Reddit reviews Gödel's Theorem

Misuse of Godel's work is a substantial topic. I highly recommend Torkel Franzen's book, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.

I did a 3 month historical and research analysis on it for one of my mathematics course :P

I read a lot of books and talked to pretty much every one of the my Mathematical Logic and Set Theory professors.

The best book that helped me was this:
Gödel’s Theorem: An Incomplete Guide To Its Use and Abuse

Seriously worth a read and would clear up everyone's misconceptions in this topic.

My 2c : How about just asking the question "why do you subscribe to the PhilosophyofScience reddit?" and then give the prize to the comment with the most upvotes? Ties being decided by you.

EDIT : I will throw in a copy of Godel's Theorem : An Incomplete Guide to its Use and Abuse as another prize.

The incompleteness theorem had incredible impact on mathematical and scientific thinking. Philosophy and Mathematics at the time was OBSESSED with grounding all of mathematics in Set Theory. See Principia Mathematica:
http://en.wikipedia.org/wiki/Principia_Mathematica

The incompleteness theorem said that this task was essentially futile. With this, objectivity of mathematics was thrown into the fire.

You're right to say completeness isn't understanding. Completeness is a mathematical property, and understanding refers to a human belief in explanatory satisfaction.

Before we get too carried away, the incompleteness theorem is also often incorrectly used by those trying to use it in arguments. In fact, books have been written about its misuse.

http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/dp/1568812388

Part of the shock regarding the incompleteness theorem comes from a field-wide belief that was proven false. It continues to be shocking because it is a belief that many mathematicians still have until they learn about it.

Gödel's incompleteness theorem is a technical statement concerning a possible formalization of mathematical reasoning known as first-order logic. There are a million variations, but basically it states that if you start with a set of axioms which is finite or even merely enumerable by some mechanical process (Turing machine), and if these axioms are consistent and contain a very minimal subset of arithmetic, then there is a statement which is "true" but you cannot prove from those axioms with first-order logic (and, in fact, it gives you an explicit such statement which, albeit "true", cannot be proven; if your axioms contain a not too minimal subset of arithmetic, one such statement is the very fact that the axioms are consistent, suitably formalized; another variant, due to Rosser, is that even if you allow for your axioms to contain false statements, there is still going to be some statement P such that neither P nor its negation ¬P follow from your axioms).

So, in essence, no matter what axioms you use to formalize arithmetic or any decent subset of mathematics, unless your axioms are useless (because they cannot be enumerated, or because they are inconsistent), or the axioms aren't sufficiently powerful to prove that they are consistent. Even if you add that as an axiom, there is still something missing (namely that with that extra axiom, the axioms are still consistent; and if you add that, then again, etc.). Interestingly, a theory cannot even postulate its own consistency (one can use a quinean trick to form a theory T consisting of usual axioms of arithmetic + the statement that T itself is consistent, but then the theory T is wrong, and inconsistent).

This is all really a technical statement concerning first-order logic. But trying a different logic will not help: another variant of Gödel's theorem (due to Church or Turing) tells us, essentially, that there can be no mechanical process (again, Turing machine) to determine whether a mathematical statement is true or false; so there can be no mechanizable, coherent and complete, logic which attains all mathematical truths, because if there were, one could simply enumerate all possible proofs according to the rules of that logic, and obtain all possible truths. All these variants of Gödel's theorem are variations around Cantor's diagonal argument: in the original variant, one constructs a statement which says something like "I am not provable" (intuitively speaking, at least), whereas in the Church/Turing version I just mentioned one would appeal to the undecidability of the halting problem.

