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Mathematics several times has statements of form $$\mathsf{Statement A}\implies\mathsf{Statement B}$$ where $\mathsf{Statement A}$ and $\mathsf{Statement B}$ are conjectures while the implication is provable.

In such cases falsity of $\mathsf{Statement B}$ implies falsity of $\mathsf{Statement A}$. However since falsity of $\mathsf{Statement A}$ does not imply falsity of $\mathsf{Statement B}$ it might be that disproving $\mathsf{Statement B}$ might be the easiest route to disproving $\mathsf{Statement A}$. However a direct disproof of $\mathsf{Statement A}$ might reveal something else not directly revealed by $\mathsf{Statement B}$ without falsifying $\mathsf{Statement B}$. Are there known good examples?

It would be nice if falsity of $\mathsf{Statement B}$ came before falsity of $\mathsf{Statement A}$ which would make for the case that falsity of $\mathsf{Statement B}$ was indeed easier.

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    $\begingroup$ Please rewrite the actual question so it is clearer exactly what you seek an example of. I can’t tell what aspect of your Mertens/RH example makes it a full or partial example of what you are interested in. And the title of the post is quite hard to parse. $\endgroup$
    – KConrad
    Commented Aug 12, 2020 at 23:07
  • $\begingroup$ Mertens/RH is a partial example. I am unable to come up with one concrete example even though the possibility exists. $\endgroup$
    – VS.
    Commented Aug 13, 2020 at 0:08
  • $\begingroup$ Both statements are conjectures and the implication is known result. $\endgroup$
    – VS.
    Commented Aug 13, 2020 at 2:05

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Here's an example from computability theory.

Statement $A$:"The Turing degrees are linearly ordered".

Statement $B$: "The $\Sigma^0_1$ Turing degrees are linearly ordered".

Statement $A$ was refuted by Kleene and Post 1954, with a construction that foreshadows the notion of forcing made famous by Cohen and the continuum hypothesis.

Statement $B$ was refuted by Friedberg and Muchnik in 1957 using another revolutionary method, the priority argument.

Even though the method for Statement $B$ is more powerful in its context, the "easier" method for Statement $A$ turns out, when viewed in the right light, to be an early version of a perhaps even more influential method.

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  • $\begingroup$ If $1957$ came before $1954$ this would have been a perfect example. $\endgroup$
    – VS.
    Commented Aug 13, 2020 at 2:04
  • $\begingroup$ I wouldn't say the Kleene-Post construction foreshadows forcing, except in the sense that the Baire category theorem foreshadows forcing. The Kleene-Post construction is essentially a category argument, showing that the Turing-incomparable pairs of reals form a comeager set. $\endgroup$
    – Andreas Blass
    Commented Aug 13, 2020 at 3:13
  • $\begingroup$ @AndreasBlass Would you say Spector's minimal degree construction foreshadows/is forcing? $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Aug 13, 2020 at 3:35
  • $\begingroup$ My (non-expert) impression is that Spector's construction is essentially what one must add to the original forcing ideas of Cohen to produce Sacks forcing and a minimal degree of constructibility. $\endgroup$
    – Andreas Blass
    Commented Aug 13, 2020 at 3:42

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