I am trying to understand Gödel's First Incompleteness Theorem. Since Imaginary numbers make mathematics closed, using Gödel's First Incompleteness Theorem it would mean there is an inconsistency. I would like to know if this thinking is correct.

$$\sqrt{x}\space\cdot\space\sqrt{x} = \sqrt{x^2} \rightarrow \lvert x \rvert $$ $$\therefore \sqrt{-1} \space \cdot \space \sqrt{-1} = 1$$ $$but \space i \space \cdot \space i = -1$$

Would this be considered an inconsistency in mathematics due to Gödel's Incompleteness Theorem?

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    $\begingroup$ What is the role of Godel's Incompleteness Theorem here ? $\endgroup$ Mar 14 '17 at 16:58
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    $\begingroup$ Your first equality only holds for $x>0$. And all this has nothing to do with Godel's theorem. $\endgroup$
    – Antoine
    Mar 14 '17 at 16:58
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    $\begingroup$ Why $\sqrt (-1) \times \sqrt (-1) = 1$ ? $\endgroup$ Mar 14 '17 at 16:59

You're wrong on several points.

  1. Imaginary numbers do not make "math complete". Adding $i$ to $\Bbb R$ makes the resulting extension algebraically closed.

  2. Yes, the theory of algebraically closed fields of characteristic $0$ is indeed complete. So by Gödel's theorem it has to either: (1) not be recursively enumerable; (2) not be strong enough to interpret arithmetic; or (3) be inconsistent.

    Of course, since $\Bbb C$ is a model of this theory, then (3) is off the table. This means either (1) or (2) must hold. Surprisingly, this theory is in fact recursively enumerable, since it simply states the field axioms, states that the field has characteristic 0, and that every polynomial has a root (these last two conditions are actually infinite schemata of axioms).

    This means that this theory does not interpret arithmetic. And now you could argue this is nonsense, since $\Bbb N$ is a subset of $\Bbb C$, and that addition and multiplication coincide with those of $\Bbb C$. This is all well and true, but nevertheless, what this means is that as a first-order structure, $(\Bbb N,+,\times,0,1)$ is simply not definable (or even interpretable, a more technical notion) in $\Bbb C$ (considered as a structure in the language of fields). This means that we cannot (in a first-order fashion) "talk about natural numbers" inside $\Bbb C$ using only $+,\cdot,0$ and $1$ and the field axioms.

So all in all, this is not going to be what renders mathematics inconsistent.

  • $\begingroup$ (I abusively did some minor editing.) $\endgroup$ Mar 14 '17 at 17:11
  • $\begingroup$ (You're an associate editor, so I trust your input on this entirely! ;-)) $\endgroup$
    – Asaf Karagila
    Mar 14 '17 at 17:29
  • $\begingroup$ (I will also make the excuse that I was on the bus that almost arrived to my stop, and my laptop was about to die out. So I had two reasons to write in haste. Thanks for the corrections!) $\endgroup$
    – Asaf Karagila
    Mar 14 '17 at 17:34
  • $\begingroup$ ( :-) )${{}{}}$ $\endgroup$ Mar 14 '17 at 20:39

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