I find this axiomatization of the naturals quite neat:
Zero is a natural number. 0∈ℕ
For every natural number there exists a succeeding natural number. ∀n∈ℕ: s(n)∈ℕ
(s denotes the successor function)
Now the neat part: If 0 is a constant, then s(0) is also a constant. So we can invent a name for that constant and call it “1.” Now s(s(0)) is a constant, too. Call it “2” and proceed to invent the natural numbers.
What’s missing here os the definition that we’re working in base 10. While it won’t be a proof, Fibbonaci has his nice little Liber Abbaci where he explains arabic numerals. A system of axioms for base 10, a definition of addition and your succession function would suffice. Probably what the originals were going for, but I can’t imagine how that would take 86 pages. Reading it’s been on my todo list, but I doubt I’ll manage 86 pages of modern math designed to be harder to read than egyptian hieroglyphs.
That ‘86 pages’ factoid is misleading. They weren’t trying to prove that 1+1=2. They were trying to build a foundation for mathematics, and at some point along the way that prove fell out of the equations.
There are non-standard models of arithmetic. They follow the original first-order Peano axioms and any theorem about the naturals is true for them, but they have some wacky extra stuff in them like you mention.
Not sure what you mean by ‘loops’ - except perhaps modular arithmetic, but there are no natural numbers that are negative - you may be thinking of integers, which is constructed from the natural numbers. Similarly, rational numbers, real numbers, and complex numbers are also constructed from the naturals. Complex numbers are often expressed as though they’re two dimensional, since the imaginary part cannot be properly reduced, e.g. 3+2i.
I find this axiomatization of the naturals quite neat:
Now the neat part: If 0 is a constant, then s(0) is also a constant. So we can invent a name for that constant and call it “1.” Now s(s(0)) is a constant, too. Call it “2” and proceed to invent the natural numbers.
What’s missing here os the definition that we’re working in base 10. While it won’t be a proof, Fibbonaci has his nice little Liber Abbaci where he explains arabic numerals. A system of axioms for base 10, a definition of addition and your succession function would suffice. Probably what the originals were going for, but I can’t imagine how that would take 86 pages. Reading it’s been on my todo list, but I doubt I’ll manage 86 pages of modern math designed to be harder to read than egyptian hieroglyphs.
That ‘86 pages’ factoid is misleading. They weren’t trying to prove that 1+1=2. They were trying to build a foundation for mathematics, and at some point along the way that prove fell out of the equations.
That axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
I think you are missing some properties of successors (uniqueness and s(n) different than any m<= n)
That would avoid “branching” of two different successors to n and loops in which a successor is a smaller number than n
There are non-standard models of arithmetic. They follow the original first-order Peano axioms and any theorem about the naturals is true for them, but they have some wacky extra stuff in them like you mention.
Not sure what you mean by ‘loops’ - except perhaps modular arithmetic, but there are no natural numbers that are negative - you may be thinking of integers, which is constructed from the natural numbers. Similarly, rational numbers, real numbers, and complex numbers are also constructed from the naturals. Complex numbers are often expressed as though they’re two dimensional, since the imaginary part cannot be properly reduced, e.g. 3+2i.
I recommend this playlist by mathematician another roof: https://www.youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rmb34ZBYe5ohrkL3t
They build the whole modern number system ‘from scratch’
I know how how natural numbers work, but the axioms in the comment i replied to are not enough to define them.
There could be a number n such that
m=s(n)andn=s(m). This would be precluded by taking the axiom of induction or the trichotomy axiom.If we only take the latter we can still make a second number line, that runs “parallel” to the “propper number line” like:
n,s(n),s(s(n)),s(s(s(n))),... 0,s(0),s(s(0)),s(s(s(0))),...I know, but the given axioms don’t preclude it. Under the peano axioms it’s explicitly spelled out:
0 is not the successor of any natural number