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I'm searching for a reference of an undecidability proof that is as simple as possible and starts "from scratch".

With "from scratch" I mean that it does not use some other undecidable problem to prove some undecidability (which is the usual case), I cannot wrap my mind about how proving undecidability that way (without a previous proof) could be possible.

This question may be inspiring: An example of an easy to understand undecidable problem

Also, I know this is probably not very objective, but it is important to me, it should be something as simple as possible, hopefully enough so that even I can understand it.

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Trylks
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    The first languages to be proven undecidable were - if I'm not mistaken - shown to be so via diagonalization. Unless you need a constructive proof, the most simple proof is that there is an uncountable number of languages but only a countable number of algorithms (since each word encodes at most one algorithm, and words are countable). If you are looking for a constructive proof, I would recommend to read Arora & Barak: Computational Complexity - A Modern Approach, p.25 under the heading "1.4 Uncomputable Functions". – G. Bach Oct 03 '13 at 15:22
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    G.Bach: diagonalization is a constructive proof, it gives an explicit language which is not decidable. – sdcvvc Oct 03 '13 at 16:49
  • What's wrong with a basic proof of the undecidability of the halting problem? Consult, for instance, http://www.comp.nus.edu.sg/~cs5234/FAQ/halt.html – Patrick87 Oct 03 '13 at 20:11
  • Since you don't give us any indication of which proof you don't get, I think this is covered by our reference question. There is, of course, always my favorite $\pi$ question for developing intuition. – Raphael Oct 04 '13 at 06:52
  • I found a very simple proof. I don't think it can get any more simple than that. – Trylks Jul 24 '14 at 10:15

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to some degree all refs on the subject have a formidable inherent complexity due to the subject matter. however, here is a classic, highly accessible ref on Gödel's thm that may be helpful, written in "popular science" style by a philosopher & mathematics historian for a broad audience, breaking it down into intuitive everyday language.

here are some other similar but more technical refs that may be helpful

there are also many refs on Turing's halting problem. one worth mentioning, a step-by-step walkthrough/tour of the original proof, but which is very advanced due to retaining all the mathematical complexity/detail:

on the other hand most of the proof complexity of the halting problem is associated with the proof of the existence of the Universal TM, wikipedia has a pretty good outline. this can be understood as a simulation or emulation of one machine by another. therefore studying the theory of machine emulation (now quite frequently applied with video games) may be helpful.

vzn
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    Since this is CS Stack Exchange, I assume the OP means "undecidability" in the sense of Turing machines, not Goedel. – David Richerby Oct 03 '13 at 19:28
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    DR to neophytes they seem dissimilar or disconnected but to theoreticians they are seen as basically instances of the same phenomenon. but agreed, wish there were more refs that were TM focused instead of Godel-focused, just citing what is available. – vzn Oct 03 '13 at 20:31