Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic. KEY WORDS: Hypercomputation; quantum mechanics; recursion theory; infinite parallelism.

this technical convenience. However, the most important result of this book is that the two senses coincide; we will prove that in the section after this one

N natural numbers: {0, 1, 2,...} C complex numbers {... ∣...} set of... such that... (a.. b), [a.. b] interval (open or closed) of reals between a and b 〈... 〉 sequence; like a set but order matters V, W, U vector spaces ⃗v, ⃗w vectors ⃗0, ⃗0V zero vector, zero vector of V B, D bases En = 〈⃗e1,..., ⃗en 〉 standard basis for Rn ⃗β, ⃗δ basis vectors RepB(⃗v) matrix representing the vector Pn set of n-th degree polynomials Mn×m set of n×m matrices [S] span of the set S

R real numbers N natural numbers: {0, 1, 2,...} C complex numbers {... ∣...} set of... such that... 〈... 〉 sequence; like a set but order matters V, W, U vector spaces ⃗v, ⃗w vectors ⃗0, ⃗0V zero vector, zero vector of V B, D bases En = 〈⃗e1,..., ⃗en 〉 standard basis for Rn ⃗β, ⃗δ basis vectors RepB(⃗v) matrix representing the vector Pn set of n-th degree polynomials Mn×m set of n×m matrices [S] span of the set S

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R real numbers

Abstract. We investigate the extent of second order characterizable structures by extending Shelah’s Main Gap dichotomy to second order logic. For this end we consider a countable complete first order theory T. We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second order logic with Henkin semantics. Contents