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The issue is the "run the turing machine for BB(748) steps" part. We don't know what BB(748) is. If the god of busy beavers came to us and told us that value, t
by sligocki 3y ago
The issue is the "run the turing machine for BB(748) steps" part. We don't know what BB(748) is. If the god of busy beavers came to us and told us that value, then we could (in theory) run the TM that long and just like you say, that would prove whether ZFC is consistent. But in order for us mere mortals to compute BB(748) we would effectively need to figure out if this specific 748-state TM ever halted (along with all the other 748-state TMs).
- dllthomas 3y agoWe don't need to know BB(748), just an upper bound on BB(748). ... which means we can't prove any upper bound on BB(748) within ZFC.
- kevinventullo 3y agoTo put it another way, an oracle telling us an upper bound on BB(748) would be strictly more powerful than an oracle telling us ZFC is consistent.
- deepsun 3y agoA lot of math draws conclusions from something not possible in practice (e.g. "let's take all the natural numbers and ..."). All the parent says: if it's possible even theoretically to learn the answer to the BB problem then we've proven something that cannot be proven, as shown by Godel incompleteness.