Conculsion of Godel's proof alternative view I've recently become very interested in Godel, having just read "Godel, Escher, Bach" and subsequently "Godel's Proof".
One thing that strikes me as a flaw/alternative explanation in the (albeit non-rigorous) argument: does not the proof just mean that there are uncountably many truths within Number Theory, and only countably many Godel Numbers to map to. Since we fail when trying to create a map between the truths and the natural numbers? The intuition here is somewhat similar to Cantor's Diagonal Proof in a way. So in theory there is a way to map to the irrational numbers, say, instead of a countable set.
I would be very grateful if someone could point me in the right direction for some extra reading or offer some kind of intuitive explanation.
 A: No, it does not mean that.
There are only countably many sentences in the language of arithmetic; the set of "truths" is therefore countable. 
What is true is that it is not computable: any time I have a computable set $S$ of (Goedel numbers for) sentences, either there is some true sentence not in $S$, or there is some false sentence which is in $S$. But the set of true sentences in the language of arithmetic is still countable.
This can be a bit subtle, so here's an observation that might help. Consider the set of computable real numbers. This set is certainly countable (there are only countably many computer programs). However, if I have a computable list $\{s_i: i\in\mathbb{N}\}$ of computable reals, I can computably diagonalize against it to get a computable real $t$ not in that list. So


*

*The computable reals are countable, 

*but they are not computably countable.
This shows that the idea of diagonalization is actually more powerful than we often give it credit for: it can distinguish, not only between sets of different cardinalities, but between sets of different complexities. This takes a lot of work to make clear, but for now just keep this in mind: Goedel's theorem is fundamentally different from Cantor's, even though there are (very deep) similarities. It's tempting to draw the kind of analogy you've written above, but it is not true.
