BB(3, 3) is Hard
I&#x27;m hoping someone can enlighten me here. My understanding is that there is a turing machine of 748 states [0], which halts iff ZFC is inconsistent (Thm 1). But this machine is a &quot;physical&quot; object, in the sense that we can materialize it on a computer and run it. Though we don&#x27;t have the computing power for this currently, there is nothing in principle stopping us from running this machine for BB(748) steps: if it halts, we have proven by Thm 1 that ZFC is inconsistent. If not, we have similarly proven that ZFC is consistent.<p>I want to stress that this is key to my confusion. This is not just some abstract result; this is a computation that <i>we can perform</i> and draw a real value from.<p>Of course, I&#x27;ll now fall back on godel&#x27;s second incompleteness theorem and say that one cannot prove, inside ZFC, that ZFC is consistent. But if the above turing machine halts, then we proved ZFC is consistent - a contradiction!<p>Where is the mistake here? My current guess is there is a technical detail in the proof of Thm 1 which uses a stronger metatheory than ZFC to show that the 758-state turing machine halts iff ZFC is inconsistent. This is not a contradiction, because yes, we can run the turing machine for BB(748) steps, but that will only show that ZFC+ proves ZFC is consistent, which is already well known - ZFC + there exists an inaccessible cardinal does the job.<p>However, I haven&#x27;t looked at the paper in detail to know whether this is the case. Does anybody who has thought deeply about this problem have insight to offer?<p>[0] <a href="https:&#x2F;&#x2F;www.ingo-blechschmidt.eu&#x2F;assets&#x2F;bachelor-thesis-undecidability-bb748.pdf" rel="nofollow noreferrer">https:&#x2F;&#x2F;www.ingo-blechschmidt.eu&#x2F;assets&#x2F;bachelor-thesis-unde...</a>