Math 3000 - Introduction and Lecture 1

I. General Class Organization

II. The nature of mathematical truth

a. Truth (with a capital T) is subject dependent.
b. Mathematical truth is a formalism.
.... true statements are those which follow from the axioms by means of logic.

This is what provides mathematics with its greatest strength - truth can be demonstrated, there is no ambiguity about a mathematical truth; and its greatest weakness - mathematics can not prove anything that is not mathematical.

c. Logic vs. Meta-logic
Meta-logic is the language and thinking process we use to talk about logic. It is not itself logic, rather it is a stepping outside of the logical system in order to examine that system. We do this in other areas as well, for instance, we learn English grammar in foreign language classes.
d. A formal logic system
MIU System - Hofstadter, Godel, Escher, Bach: an eternal golden braid. Basic Books, 1979.

Symbols: M, I, U
Propositions: Strings of these symbols. (may be empty)
Axioms: MI
Grammar: (Rules of Production) x is any proposition.

  1. If xI then xIU
  2. If Mx then Mxx
  3. If xIIIy then xUy
  4. If xUUy then xy

A theorem is a proposition which can be obtained from the axioms by applying the rules of the grammar.

For instance, MUIU is a theorem.

MI --(2) MII --(2) MIIII --(3) MUI --(1) MUIU.

MUIIU is also a theorem.

MI --(above) MUIU --(2) MUIUUIU --(4) MUIIU.

But clearly U is not a theorem.

Question: Is MU a theorem?