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APPENDIX D
THE AXIOMS OF GOOGOLOGY

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. Axiom of the Progenitor [AxP]

1 is a number

Axiom 2. The Existential Axiom of the Successor [AxE]

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. Axiom of Identity [AxId]

n = n

Axiom 4. Axiom of equal symmetry [AxeS]

a = b <--> b = a

Axiom 5. Axiom of equal transitivity [AxeT]

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

Axiom 6. Axiom of Successorship [AxS]

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

Axiom 11. Axiom of Part and Sum [AxP&S]

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. Axiom of Minimum Successorship [AxMS]

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