Set-Theoretic Paradoxes and their Resolution in Z-F

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By Samuel Horelick

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By Samuel Horelick

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9. Essentials of Axiomatic method

10. Continuum Hypotheses revisited

11. Unlimited Abstraction Principle and Separation Principle

12. Undecidability of Continuum Hypotheses in Zermelo-Fraenkel system

13. Note on the objections to Zermelo’s system.

How to decide if two sets are of the same size without the appeal to a concept of “size”?

Well,
if we could find a way to assign an element of one set to an element
of the other set so that each element of one set is assigned to
exactly one, unique element of another set and no elements are left
over unassigned then these two sets are of the same size. That is, if
we can show a one-to-one and onto (*i.e.,*
bijective) function between the sets then we know that these two sets
have exactly the same number of elements. The concept of being “of
the same size” is quite irrelevant here. This method is obviously
true for any given finite set of things. Such a finite set can be put
into one-to-one correspondence with some subset of integers, say,
from 1 to some integer n. This number n is called *cardinality*
of the set.

If
the set is infinite, we can attempt to put it into one-to-one
correspondence with the set all of integers. If we can show some
methodical and well-defined way of doing it (that is, if can show a
bijective function between N and a given set A) then we call this set
A *countable*
(because we can count it, literally, even if it will take eternity).
For example, the set of all even numbers is infinite, but we can
define a function f on N such f(n) = 2n. It is a bijective function
and so the set of all even integers is of the same size (*cardinality*)
as the set of all integers – countably infinite.