# What does a Godel sentence actually look like?

My understanding is that Godel's first theorem says that there is a sentence that is in a sense true in a formal system F but cannot be derived in that system.

I then hear that Godel actually goes on to construct such a sentence.

My question is, what would this even look like?

For example, in Peano Arithmetic would it be something like:

5 + 3 * 15 ... = ...? Is it long? Is it short? Could it be written on a piece of paper?

If not what is wrong with my thinking?

Thanks

• It is very very very long. The sentence was not constructed, a (partial) recipe for constructing the sentence was given. Since then some "natural" sentences have been constructed, not using the recipe. Oct 10, 2015 at 3:39
• See the following post for the construction of the Gödel sentence: math.stackexchange.com/a/34927/137499 Oct 10, 2015 at 3:41
• Thanks for that, just how long is the actual formula though out of curiosity. My original question stands on whether it could be written on a piece of paper (however large). Another question is whether we know the first and/or last symbols in it Oct 10, 2015 at 8:06
• Someone on this site has tried to write it; see Hagen von Eitzen's answer to this post. Oct 10, 2015 at 9:45
• For a brief exercise on "encoding", see this post. If you perform the first steps in the arithmetization process, you can "taste" how it works ... and "measure" how far you have to go to produce an example of Gödel sentence. Oct 10, 2015 at 9:50