3
votes
2answers
1k views

Set theory definition of addition, negative numbers, and subtraction? [closed]

Using the definition of natural numbers $0 = \emptyset$ and $S(n) = n \cup \lbrace n \rbrace$ where S is the successor function, what is the definition of addition on natural numbers? Concerning the ...
16
votes
1answer
757 views

Surreal exponentiation — are the varying definitions equivalent? If not, is there agreement on which ones are better?

The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
3
votes
2answers
299 views

Definition modifications without choice

What definitions or equivalencies between definitions for standard set theory objects (such as large cardinals) do not hold or do not carry through in the expected manner to the world without choice? ...