D
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
ADDITION AXIOMS
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