We show that the arithmetical theory T 0 2 +Σ̂ b 1-IND |x|5 , formalized in the language of Buss, i.e. with bx/2c but without the MSP function bx/2yc, does not prove that every nontrivial divisor of a power of 2 is even. It follows that this theory proves neither NP = coNP nor S0 2 . Some arithmetical theories are not merely weak but very weak, in the sense… (More)