**Introduction**

*Many of the axioms listed here are borrowed from the Peano Axioms, as well as the axioms of arithmetic. A few axioms here are especially tailored to googology and can not be found elsewhere. Together they form a fairly easy to use axiomatic base for working in googology. There are only axioms for the successorship function, addition, and multiplication. Further operations such as exponents, factorials, hyper-operators, etc. must be defined independently of these axioms. Their properties must be derived as lemmas and theorems from the axioms listed here about addition and multiplication. I make frequent reference to these axioms throughout my writing. Here they are gathered into one place as a reference, along with an established order and shorthand names.*

**SUCCESSORSHIP AXIOMS**

**Axiom 1.**

*[AxP]*

**Axiom of the Progenitor**

1 is a number

**Axiom 2.**

*[AxE]*

**The Existential Axiom of the Successor**

If "n" is a number than S(n) is also a number.

Alternatively:

*every number has a successor, and every successor is a number*.**Axiom 3.**

*[AxId]*

**Axiom of Identity**

n = n

**Axiom 4.**

*[AxeS]*

**Axiom of equal symmetry**

a = b <--> b = a

**Axiom 5.**

**Axiom of**

*[AxeT]*

**equal transitivity**

a = b & b = c --> a = c

**Axiom 6.**

*[AxS]*

**Axiom of Successorship**

n < S(n)

for all n

**Axiom 7.**

**Axiom of successor symmetry**[AxSS]

a < b <--> b > a

**Axiom 8.**

*Axiom of successor transitivity*[AxST]

if a<b & b<c then a<c

**Axiom 9.**

*Axiom of Substitution*[AxSB]

a=b --> E(a) = E(b) ; where E(x) is an expression of "x"

a=b --> ( P(a) <--> P(b) ) ; where P(x) is a property of "x"

**Axiom 10.**

*Axiom of Induction*[AxIn]

If P(1) holds, and P(k) --> P(k+1), then P(n) is TRUE for all n

**ADDITION AXIOMS**

**Axiom 11.**

*[AxP&S]*

**Axiom of Part and Sum**

a < a+b

for all numbers a,b

**Axiom 12.**

*Axiom of commutativity of addition*[AxCA]

a+b = b+a

**Axiom 13.**

**Axiom of associativity of addition**[AxAA]

(a+b)+c = a+(b+c)

**Axiom 14.**

*[AxMS]*

**Axiom of Minimum Successorship**

a < b --> a+1 <= b

**Axiom 15.**

**Axiom of Greater Sums**[AxGS]

*"greater parts, greater sum"*

a<b & c<d --> a+c < b+d

**MULTIPLICATION AXIOMS**

**Axiom 16.**

*Axiom of product and factor*[AxP&F]

a < a*b : b>1

**Axiom 17.**

*Axiom of commutativity of multiplication*[AxCM]

a*b = b*a

**Axiom 18.**

**Axiom of associativity of multiplication**[AxAM]

(a*b)*c = a(b*c)

**Axiom 19.**

*Axiom of Greater factors*[AxGF]

a<b & c<d --> a*c < b*d

**MIXED AXIOMS**

**Axiom 20.**

*Axiom of distribution*[AxD]

a(b+c) = a*b+a*c