The Kurt G?del Thread
This thread is devoted, to one of my heros...
Kurt G?del, 20th century Logician
1906 - 1978
Godel proved in his doctoral dissertation the completeness of first-order logic. What this means is that if you start with the syntax of a formal system you can get the semantic notion and vice versa. In other words, both notions produce the same results.
A year later, he discovered an interesting thing about Mathematics...that it was incomplete. He is known for his Incompleteness Theorems, which are:
G?del's first incompleteness theorem
For any consistent formal theory that proves basic arithmetical truths, it is possible to construct an arithmetical statement that is true but not provable in the theory. That is, any consistent theory of a certain expressive strength is incomplete.
Basically what this says, is that in any formal axiomatic system in which a certian amout of arithematic is carried out, there will be propositions cannot be proven if the theory is consistent...hence, there are some propositions for which the theory is undecided on. Thus, you can have a complete or a consistent system, but not both. (Complete in this context simply means that given the pair [p, ~P] the system will prove one of these. Consistency means that the system will ONLY prove ether P or ~P)
G?del's second incompleteness theorem
For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.
In other words, if a formal axiomatic system is consistent it cannot prove that it is consistent.
These two theorems are wicked cool. Think about them for a minute.
G?del also discovered facts about the axiom of choice and the continuum hypothesis of set theory, but this would take alot of explaining on my part.
These are G?del's main achievements. If you have any questions, let me know!
BTW: April 28th is his hundreth birthday!
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions