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# Tag Info

Accepted

### To what extent can the mathematics of Reals be applied to Computable Reals?

The real numbers may be characterized in a couple of ways, let us work with the Cauchy-complete archimedean ordered field. (We need to be a bit careful how exactly we say this, see Definition 11.2.7 ...

### What is the axiomatic (set theory) context of the P vs NP and NP=EXPTIME conjectures?

It's not specified. When there is a serious enough candidate paper purporting to resolve P ≟ NP, a Special Advisory Committee will be formed to decide whether (and to whom) to award the prize. I ...
Accepted

### Standard reference for basic model theory definitions

Here is one possibility, but other people might use different words. I will use first-order logic as a running example. Language The language is a collection of expressions, which are syntactic ...
Accepted

### Is it decidable that a computable analytic function over $\mathbb{R,C} ,$ equals $0$

No, it is not decidable. A good heuristic to answer such questions is the following: every computable map is continuous. If you could decide whether $f(x) = 0$ for all $x \in \mathbb{C}$, then the ...
Accepted

### Decidability of first-order theory of real closed fields with functions

It's undecidable because we can interpret natural numbers (with addition and multiplication). For example, let $Ind(f)$ be the formula: $$f(0)=1 \land \forall x \ \big(f(x)=1 \to f(x+1)=1\big)$$ Now ...