In 1931 Kurt Godel published his fundamental paper, "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Godel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement i...more
Being relatively short, this book does not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing (in theory of computability, ...more
How can anyone wrap oneself around that ancient paradox? A true statement whose unprovability resulted precisely from its truth!
With this self-undermining formula as his hammer, Godel knocked down the fortress of Principia Mathematica (PM), so painstakingly constructed by Russell and Whitehead.
He also showed that his method applied to any system whatsoever that tried to accomplish the goals of Principia Mathematica. In effect, then, Godel destroyed the hopes of those who ...more
Higher Order Languages: "Gödel's Proof" by Ernest Nagel, James Newman
What Gödel's Theorem really says is this: In a sufficiently rich FORMAL SYSTEM, which is strong enough to express/define arithmetic in it, there will always be correctly built sentences which will not be provable from the axioms. That, of course, means their contradictions will not be provable, either. So, in a word, the sentences, even though correctly built, will be ...more
The long and the short of it is that once upon a time, I sorta understood Godel's incompleteness theorem, and after this modest reading, I sorta understand it again. Success!
More importantly for me, it was fun to try to connect neurons in my poor fuzzy ...more
Here's my more personal review.
I'm a functional progr ...more
Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true arithmetical statements that cannot be derived from the set.
Godel showed that it is impossible to give a meta-mathematical proof of the consistency of a system comprehensive enough to contain the whole of arithmetic—unless the ...more
P.S. Remember to not miss-use the incompleteness proof to give sweeping and profound statements about nature of the world or other mumbo jumbo. Godel went to a great deal of trouble to write a very exact and general proof, do not desecrate his work to prove weird stuff.
On the plus side, it was a very involved and difficult topic, and it was a bol ...more
The book dumbs down the proo ...more
I found this book fairly easy to read with ...more
"Gödel's paper is difficult."
After reading this book, and the masterwork Gödel, Escher, Bach: An Eternal Golden Braid, I can honestly say that my mind refuses to accept the ideas of Gödel's paper, let alone understand what his results imply.
Gödel, at the age of 25, was able to prove that any consistent formal system able to model basic arithmetic is incomplete. In other words, the ...more
Highly recommend for intense logic formalism-heavy fun.
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