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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?

enter image description here

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    $\begingroup$ Why the vote to close? $\endgroup$ – Noah Schweber Jun 10 '15 at 6:14
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    $\begingroup$ To be honest, I can't get this particular explanation even though I have a significant amount of education in mathematics (CS, but oh well, still more than your average "layman"). $\endgroup$ – Davor Jun 10 '15 at 8:50
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    $\begingroup$ Usually, the typical question that comes in this context is "How/why is X true if we can't prove it?" and you don't really address this question. You can probably be a bit clearer about the distinction between true, wrong, provable, provably true etc. Otherwise it's a nice visual. $\endgroup$ – Alexandre Halm Jun 10 '15 at 12:32
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Well, first of all I wouldn't call them "theorem" $\infty, ?$, since by definition something's only a theorem if it's provable. :) But this is a very minor criticism.

I like this picture a lot, partly because of the suggestive feel of "if we could make our proofs infinitely long, then we could prove these things!" This can be made precise in a variety of ways, and is true to different degrees depending on how it is made precise, but it is always somewhat true: if we allow "infinitely long proofs" (whatever those may be) then certain at least every true $\Pi^0_1$ statement - such as "PA is consistent" - will be provable.

There are two tiny criticisms I have, though they obviously don't mean it's not cool (like I said above, I like it a lot):

  • One, it addresses what it means for something to be not provable; it doesn't explain how one would possibly show that something's not provable, or what such a statement might look like. (Of course, that may well be a job for another picture . . .)

  • More subtly, the question "Which kinds of true sentences can be proved if we allow infinitely long proofs?" is incredibly deep and subtle, and a picture like that suggests that the answer is "all of them," which (in most interpretations) it is not.

However, these are very much not big problems. The second one in particular is definitely something I wouldn't worry about until well after one has understood Godel's theorem.

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    $\begingroup$ I would emphasize the first bullet, though: make it clear that this is a picture of the statement, not proof, of incompleteness. $\endgroup$ – Noah Schweber Jun 10 '15 at 3:51
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All theorems are provable: that is what the word "theorem" means. The point is that not all truths are provable, that is, not all truths are theorems. IMO your diagram would be more helpful if you replaced the word "theorem" everywhere by "truth" (or something synonymous). Perhaps the box at the top could be labelled "theorems" as well.

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    $\begingroup$ It's true because it's provable in some other system. For example, it is true that every Goodstein sequence terminates at zero because this can be proven in second-order arithmetic. But it cannot be proven in Peano arithmetic. So, within the world of PA, the convergence of Goodstein sequences is something that is true but unprovable. $\endgroup$ – David Richerby Jun 10 '15 at 9:21
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    $\begingroup$ @HowDoIMath - another way to think about it is that the logical rules of PA leave some questions open, but those questions (though not others) are answered by other rules. It would be reasonable (though very unfashionable) to wonder whether that made the original statements "true" rather than "provable in some other system". You might also read up on Tarski's closely related proof that arithmetic truth is undefinable. $\endgroup$ – Francis Davey Jun 10 '15 at 10:01
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    $\begingroup$ Thank you for your answers. I'm not trying to be difficult, but if one calls an unprovable statement "true", if it can be proven in another system, wouldn't it be equally reasonable to call it false, since its negation can be proven within some third system (e.g. one where you include the negation as an axiom)? @FrancisDavey (can only tag one) $\endgroup$ – Mankind Jun 10 '15 at 10:38
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    $\begingroup$ "In my world, a true statement is something you can prove." It might be said that the whole point of Gödel's theorem (it was for Gödel himself!) is to show that how and why this is a confusion .... $\endgroup$ – Peter Smith Jun 10 '15 at 11:01
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    $\begingroup$ @HowDoIMath - you might want to make your question into a proper question and submit it rather than having it debated here. The point I was alluding to is that there is some philosophy going on here. If it helps: there are models of PA in which the Godel sentence is false and others in which it is true. The axioms of PA don't nail that down. To say it is a true sentence is to say "I am thinking of something else that I haven't described and this is what I really meant". Since you can't define arithmetic truth this is unsurprisingly unsatisfying to some. $\endgroup$ – Francis Davey Jun 10 '15 at 13:49
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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.

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(The best source for this subject is the book "Gödel, Escher, Bach" by Douglas Hofstadter)

First, I agree with the distinction the other answers make about truths and theorems.

Second, I am not sure I like the red dotted lines. To my eye they suggest that there are proofs leading to those statements. If you want to talk about infinitely long proofs, leave them in, otherwise take them out. Gödel's work assumes that proofs are finite, so infinite proofs would be a detour.

Third, I think you might want to add a second half to the figure, with falsehoods and anti-theorems. Anti-theorems being the negation of theorems, statements that are provably false. This could lead to a figure that is too cluttered, though. Your call.

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