Abstract
Emergence and development of recursion theory and computer science enable us to rigorously address the question of characterising the class of mathematical concepts that are cognitively accessible to computational devices such as human minds. The answer to this question would give reasons for which some concepts are epistemically easy (e.g. provable within first-order theories or possessing certain combinatorial properties) and the others are cognitively hard for the human mind.
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- 1.
The author was supported by The National Science Centre, Poland (NCN), grant number 2013/11/B/HS1/04168.
- 2.
We obviously read \(\forall x \leq y\varphi (x)\) as \(\forall x(x \leq y \Rightarrow \varphi (x))\) and \(\exists x \leq y\varphi (x)\) as \(\exists x(x \leq y \wedge \varphi (x))\).
- 3.
Instead of this we can denote it more easily: \(Cn(PA+\{\varphi \in \Pi _{1}^{0}: \mathbb{N}\models \varphi \})\) is not \(\Delta _{2}^{0}\).
- 4.
Another way to see that X is \(\Pi _{1}^{0}\)-hard – explicitly using diagonalization – would be as follows (the argument below is a quotation of E. Jerabek – a proof given in the communication via Internet, see: www.mathoverflow.net/questions/63690):
Let \(\sigma (x) =\exists v\:\theta (x,v)\) be a complete \(\Sigma _{1}^{0}\)-formula (such that it is not equivalent to any \(\Delta _{0}^{0}\)-formula, where \(\theta \in \Delta _{0}^{0}\), and find a formula π(x) such that PA proves
$$\displaystyle{ \pi (x) \equiv \forall w\:(\mathrm{Prov_{PA}}(w,\ulcorner \pi (\dot{x})\urcorner ) \Rightarrow \exists v \leq w\:\theta (x,v)) }$$by the diagonal lemma. Let n ∈ ω. Since \(\neg \pi (\bar{n})\) is equivalent to a \(\Sigma _{1}^{0}\) sentence, PA proves \(\neg \pi (\bar{n}) \Rightarrow \mathrm{ Pr_{PA}}(\ulcorner \neg \pi (\bar{n})\urcorner )\). By definition, \(\neg \pi (\bar{n}) \Rightarrow \mathrm{ Pr_{PA}}(\ulcorner \pi (\bar{n})\urcorner )\), hence PA proves \(\mathrm{Con_{PA}} \Rightarrow \pi (\bar{n})\). We claim that
$$\displaystyle{ ({\ast})\:\:\:\mathbb{N}\models \sigma (n)\Longleftrightarrow\mathrm{PA} \vdash \pi (\bar{n}), }$$which means that \(n\mapsto \ulcorner \pi (\bar{n})\urcorner\) is a reduction of the \(\Pi _{1}^{0}\)-complete set \(\{n: \mathbb{N}\models \neg \sigma (n)\}\) to X.
To show (∗), assume first that \(\mathcal{M}\models \mathrm{PA} +\neg \pi (\bar{n})\). Then there is no standard PA-proof of \(\pi (\bar{n})\), hence the witness \(w \in \mathcal{M}\) to the leading existential quantifier of \(\neg \pi (\bar{n})\) must be nonstandard. Then ¬ θ(n, v) holds for all v ≤ w, and in particular, for all standard v, hence \(\mathbb{N}\models \neg \sigma (\bar{n})\).
On the other hand, assume that PA proves \(\pi (\bar{n})\), and let k be the code of its proof. Since PA is sound, \(\mathbb{N}\models \pi (\bar{n})\), hence there exists v ≤ k witnessing \(\theta (\bar{n},v)\), i.e. \(\mathbb{N}\models \sigma (\bar{n})\), which ends the proof.
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Godziszewski, M.T. (2017). Experimental Logics as a Model of Development of Deductive Science and Computational Properties of Undecidable Sentences. In: Hofer-Szabó, G., Wroński, L. (eds) Making it Formally Explicit. European Studies in Philosophy of Science, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-55486-0_13
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