TITLE: Tarski's decidability proof on real closed field and Peano arithmetic
QUESTION [23 upvotes]: It seems very confusing that real closed field (which also can be used as the theory of real number) is decidable, while Peano arithmetic, which seems to be a subset of real closed field is undecidable.
What am I getting wrong?

REPLY [6 votes]: As others point out, the main reason for this seemingly paradoxical situation (all natural numbers are real numbers but the theory of the first one is much more complicated than the second one) is because natural numbers are not definable in the language of RCF.
We can use the same language to talk about them, say $\{0,1,+,\cdot\}$ and natural numbers are a substructure of real numbers. However being a subsructure doesn't make things simpler. Take for example prime numbers, do you expect that their theory in the same language is simpler than the theory of natural numbers? It doesn't need to be the case. The main issue as others have point out is quantification. Quantification over natural numbers can generate very complicated sets (arithmetical hierarchy) where as quantification over reals does not, in fact if we simply add order to the language then every quantified formula is equivalent formula with no quantifiers for reals, i.e. quantifiers do not increase the complexity of sets we consider for reals and can be eliminated (and that is how it is proven that RCF is decidable). We can show that this is not possible for natural numbers, quantifiers can increase the complexity of sets. The undecidability follow essentially from the fact that we can talk about computation of machines in the theories like PA and much weaker theories about natural numbers. 
It might be helpful to look at something in between: consider integers. We can define natural numbers by using the fact that every natural number is equivalent to sum of four squares. Therefore they are going to be as complex as natural numbers. Another more tricky set in the middle: consider rational numbers. Their theory is similarly undecidable, however showing that we can define natural numbers in the their theory is much more complicated (this nice result is originally due to Julia Robinson). Another example: consider complex numbers. Their theory ACF is kind of even simpler than real numbers.
To make it even more clear, if we just add a predicate to the language that tells us which real numbers are natural numbers, then theory of real numbers with this predicate becomes undecidable. The main issue is that without such predicate natural numbers are indistinguishable from other real numbers in the language.