Assumptions at Primordial Math I'm a beginning math student and was wondering what assumptions can be made at the absolute lowest level of math. Disregarding all proofs and previous work, what can you truly assume about numbers? Are assumptions of numbers even logically possible? If someone has asked a question like this, I apologize and would really like to see the question. 
 A: The standard way to do it today is to begin with sets. We do this by taking for granted certain axioms of set theory, which serve to tell us more or less what properties sets have. From this, we use the language of set theory to actually define what a real number is. Taking a course in real analysis will hopefully help you, but the general steps to construct real numbers in such a way that we can still talk about the arithmetic on them is:
(i) Begin with constructing the natural numbers $\mathbb{N}$, see here (see especially the Von Neumman construction). We can use this to endow the naturals with arithmetic, i.e. addition, multiplication, even exponentation.
(ii) Define from here the integers $\mathbb{Z}$, which we can do using the language of abstract algebra to extend $\mathbb{N}$ to an additive group.
(iii) We then consider the field of rationals $\mathbb{Q}$, i.e. the set of "fractions", which we do by considering a rational number to be an ordered pair of an integer and a (non-zero) natural. We use some convenient ways to transfer the arithmetic of the integers into the rationals, ways which you can probably figure out with a little thought and meditation upon your grade-school math classes when you learned how to add and multiply and divide fractions.
(iv) Here comes the big question: How do you go from $\mathbb{Q}$ to the field of all real numbers $\mathbb{R}$? First, we consider what it is that make $\mathbb{Q}$ unsatisfying. The property that discerns the reals is what we call completeness. Intuitively, this means that there aren't any "holes" in $\mathbb{R}$. The same cannot be said of $\mathbb{Q}$. For example, we don't have $\sqrt{2}$, i.e. we have a "hole" where we wanna have $\sqrt{2}$, as $\sqrt{2}$ is not rational. There are two common ways to "fill in the holes" that maintain the arithmetic. One way is by what we call Dedekind cuts: We consider the family of partitions of $\mathbb{Q} = L \cup U$, where $L$ has the property that if $p \in L$, and $q \leq p$, then $q \in L$, and $U$ has the property that if $p \in U$, and $q \geq p$, then $q \in U$. We consider each of these cuts to be a "real number" (essentially), where the Dedekind cut serves to "cut" the rationals in two parts, and each number is where the pieces "meet". To understand how we use this to describe $\sqrt{2}$, consider the following. Say that $L = \{ p \in \mathbb{Q} : p \leq 0 \textrm{ or } p^{2} < 2 \}, U  = \{ p \in \mathbb{Q} : p \geq 0, p^{2} > 2 \}$. Every rational falls in one of these, and each partition element satisfies the needed properties. We can "locate" the rationals by associating the rational number $q$ with $L = \{ p \in \mathbb{Q} : p < q \}, U = \{ p \in \mathbb{Q} : p \geq q \}$.
Another more popular way (though I think Dedekind cuts have their advantages) is to talk about what are called Cauchy sequences of rationals. The rationals have an obvious distance to them, so we say that a sequence $(q_{k})_{k \in \mathbb{N}}$ of rationals (note that we have only used the numbers we defined in (i) - (iii)) is Cauchy if for every natural number $N$, there is a number $K_{N}$ such that if $k_{1}, k_{2} \geq K_{N}$, then $|q_{k_{1}} - q_{k_{2}}| < 1/N$. This basically means that they eventually get arbitrarily close. We say that a space is complete if every Cauchy sequence converges to some point. This is not true in $\mathbb{Q}$. Take for example the sequence $1.4, 1.41, 1.414, \ldots$, which converges to where we should hope to find $\sqrt{2}$ (but we don't have it!). We say that two sequences $(p_{k})_{k \in \mathbb{N}}, (q_{k})_{k \in \mathbb{N}}$ are equivalent if for every $N \in \mathbb{N}$, there exists $M_{N}$ such that if $k_{1}, k_{2} \geq M_{N}$, then $|p_{k_{1}} -q_{k_{2}}| < 1 / N$. This means they eventually become very close to each other, and so would converge to the same place (even if there's a hole there). We then look at $[(q_{k})_{k \in \mathbb{N}}]$, the set of all sequences equivalent to $(q_{k})_{k \in \mathbb{N}}$. We consider each of these classes to be a distinct real number. Again, if we want to find the rationals in it, we can associate a rational $q$ with the class $[(q_{k})_{k \in \mathbb{N}}]$, where we say $q_{k} = q$ for all $k \in \mathbb{N}$.
EDIT: Many real analysis texts will go over the construction of the real numbers. Rudin's Principles of Mathematical Analysis takes the Dedekind cut approach. Both the Cauchy sequence and Dedekind cut approaches yield the same system.
A: Before you can write formal proofs about numbers, you need rules of logic and some kind of set theory. The five essential properties of the set of natural numbers (see Peano's Axioms) can be expressed in the language of set theory. For the set $N$ of natural numbers, successor function $S$ and $0$ we have the following essential properties or axioms:


*

*$0\in N$

*$S: N\to N$

*$S$ is injective

*$\forall x\in N: S(x)\neq 0$

*$\forall P\subset N: [0 \in P \land \forall x\in P: S(x)\in P \implies P=N]$


From these five axioms, it is widely believed that all of number theory and classical analysis can be derived.
