With a very basic knowledge of the rules of formal logic, you can define a simple type of arithmetic- Robinson Arithmetic.
Robinson Arithmetic is not complete, not even close. Peano Arithmetic comes closer, but have a talk with Mr. Gödel if you want to pursue this topic further. RA is simply a neat way for people with an undergraduate knowledge of logic to "prove" that 2+2=4.
Some knowledge of predicate logic is required to fully understand this, but it is honestly quite simple. With these axioms, you can take a limited set of arithmetical statements (phrased correctly: you would have to write "3" as sss0, for example) and prove or disprove them according to the rules of inference.
The area over which these axioms are considered true is the realm of natural numbers: its domain is said to be the set of N = {0,1,2,3,4,5.........}.
In formal logic, you must specify everything, so here is the definition of every function and fixed symbol I use (the variables are just x and y):
0 = the number zero
s = +1 (stands for "successor"), that is, sN = N + 1
+ means plus (addition)
* means times (multiplication)
→ means if-then, x → y means if x, then y.
A note on + and *. In logic, the "standard" way to define all functions, including basic ones like addition, etc, is like this: Where a is the addition function, axy means x + y, where m is multiplication, mxy means x * y. However, since common symbols already exist for these particular functions, doing it the correct way would simply make the axioms harder to understand- so I'm doing it this way in the interest of clarity. The successor function is done the standard way. It is more important to understand the concepts than to slavishly follow some little scheme.
THE AXIOMS
If x does not equal y, then the successor of x is not the same as the successor of y:
∀x ∀y (x ≠ y → sx ≠ sy)
Zero doesn't follow anything:
∀x0 ≠ sx
Anything plus zero is the same thing:
∀x(x + 0)=x
If x isn't zero, then x follows something:
∀x(x ≠ 0 → ∃y x=sy)
x plus whatever follows y is the same as whatever follows x + y:
∀x∀y(x + sy) = s(x + y)
Anything times zero is zero:
∀x(x * 0)=0
A number times whatever follows another number is the same as multiplying the two numbers together, and then just adding the first number:
∀x ∀y(x * sy) = ((x * y) + x)
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