User:Wvbailey/Explication of Godel's incompleteness theorems
The following is an exploration of how to explain the first Gödel incompleteness theorem (Theorem VI, 1931).
Rosser 1939
I. Notion of "two logics" --
- (1) "logic of ordinary discourse"
- "formal logic L"
- "propositions of L" are formulas built per "certain rules of structure"
- symbols, interpretations, rules of structure interpretations will be declarative sentences (not necessarily true) of "ordinary discourse" -- "expressions in L".
II. "Amonst the symbols of L must be one, ~, which is interpreted as "not", the 'contradictory' when the symbol is applied before a sentence.
III.1. For each integer, there must be a formula in L which denotes that integer.
III.2 "variables"
III.3 "substitution" or "replacement"
IV. A process whereby certain of the propostions of L are specied as "provable". [notion of simple consistency and ω-consistency]
V. If F and ~F are provable in L, then all propositions of L are provable. The non-provability of any formula whatever of L implies the simple consistency of L. ω-consistency implies simple consistency; but if L is not simply constent then not ω-consistent.
VI. Modus ponens: Symbol ⊃ of L such that if formula A expresses sentence S and formula B expresses sentence T and A ⊃ B expresses "IF S THEN T".
VI.1 "Provable means that if A & A⊃B are provable then so is B.
VII. Notion of arithmetization of formulas: "For every formula, a number is assigned. However, not all numbers are assigned to formulas. If a number is assigned to a formula, the formula can always be found...". (p. 226). Thus, when numbers have been assigned to formulas, statements about formulas can be repalced by statements about numbers. That is, if P is a property of formulas,we can find a property of numbers, Q, such that the formula A has the property P if and only if the number of A has the property Q.
- Lemma 1. Let "x has the property Q" be expressible in L.