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Z is designed to loop through its 7918 instructions forever, but if it did eventually stop, it would prove ZFC inconsistent. Mathematicians wouldn’t be too pani

Z is designed to loop through its 7918 instructions forever, but if it did eventually stop, it would prove ZFC inconsistent. Mathematicians wouldn’t be too panicked, though – they could simply shift to a slightly stronger set of axioms. Am I correct in thinking that this is completely wrong? Strengthening a set of inconsistent axioms -- taking a "stronger" set of axioms to be a set of axioms that can prove more statements -- will just produce another inconsistent set of axioms. (The inconsistent set <A, B->~A, B> cannot be salvaged by extending it to <A, B->~A, B, C> -- no matter what C is, you can always derive <A, ~A> from the original three axioms.) Incompleteness is a whole other thing, of course.

toth 10y ago
You are correct. They would have to drop an axiom to hope to go from inconsistent to consistent. E.g., go from ZFC to ZF (drop the axiom of choice). If ZF is inconsistent... I think that would make a lot of mathematicians panic.
saghm 10y ago
I noticed the same thing while reading it. I assume that this is just a mistake or misunderstanding by the author, and they actually meant "weaker". Of course, switching to a weaker set of axioms would mean that some of the things proved by the old set might not be provable anymore, which still seems like an issue...
- dnautics 10y ago
  As a mathematician, I a appreciate the notion that a stronger set of axioms means more power and a smaller and more defined working universe, but I think the author meant "stronger" against contradiction, which is consistent with colloquial use, if not jargon.
spott 10y ago
I think this turing machine will only halt if ZFC is inconsistent, not if some other set theory with different axioms is inconsistent.
euyyn 10y ago
I interpreted "stronger" as "more constraining", which would mean the opposite: able to prove less statements.

toth 10y ago

You are correct. They would have to drop an axiom to hope to go from inconsistent to consistent. E.g., go from ZFC to ZF (drop the axiom of choice). If ZF is inconsistent... I think that would make a lot of mathematicians panic.

saghm 10y ago

I noticed the same thing while reading it. I assume that this is just a mistake or misunderstanding by the author, and they actually meant "weaker". Of course, switching to a weaker set of axioms would mean that some of the things proved by the old set might not be provable anymore, which still seems like an issue...

dnautics 10y ago
As a mathematician, I a appreciate the notion that a stronger set of axioms means more power and a smaller and more defined working universe, but I think the author meant "stronger" against contradiction, which is consistent with colloquial use, if not jargon.

spott 10y ago

I think this turing machine will only halt if ZFC is inconsistent, not if some other set theory with different axioms is inconsistent.

euyyn 10y ago

I interpreted "stronger" as "more constraining", which would mean the opposite: able to prove less statements.