But one thing to keep in mind is that almost every attempt to draw philosophical or epistemological consequences from Gödel's theorem has been sheer nonsense. Explanations àla handwaving such as "every formal mathematical system is necessarily incomplete" or "formalization of mathematics is inherently impossible" or whatever, are perhaps nice for giving a vague intuitive idea of what it's all about, but the actual mathematical theorem is not so lyrical. (For example, I have used the word "truth" in quotation marks once or twice in the above. This is because I don't have the patience to write down all the caveats about the meaning of "truth" in the mathematical sense in this context. So while what I have written is correct, attempts to draw metaphysical consequences from it will not be. :-)

For further reading, besides what others have already suggested, I believe Torkel Franzén's book on Gödel's theorem (destined for a general audience) is excellent.

No, it wouldn't. It doesn't say anything at all about the physical universe. See Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.

If I remember Godel's theorems correctly, then we can not provide such examples. There are some accompanying statements to (1) and (2) above, that go something like

(1) If there is an unprovably true statement A, we can not ever prove that A is unprovably true.

and

(2) If there is a provable but false statement B, we can not ever prove that B is provable but false.

It is extremely handy to know these when faced in debate by someone who has only read very hand-wavy accounts of Godel's theorems. A fallacious argument based on Godel's theorems is "Statement A seems to be true, but I have no logical proof. But, Godel tells me that many things that are true can not be proved to be true. Therefore A." One response might be "Godel also tells us that unprovably true statements can never be known to be true, so if A is unprovably true, we can never logically conclude that it is true. Thus your statement is fallacious."

Honestly, I haven't run into this argument in use that often myself, but I don't spend a lot of time reading New Age drivel. Visit the Amazon page for the book Godel's Theorem: An Incomplete Guide to Its Use and Abuse . I have not read all of this book myself (just a selection), but it should provide a broader answer to your question than I have given here.

This might be the book for you:

https://www.amazon.com/G%C3%B6dels-Theorem-Incomplete-Guide-Abuse/dp/1568812388

There are two theorems. Your intuition captures very bluntly the first theorem. Your intuition is nowhere near the second one. The theorem applies to certain formal systems.
A good book:
Godel's Theorem: An Incomplete Guide to Its Use and Abuse
http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/dp/1568812388

I skimmed the comments to see if anyone mentioned this related book:

Gödel's Theorem: An Incomplete Guide to Its Use and Abuse

I recently checked it out of our math library. Pretty heady stuff for a small fry like me!

Peter Smith is a philosopher and his Godel's Theorems book seems to me like what you are looking for. It considers the implications of the theorems and comes with a careful bibliography. Most university libraries would have it.

A shorter book is Torkel Franzén's.

> Kurt Gödel in 1931 with his incompleteness theorems demonstrated mathematically that only the simplest of arithmetic calculations can be complete [6].

Well, you sort of can say that in broad strokes so CSW lovers won't say I'm nitpicking, but it is not the calculations what matters but the system and axiomatics. For instance, presburger arithmetic is decidable, unlike peano arithmetics, and it is weaker than PA, it doesn't have multiplication operation, only addition and equality. But one can express multiplication using only addition, this is by the way essentially what a computer does!

> Science is all about models. We like to believe we can know it all, but this grasp of unbounded knowledge something that will always lie outside our grasp. Gödel proved that.

Gibberish, gödel didn't prove such thing. I suggest u/craig_s_wright reads this book: https://www.amazon.com/G%C3%B6dels-Theorem-Incomplete-Guide-Abuse/dp/1568812388

ps: TIL -- roughly speaking, gödel's formal system (or PA if you like) becomes decidable with the addition of transfinite induction.

It is pleasing to use Gödel's theorems metaphorically when speaking of the unknowable but Gödel's theorems actually make very specific statements about formal mathematical systems that are not really applicable in this context.

If you are interested, I suggest Gödel's Theorem: An Incomplete Guide to its Use and Abuse. This goes for the OP as well.

Godel's theorem: An incomplete guide to its use and abuse

Anyone who thinks Peterson's statement is anything more than pseudo-intellectual gibberish, please get a copy of this book and educate yourself. It's aimed at a non-technical audience and is very well written.

This book applies.

perhaps Gödel's Theorem: An Incomplete Guide to Its Use and Abuse?