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Elements of Set Theory: Ordering on Natural Numbers

# Ordering on Natural Numbers Previously, we have defined natural numbers that was a side effect of our spurious definition, e.g. . Also, we have the following simple definition of order on natural n…

Elements of Set Theory: Arithmetic

# Arithmetic We can apply recursion theorem, from the previous section, to define addition and multiplication on . Suppose we want a function such that is the result of adding 5 to n. Then must satis…

Elements of Set Theory: Recursion on Natural Numbers

Consider the situation where I want you to guess the function such that I only give you two information; that is, 1. a starting value 2. a function such that for all This gives away all the informati…

Elements of Set Theory: Peano's Postulates

Hi! I'm Giuseppe Peano, I introduce Peano system to make the life of math students miserable. Source In 1889, Peano published a study giving an axiomatic approach to the natural numbers, showing how…

Elements of Set Theory: Inductive Sets

In mathematics, there are two ways of introducing new objects for study, 1. axiomatic approach (e.g. the one we have used for sets) The concept of set is one of our primitive notions, and we have ado…

Elements of Set Theory: Ordering Relations

### Elements of Set Theory: Ordering Relations From the first section of this chapter, we've covered about the ordering relation of the ordered set Now we want to consider the ordering relation on ot…

Elements of Set Theory: Equivalence Relations

Consider a set A given by a figure in (a), say we want to partition it into six boxes as in (b). For example, take, we can partition into six parts: By partition we mean, dividing the box into a simi…

Elements of Set Theory: Infinite Cartesian Products

We've encountered in previous section Cartesians products of two sets, this time we will show that we can form something like the Cartesian product but of infinitely many sets, provided that the sets…