Abstract
We analyze N. C. A. da Costa and F. A. Doria’s “exotic formalization” of the conjecture P = NP [3–7]. For any standard axiomatic PA extension T and any number-theoretic sentence \({\varphi }\) , we let \({\varphi ^{\star} := \varphi \vee \lnot \mathsf{Con}\left( \mathsf{T}\right)}\) and prove the following “exotic” inferences 1–3. 1. \({\mathsf{T}+\varphi ^{\star}}\) is consistent, if so is T, 2. \({\mathsf{T}+\varphi}\) is consistent, provided that \({\mathsf{T}+\varphi ^{\star}}\) is ω-consistent, 3. \({\mathsf{T}+\varphi}\) is consistent, provided that T is consistent and has the same provably total recursive functions as \({\mathsf{T}+\left( \varphi \leftrightarrow \varphi ^{\star }\right) }\) . Furthermore we show that 1–3 continue to hold for \({\varphi ^{\star} := \varphi _{S} :=\varphi \vee \lnot S}\) , where \({S=\forall x\exists yR\left( x,y\right)}\) is any \({\Pi _{2}^{0}}\) sentence satisfying: 4. \({\left( \forall n\in \omega \right) \left( \mathsf{T}\vdash S_{x}\left[ \underline{n}\right] \right) }\) , 5. \({\mathsf{Con}\left( \mathsf{T}\right) \Rightarrow \mathsf{T}\nvdash S}\) . We observe that if \({\varphi :=\left[ \mathsf{P}=\mathsf{NP}\right] }\) and \({S:= \left[\digamma total\right] }\) , where \({\digamma=\digamma _{\mathsf{T}}}\) is da Costa-Doria “exotic” function with respect to T, then 4, 5 are satisfied for most familiar (presumably) consistent T in question, while \({\varphi _{S}}\) becomes equivalent to da Costa-Doria “exotic formalization” \({\left[ \mathsf{P}=\mathsf{NP}\right]^{\digamma}}\) . Moreover, the corresponding “exotic” inferences 1–3 generalize analogous da Costa-Doria results. Hence these “exotic” inferences are universal for all number-theoretic sentences and not characteristic to the conjecture P = NP. Nor do they infer relative consistency of P = NP (see Conclusion 15 in the text).
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References
Aaronson S.S.: Is P versus NP formally independent?. Bull. EATCS 81, 109–136 (2003)
Ben-David, S., Halevi, S.: On the Independence of P versus NP. Technion, TR 714 (1992)
da Costa, N.C.A., Doria, F.A.: Consequences of an exotic formulation for P = NP. Appl. Math. Comput. 145 (2003) 655–665 (2003) and Addendum. Appl. Math. Comput. 172, 1364–1367 (2006)
da Costa, N.C.A., Doria, F.A.: Computing the future, In: Computability, Complexity and Constructivity in Economic Analysis—Vela Velupillai K, ed. (2005)
da Costa, N.C.A., Doria F.A.: Some folklore about the P vs. NP question. (incorporated into [7])
da Costa N.C.A., Doria F.A.: Some thoughts on hypercomputation. Appl. Math. Comput. 178, 83–92 (2006)
da Costa N.C.A., Doria F.A., Bir E.: On the metamathematics of the P vs: NP question. Appl. Math. Comput. 189, 1223–1240 (2007)
Gordeev, L.: Toward combinatorial proof of P < NP. In: Proc. CiE 2006, Report #CSR 7-2006 119–128 (2006)
Kurtz S., O’Donnell M.J., Royer S.: How to prove representation-independent independence results. Inf. Proc. Lett. 24, 5–10 (1987)
Schindler R.D.: Review. Bull. Symb. Log. 10, 118 (2004)
Weiermann A.: Classifying the provably total functions of PA. Bull. Symb. Log. 12(2), 177–190 (2006)
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Gordeev, L. A note on da Costa-Doria “exotic formalizations”. Arch. Math. Logic 49, 813–821 (2010). https://doi.org/10.1007/s00153-010-0203-x
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DOI: https://doi.org/10.1007/s00153-010-0203-x