# A Book of Set Theory by Charles C. Pinter

This is a rather interesting book that I am slowly making my way through. The ideas contained in the book, the authors tone, and care of writing are remarkable. What comes through is a clear demonstration of how much the Author cares about Mathematics, and this makes you care too.## Notes, Ideas & Links

- Georg Cantor
- Virtual & Actual Infinites

**Actual:**Many objects are conceived as existing simultaneously

**Virtual:**The**Potential**to exceed any finite quantity - Mathematicians dreamed of a "Unifying branch of mathematics "
- The Ancient schools thought this unification would lie in geometry
- 19th Century Mathematicians such as Weierstrass & Dedekind thought unification could be achieved under the rubric of Arithmetic.
- It was shown that every Real Number could be represented as a sequence of rational numbers. And Rationals as pairs of Integers.
- We can in turn show how Natural numbers can be
represented by
**sets** - Set Theory Paradoxes, there are two main types:

- Logical - Russell's Paradox
- Semantic - Berry's Paradox

- The paradoxes arise because in
**Naive Set Theory**we take the position that we can describe properties of an object and the decribe sets of objects based on these**properties**. - However certain properties lead to paradoxes
- So we must ask, what are legitimate properties? How
do we decribe the
**existence of sets**. - There have been three major schools for solving these problems:
- Axiomatic Axioms are not Universal statements of truth but rather what we which to use as premises of statements and deductions are Independant of meaning. They are mechanical, following rules.
- Logistic
- Intuitionist

We need

**predicates, logical connectives & quantifiers**

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07dec16 | admin |