TITLE: Is this visual analogy to Gödel's incompleteness theorem accurate?
QUESTION [17 upvotes]: Today I was trying to explaining the Gödel's theorem to a layman, I've drawn a figure similar to the one below and said that:

A truth is a consequence of the axioms (with the axioms also being truth).
The lines between the axioms and the theorems and the lines between theorems and theorems are the employed notions to show the truth of that theorem.
And that there are theorems that are true (red diamonds) but unreachable by any arrangement of lines from the axioms to the theorems to them. The red lines are meant to show that there is no line that reaches there.

Is this visual analogy accurate? I know that perhaps I'm oversimplifying, but does it captures the big picture or is there something else I should add?

REPLY [2 votes]: I don't like this:

A truth is a consequence of the axioms (with the axioms also being truth).

You're trying to convey that there are truths which are not consequences of the axioms.  So this definition of truth is counterproductive.  We could try to come up with another, but I doubt you can define whether (say) the Continuum Hypothesis is true or false without annoying someone.  Worse, the CH is precisely the kind of statement we're interested in!
I'm not sure it's a great idea to talk about "truth" here.  Instead, focus on the fact that there are some statements which we cannot prove, and whose negations we also cannot prove.  Combined with the law of the excluded middle, that helps to suggest the intuition that some of these statements are "true" in some sense, while avoiding a direct definition of truth.