc)

also, to me : it is not "theory of 'natural' numbers" , it is "theory of numbers" or "numbers theory"

Edited March 10Mar 10 by ahmet

This is why I've clarified that I specifically mean natural numbers.

I guess that you don't mean just the Peano axioms? That would be too easy.

2 minutes ago, TheVat said:

I guess that you don't mean just the Peano axioms?

That's a good system.

2 minutes ago, TheVat said:

That would be too easy.

I want to see.

Euclid's 'The Elements' lists 5 Axioms ( which can easily be looked up, and related to Geometry ), and 5 'common notions', which are ...

1. Things that are equal to the same thing are also equal to one another (the transitive property of a Euclidean relation).

2. If equals are added to equals, then the wholes are equal (Addition property of equality).

3. If equals are subtracted from equals, then the differences are equal (subtraction property of equality).

4. Things that coincide with one another are equal to one another (reflexive property).

5. The whole is greater than the part

Is this what you mean ?

16 minutes ago, MigL said:

Euclid's 'The Elements' lists 5 Axioms ( which can easily be looked up, and related to Geometry ), and 5 'common notions', which are ...

1. Things that are equal to the same thing are also equal to one another (the transitive property of a Euclidean relation).

2. If equals are added to equals, then the wholes are equal (Addition property of equality).

3. If equals are subtracted from equals, then the differences are equal (subtraction property of equality).

4. Things that coincide with one another are equal to one another (reflexive property).

5. The whole is greater than the part

Is this what you mean ?

No, I mean axioms of natural numbers, the ones that govern the arithmetic of 0, 1, 2, 3, ...

Is this topic somehow about Gödel's incompleteness theorems?

Here are

How many axioms are here?

If the answer were "seven,"

8 hours ago, TheVat said:

That would be too easy.

22 hours ago, Genady said:

How many axioms are there in a theory of natural numbers?

I don't understand why this question is being set in a recreational thread ?

This has led to repeated requests for clarification.

It is just too vague

The actual answer will depend upon your formulation since some axioms/definitions/properties may be inherited.

If it is really about the Peano system, which is first order logic, then peano arithmetic with only addition is Godel decidable; peano arithmetic with both addition and multiplication is undecidable.

Peano axioms also refer to axioms for the construction of the natural numbers, as opposed to their arithmetic.

Good discussion of these mathematical matters can be found in

Computability and Logic
Boolos and Jeffrey
Cambridge University Press

Readings in the Philosophy of Science
Feigl and Brodbeck
Appleton-Century-Crofts
New York

16 minutes ago, studiot said:

I don't understand why this question is being set in a recreational thread ?

This has led to repeated requests for clarification.

It is just too vague

also, it is pointless. What will the OP do with the number of those axioms,I wonder?

Edited March 11Mar 11 by ahmet

Well, the answer is simple albeit probably surprising.

There are infinitely many axioms in that list, three posts above.

Edited March 11Mar 11 by Genady

I don't get it. But I was going to guess nine. So I was a little off.

Here is an explanation:

Peano axioms - Wikipedia