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I read that the reason Godel's incompleteness theorem doesn't apply to reals is that the axioms of real numbers aren't strong enough to produce statements about natural numbers arithmetic. And if you can't form such statements, you can no longer form the statement "This statement is non-demonstrable" using the theory. So Godel's proof doesn't apply.

I think one may use trigonometric functions to make statements about natural numbers in the real number system. For example, Fermat's last theorem can be stated as:

$$\nexists (x, y, z) : (x, y, z) \in R \wedge (x^3+y^3=z^3) \wedge (\sin(x\pi) =\sin (y\pi) =\sin(z\pi) =0)\wedge (x, y, z) >0$$

$\wedge$ stands for AND. The AND clause with the $\sin$ functions ensures that $x$, $y$ and $z$ are integers. $x, y, z>0$ ensures that all three numbers are positive integers, i.e. they are natural numbers.

Similarly, we can also make statements about divisibility. To declare that the quotient $q$ must be an integer, we can use the condition $\sin (q\pi) =0$

What's wrong with doing this? This shows that we can make statements about naturals in the real number system

Ryder Rude
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  • Where did you read that? The real numbers are an ordered field, so they have ${0,1,1+1,1+1+1,\ldots}$ as a subset, corresponding to the natural numbers – Henry Feb 24 '23 at 13:11
  • @Henry They are a subset in the trivial sense but not in the context of completeness of axioms. Natural numbers supposedly have a richer structure. See the answer to: https://math.stackexchange.com/questions/362837/are-real-numbers-axioms-a-consistent-or-complete-system – Ryder Rude Feb 24 '23 at 13:14
  • See this question – lulu Feb 24 '23 at 13:59
  • The axioms of real closed fields alone can't show the trig functions exist (which is good because they don't exist as functions $\mathbb A\rightarrow\mathbb A$). However, I remember reading that adding a function symbol $\sin$ that works as you'd expect does make the theory undecidable for essentially the reason you lay out here. It is an open problem whether adding $\exp$ makes the theory undecidable. – eyeballfrog Feb 24 '23 at 14:35
  • @eyeballfrog I don't understand. $\sin$ is just a formula applied on reals. It's just multiplication and addition and limits. How do the real number axioms not allow us to talk about $\sin$? Do the axioms not include the concept of limits? – Ryder Rude Feb 24 '23 at 14:53
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    OK, I think I see the confusion now. The decidable theory is that of real closed fields. That is, all statements expressible in first-order logic using the language $(0, 1, +, *, <)$ that are true for the real numbers. This is not the full theory of the real numbers, though, because the least upper bound axiom (which is needed for limits) is not a statement of first-order logic. Adding that axiom makes the theory undecidable. – eyeballfrog Feb 24 '23 at 15:09
  • @eyeballfrog is the discipline of calculus not complete because it requires limits? This means Godel's theorem applies to physics which uses calculus? – Ryder Rude Feb 24 '23 at 15:12
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    The theorem that adding sin to the reals makes its first order theory undecidable is https://en.wikipedia.org/wiki/Richardson%27s_theorem . You are correct that (real) limits are describable in the first-order logic of (R,+,*). What formula are you using to define sin for the reals? The limit of the Taylor series? This has other pieces which are not describable in this first-order logic (natural number indexes, arbitrary length sums/products). – TomKern Feb 24 '23 at 15:21
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    Godel's theorem definitely applies to calculus. As to physics, that's more complicated. While physics is usually stated in mathematical terms, the incompleteness theorem could only apply if the fragment of mathematics needed to formulate the laws of physics is itself undecidable. It is not known how much of mathematics is needed for all current physical theories. Fields' axiomatization of Newtonian gravity without any mathematics suggests it may be less than you'd think. – eyeballfrog Feb 24 '23 at 15:31

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