The surreal numbers are a generalization of the reals. Each surreal number consists of two parts (called the left and right), each of which is a set of surreal numbers. For any surreal number , these parts can be called and . (This could be viewed as an ordered pair of sets, however the surreal numbers were intended to be a basis for mathematics, not something to be embedded in set theory.) A surreal number is written .
Not every number of this form is a surreal number. The surreal numbers satisfy two additional properties. First, if and then . Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the relation by mutual induction:
, which has both left and right parts empty, is .
Given two (possibly empty) sets of surreal numbers and such that for any and , , .
Define if there is no such that and no such that .
This process can be continued transfinitely, to define infinite and infinitesimal numbers. For instance if is the set of integers then . Note that this does not make equality the same as identity: , for instance.
It can be shown that is “sandwiched” between the elements of and : it is larger than any element of and smaller than any element of .
Addition of surreal numbers is defined by
It follows that .
The surreal numbers satisfy the axioms for a field under addition and multiplication (whether they really are a field is complicated by the fact that they are too large to be a set).
In general, is the simplest number between and . This can be easily used to define the dyadic fractions: for any integer , . Then , , and so on. This can then be used to locate non-dyadic fractions by pinning them between a left part which gets infinitely close from below and a right part which gets infinitely close from above.
Ordinal arithmetic can be defined starting with as defined above and adding numbers such as and so on. Similarly, a starting infinitesimal can be found as , and again more can be developed from there.
|Date of creation||2013-03-22 12:58:49|
|Last modified on||2013-03-22 12:58:49|
|Last modified by||Henry (455)|