# Gödel -- What Does this Mean (DrMatrix) **URL:** \ **Category:** Factual Questions\ **Created:** [May 25, 2003, 5:08am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365 "2003-05-25T05:08:08Z")\ **Posts on this page:** 13\ **Page:** 1
**Author:** ![Major\_Kong](https://avatars.discourse-cdn.com/v4/letter/m/b9bd4f/32.png) [@Major\_Kong](https://boards.straightdope.com/u/Major_Kong)\ **Post date:** [May 25, 2003, 5:08am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/1 "2003-05-25T05:08:08Z")
In this thread: [http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=185649](http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=185649) DrMatrix said: > [@](#): > > Gödel has shown that we cannot prove that any (sufficiently complex) mathematical system is consistent. (Well, actually he showed that if we can prove a system is consistent, then it is not consistent.) What does that mean? If it is consistent, it is not consistent? ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [May 25, 2003, 5:16am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/2 "2003-05-25T05:16:41Z")
[http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html](http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html) > [@](#): > > A statement sometimes known as Gödel’s second incompleteness theorem states that if number theory is consistent, then a proof of this fact does not exist using the methods of first-order predicate calculus. Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency if and only if it is inconsistent. ---
**Author:** ![daniel801](https://avatars.discourse-cdn.com/v4/letter/d/b2d939/32.png) [@daniel801](https://boards.straightdope.com/u/daniel801)\ **Post date:** [May 25, 2003, 5:21am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/3 "2003-05-25T05:21:00Z")
[Aside] “A “complete” system will generate all true statements. A “consistent” one will not generate contradictions.” -Michael Macrone Anyways, what an ironic post. [/Aside] ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [May 25, 2003, 7:13am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/4 "2003-05-25T07:13:08Z")
An inconsistent system can prove any statement, whether the statement is true or false. So, if a system in not consistent, it can prove that it is consistent. What Gödel showed is that if system can prove that it is consistent (by demonstrating that say 1=0 has no proof) then it can prove any statement and is therefore not consistent. He also showed that number theory is incomplete. That is, it cannot prove every true statement. He did this by showing that it is possible to construct a statement that asserts that it has no proof. If number theory is consistent, then this statement cannot have a proof and is therefore obviously true, but number theory cannot produce a proof. It can take a while to wrap your head around these two results. I recommend _Gödel, Escher, Bach, an Eternal Golden Braid_ by Douglas Hofstadter. ---
**Author:** ![Homer](https://avatars.discourse-cdn.com/v4/letter/h/df788c/32.png) [@Homer](https://boards.straightdope.com/u/Homer)\ **Post date:** [May 25, 2003, 11:49pm UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/5 "2003-05-25T23:49:39Z")
GEB is a great book! ---
**Author:** ![Major\_Kong](https://avatars.discourse-cdn.com/v4/letter/m/b9bd4f/32.png) [@Major\_Kong](https://boards.straightdope.com/u/Major_Kong)\ **Post date:** [May 26, 2003, 12:18am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/6 "2003-05-26T00:18:08Z")
Thanks. I will check out the book, but it is probably beyond me. ---
**Author:** ![AHunter3](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ahunter3/32/368_2.png) [@AHunter3](https://boards.straightdope.com/u/AHunter3)\ **Post date:** [May 26, 2003, 3:06am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/7 "2003-05-26T03:06:46Z")
You’d be surprised how much of it is accessible to the mathematically challenged 🙂 ---
**Author:** ![Trigonal\_Planar](https://avatars.discourse-cdn.com/v4/letter/t/0ea827/32.png) [@Trigonal\_Planar](https://boards.straightdope.com/u/Trigonal_Planar)\ **Post date:** [May 26, 2003, 3:15am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/8 "2003-05-26T03:15:31Z")
Its an excellent book. Don’t worry - its pretty understandable. And the few parts that aren’t you can generally skip over without losing anything 🙂 Anyway, I guess I understood wrong because I thought that that premise was a formal system cannot be consistent AND complete. It can be one or the other, but not both. ---
**Author:** ![Smeghead](https://avatars.discourse-cdn.com/v4/letter/s/f1d935/32.png) [@Smeghead](https://boards.straightdope.com/u/Smeghead)\ **Post date:** [May 26, 2003, 4:14am UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/9 "2003-05-26T04:14:26Z")
His _Metamagical Themas_ is good stuff, too. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [May 26, 2003, 4:42pm UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/10 "2003-05-26T16:42:30Z")
GEB is a good book, but it’s a layman’s treatment. You can get a rough idea of the things he’s talking about, but to do any work with them, you need a more specific treatment. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [May 26, 2003, 4:51pm UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/11 "2003-05-26T16:51:22Z")
> [@](#): > > \*Originally posted by Major Kong \* > \*\*What does that mean? If it is consistent, it is not consistent? \*\* There are some systems that can be proven to be consistent, but they are not complex enough to be interesting. Once you are able, within a system, to prove two things that are inconsistent, then–logically–you can prove _anything_. So, for inconsistent systems, you can indeed “prove” that they are consistent. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [May 26, 2003, 4:56pm UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/12 "2003-05-26T16:56:42Z")
\>\> There are some systems that can be proven to be consistent, but they are not complex enough to be interesting. Define “interesting”. I used to know this one girl who was quite simple and yet very interesting. At least parts of her were very interesting. To me at least. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [May 27, 2003, 3:24pm UTC](https://boards.straightdope.com/t/godel-what-does-this-mean-drmatrix/177365/13 "2003-05-27T15:24:34Z")
> [@](#): > > \*Originally posted by sailor \* > Define “interesting”. “Interesting” means powerful enough to prove the common known results in Number Theory (or Math in general). Quantified Boolean Algebra is complete and consistent, but is too weak to express many standard theorems in Number Theory. (BTW, I strongly recommend _against_ reading GEB. It’s horribly long winded and boring. A better writer could have done a better job in a 100 pages. An awful, awful book. His Scientifc American column didn’t last at all long for the same reason.)