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I am trying to pin down: who first proved that $\mathsf{WKL}_0$ has an $\omega$-model in which every set is of low degree? As shown in Simpson's Subsystems of Second Order Arithmetic (Theorem IX.2.17), one doesn't have to travel too far to get to this result when starting from the low basis theorem of Jockusch and Soare. But Simpson's typically extensive bibliographic remarks on the section in which this result is proved offer no indication of when this fact about $\mathsf{WKL}_0$ was first noted. Thanks.

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$WKL_0$ was introduced by Friedman in Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp. 235–242. The observation you mention is not explicit there, but he proves that the sets in any $\omega$-model of $WKL_0$ form a Scott system. From this, it is literally just the low basis theorem to obtain the result. (Cont.) –  Andrés Caicedo Aug 9 '13 at 3:14
This leads me to believe there is no paper where this was formally stated before Subsystems, and it was considered part of the folklore, due to either Harvey of Stephen. –  Andrés Caicedo Aug 9 '13 at 3:15
@Andres: In your first comment, do you mean "he proves that any Scott system is the collection of sets in some $\omega$-model of $WKL_0$?" Just showing that each $\omega$-model of $WKL_0$ is a Scott set doesn't help - the same is true of $ACA_0$, or any stronger theory. (That said, you should post this as an answer.) –  Noah Schweber Aug 9 '13 at 5:28
Oh, yes. I meant to say that the $\omega$-models of $WKL_0$ are precisely the Scott systems. I will look a bit more in the literature to see if there is something I'm missing. If in a few days nothing is unearthed, I'll expand the comments into an answer. –  Andrés Caicedo Aug 9 '13 at 6:07
I edited the question primarily to add the reverse-math tag –  Carl Mummert Aug 10 '13 at 12:33

1 Answer 1

up vote 4 down vote accepted

The general issue here is that there are some results in Reverse Mathematics that are very little besides rephrased computability theory results. In such cases, it is common for nobody to take credit for the Reverse Mathematics result.

Another example of this is the theorem that $\mathsf{RT}^3_2$ implies $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. Jockusch's paper on Ramsey theory (from 1972) is phrased only in terms of computability and predates the definitions of Reverse Mathematics. So there is no canonical source for the Reverse Mathematics result about $\mathsf{RT}^3_2$, although it would have been obvious to anyone who knew the definitions and Josckush's paper.

Friedman's theorem 1.3 from his 1974 paper "Some Systems of Second Order Arithmetic and Their Use" is certainly enough, with the common knowledge about Scott systems at the time, for someone from that era to prove the existence of a low $\omega$-model of $\mathsf{WKL}_0$. The low basis theorem was published by Jockusch and Soare in 1972. But Friedman seems to have had something else in mind, because in his comments after the theorem he refers to a "$\Delta^0_2$-complete" completion of PA rather than a low completion of PA.

The existence of low $\omega$-models of $\mathsf{WKL}_0$ is also closely related (in my mind at least) to the density of the PA-over relation. Recall $C \ll B$ ("$B$ is PA over $C$") if $B$ computes a path through each infinite subtree of $2^{<\omega}$ that is computable from $C$ (this is equivalent to: $B$ computes a completion of PA with $C$ as an additional predicate). Simpson states in his article in the Handbook of Mathematical Logic (1978) that if $C \ll B$ there is a $D$ with $C \ll D \ll B$. This also immediately gives an $\omega$-model of $\mathsf{WKL}_0$ where every set is low: let $0 \ll B$ where $B$ is low, make a sequence $0 \ll C_1 \ll C_2 \ll \cdots \ll B$, and let $M = \{ X : (\exists i)[X \leq_T C_i]\}$. I have no idea exactly when in the 1970s Simpson proved that density result.

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