Is it known to be consistent with ZFC for there to exist a Turing degree invariant projective set which neither contains nor is disjoint from a cone? What about in $L$, i.e., is it known that (the consequence of) Martin's cone theorem fails in $V=L$
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- 6$\begingroup$ In L, you could get an example by considering whether the least ordinal not recursive in $x$ is a successor or limit admissible. $\endgroup$Theodore Slaman– Theodore Slaman2025-07-01 07:29:00 +00:00Commented Jul 1 at 7:29
- 1$\begingroup$ @TheodoreSlaman Since comments are ephemeral I've taken the liberty of expanding your comment into an answer. Feel free to modify in any way (also if you add an answer of your own I'll delete mine). $\endgroup$Noah Schweber– Noah Schweber2025-08-24 19:21:49 +00:00Commented Aug 24 at 19:21
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I'm belatedly turning Ted Slaman's comment into an answer for posterity; if he adds an answer I'll delete this one.
Say that a real $x$ is nice if $\omega_1^{CK}(x)$, the smallest ordinal with no $x$-computable copy, is a limit of admissible ordinals. Since every admissible ordinal is $\omega_1^{CK}(x)$ for some $x$, nice ordinals certainly exist. The claim now is that in $L$ or similar models, the set $N$ of nice reals neither contains nor avoids a cone.
$N$ does not contain a cone (assuming $V=L$): given a real $x$, let $\alpha$ be the smallest admissible ordinal such that $x\in L_\alpha$ and let $\beta$ be the next admissible above $\alpha$. Letting $g$ be generic over $L_\beta$ for $Col(\omega,\alpha)$, we get that $\omega_1^{CK}(g\oplus x)=\beta$: that the left hand side is $>\alpha$ is trivial, and that the left hand side is $\le\beta$ follows from the fact that $L_\beta[g]$ is admissible since set forcing preserves admissibility. So $x\le_T x\oplus g\not\in N$.
- This argument can be used to motivate the notion of the admissibility spectrum, and in particular the existence of a real whose admissibility spectrum contains only limit admissibles is a "strong anti-$L$" property (cf. "Harrington's $\star$").
$N$ does not avoid a cone (in ZFC alone): The proof that there is a nice real in the first place relativizes to get a nice real Turing-above any fixed $x$. (So while the construction of a nice real is much trickier than the construction of a non-nice real, it's also a much more robust argument in a sense.)
Besides canonicity, what I really like about this example is that it's clear which way the cone is tilting so-to-speak rather than having both parts rely on V=L.
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If you have a projective well-ordering of the reals in order type $\omega_1$, as you do in $L$, then you can simply build one by transfinite recursion. Well order the reals, which gives you also a well-ordering of the Turing degrees. There are continuum many cones, and so we can build our desired set in a process of length $\omega_1$ by adding a degree to our set and a degree to the complement of our set. At any stage, we've made only countably many promises, so there are more available. This whole construction is projectively definable if the underlying well-order is projective.
To be more concrete, if there is a projectively definable well order of the real numbers in type $\omega_1$, then there is a projectively definable $\omega_1$ sequence of reals that is cofinal in the Turing degrees. And now one can simply take every other point of this sequence or more generally divide it into two cofinal sets (such as limits/successors, or compound limits/simple limits, etc.), and neither set will contain or omit a cone. The other answers on this question can be seen as instances of this.
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1 If $V=L$ the $\Pi^1_1$ set $C_1$ of "quickly constructible" reals $$ C_1 := \{\,x\in\mathbb{R}\,:\, x\in L_{\omega_1^x}\,\} $$ gives a counterexample. Every real in $L$ is recursive in a member of $C_1$. So $C_1$ meets every Turing cone. On the other hand, for every real $x$ you can find a Cohen real $c$ over $L_{\omega_1^x}[x]$, so that $\langle x,c\rangle$ is a real Turing-above $x$ and $\langle x,c\rangle\notin C_1$.
- $\begingroup$ Another nice and canonical example! $\endgroup$Noah Schweber– Noah Schweber2025-09-12 00:55:20 +00:00Commented Sep 12 at 0:55