Why is Godel's first theorem not a proof for the truth of the so called undecidable proposition? You may say it's a proof from the outside, but if not all proofs from the outside be formalized inside the system then the system is really not powerful.
Why is Godel's first theorem not a proof from the inside?
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$\begingroup$
logic
incompleteness
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1@Zirui Wang: (cont...) Finally, as I noted above, the argument Goedel makes is that **if** the theory is consistent, then $G$ must be true (in the standard model). This is in fact the statement $\mathrm{Con}(T)\rightarrow G$. And the Second Incompleteness Theorem shows that in fact you **can** formally prove $\mathrm{Con}(T)\rightarrow G$ in the theory; this is why we know that you cannot formally prove $\mathrm{Con}(T)$ (because we know we cannot formally prove $G$, and if you could formally prove $\mathrm{Con}(T)$, then by modus ponens you would be able to formally prove $G$). – 2011-01-12
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Because by Tarski's undefinability of truth theorem one can not even define the truth of formulas of a strong enough system inside it (because it leads to the famous liar's paradox). You can say that the system is not "powerful", but then there is no reasonable "powerful" systems at all.
(One can say the same thing about any impossible thing but that is not a good argument, untenability to do an impossible thing is not a weakness.)