請注意這是由另外的假設體係導出交換律， 與此同時， 算術公理體係也可以參考這裏 http://en.wikipedia.org/

There are many different, but equivalent, axiomatizations of Peano arithmetic. While some axiomatizations, such as the one just described, use a signature that only has symbols for 0 and the successor, addition, and multiplications operations, other axiomatizations use the language of ordered semirings, including an additional order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.^[8]

1. . , i.e., addition is associative.
2. . , i.e., addition is commutative.
3. . , i.e., multiplication is associative.
4. . , i.e., multiplication is commutative.
5. . , i.e., the distributive law.
6. . , i.e., zero is the identity element for addition.
7. . , i.e., one is the identity element for multiplication.
8. . , i.e., the '<' operator is transitive.
9. . , i.e., the '<' operator is irreflexive.
10. . .
11. . .
12. . .
13. . . .
14. . .
15. . .

The theory defined by these axioms is known as PA^–; PA is obtained by adding the first-order induction schema.

所有跟帖： 

• 數學專家，鑒定完畢~~ -隨意- ♀ (0 bytes) () 07/11/2014 postreply 10:44:24

• 專家談不上，隻是懂一點皮毛而已。。。 -littlecat8- ♂ (0 bytes) () 07/11/2014 postreply 10:49:37

• 你好像是鑒定專家啊，你上一次那個鑒定我覺得很有水平啊，哈哈 -醫者意也- ♂ (0 bytes) () 07/11/2014 postreply 10:52:55

• thx for noticing my finding on the couple relationship -隨意- ♀ (61 bytes) () 07/11/2014 postreply 11:43:34

• Can you challenge those? -26484915- ♂ (0 bytes) () 07/11/2014 postreply 10:50:29

• 公理的意思是我們不能證明是正確，也不能證明它是錯的 -littlecat8- ♂ (260 bytes) () 07/11/2014 postreply 10:58:51