The Kurt G?del Thread
Offline This thread is devoted, to one of my heros...
Kurt G?del, 20th century Logician
1906 - 1978
accomplishments:
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
- Login to post comments
... and the day you appear on the Rational Response Squad, puttin a hurtin on Ray Comfort.
Vote for Democrats to save us all from the anti-American Republican party!
Please become a Patron of Brian Sapient
Godel was a theist too. See, not all theists are evil.
Indeed. April 28th will be a good day
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
Not a theist in the usual sense of the word. Godel was more a theist akin to Spinoza, than to the monotheistic conception of God.
If you look at Godel's Ontological Argument, it is designed to prove that SOMETHING is necessary, not the God of the monotheistic religions.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
For those interested in Godel's impact on logic, vist these sites:
The Popular Impact Of Godels Incompleteness Theorems
How Godel Transformed Set Theory
Books on Godel's Incompleteness Theorems:
Books on Godels life:
Time article regarding Godel:
Time 100: Kurt Godel
These are but a few. I expect all of you to study up on Godel. It would be immoral for you to ignore this mans work
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
Keen readers will notice that I did not post a picture of his famous paper "On Formally Undecidable Propositions In Principia Mathematica and Related Systems."
This is for good reason. I am reading it right now for one of my logic classes...and man, it is a hard read. The paper is hardly intellegable
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
Happy 100th birthday Kurt G?del!
By the by, remember I was having those difficulties with the use of Free Formula's in his proof?
I've managed to get my head around the diagonalization lemma now.
So I understand the outline to the proof.
There's still a lot that I wouldn't be able to prove, like how his GodelNumber Function and Diag Function (for diagonalization lemma) are recursive and that recursive functions are representable in arithemetic theorems.
But I think that such an outline is sufficient for those of us who aren't Godel obsessives!
Try reading it in German