On Silver’s Remark on the Ramseyness of Σ_{1}^{1} and Σ_{2}^{1}

{岩田竜之 \atop 学籍番号 31960105-6}

Summary

Our topic in this paper is subsets of reals (⊆ ^{ω}2) and their Ramseyness.

We think of ^{ω}2 as the product topological space of ω-many discrete spaces {0, 1}, which is a perfect Polish space, that is, a complete separable metric space without isolated points.

For X ⊆ ω, [X]^{ω} denotes the set of infinite subsets of X, and [X]^{<ω} denotes the set of finite subsets of X.

A ⊆ ^{ω}2 is said to be Ramsey when there is some infinite S ⊆ ω such that either [S]^{ω} ⊆ A or [S]^{ω} ∩ A is empty. If this holds, S is said to be homogeneous for A.

A ⊆ ^{ω}2 is said to be completely Ramsey when for every continuous function f : ^{ω}2 → ^{ω}2, f^{−1}(A), the inverse image of A, is Ramsey. For s ∈ [ω]^{<ω} and S ∈ [ω]^{ω} with min s < max S, [s, S] denotes {X ∈ [ω]^{ω} | s ⊆ X ⊆ s ∪ S}. Then if A is completely Ramsey and s, S are as above, there is S′ ⊆ S such that [s, S′] ⊆ A or [s, S′] ∩ A = 0.

Erdös and Rado [1] showed the existence of non-Ramsey subsets using the Axiom of Choice (AC). Mathias showed [4] that the use of AC is necessary. Thus this non-Ramsey subset is non-constructible. It is natural to ask how complex subsets must be to be Ramsey.

Galvin and Prikry [3] showed that every Borel subset is completely Ramsey.

J. Silver [2] proved the following:

1. Every Σ_{1}^{1} subset of ^{ω}2 is Ramsey.

2. If there is a measurable cardinal then every Σ_{2}^{1} subset of ^{ω}2 is also Ramsey.

In his paper Silver used the forcing notion making MA(ω_{1}) and remarked that instead of this forcing we can use a less complicated forcing notion.

In this paper we give the definitions of these concepts and a more detailed proof of Silver’s remark. We argue as follows in the case of Σ_{1}^{1}: Let A be a Σ_{1}^{1} set. It is known that every Σ_{1}^{1} set is split into ℵ_{1} Borel sets in a canonical way. We write this situation as A = ⋃_{α<ω_{1}} A_{α}. We start from a countable transitive model N of a sufficiently large fragment of ZFC. Next we show that there exists a p.o. set P_{F} ∈ N such that if G is any P_{F}-generic over N, then in N[G] there is a generic X ∈ [ω]^{ω} such that each A_{α} is open in [X]^{ω}. Then by the second countability axiom, there exists γ < ω_{1} such that

⋃_{α<γ}(A_{α} ∩ [X]^{ω}) = A ∩ [X]^{ω}

and this is open in [X]^{ω}. Using Galvin-Prikry’s results, there exists Y ⊆ X such that

[Y]^{ω} ⊆ ⋃_{α<γ} A_{α} , or

[Y]^{ω} ∩ A = ∅

In fact, it is in N[G] that this homogeneous situation holds. But this is described as a Π_{1}^{1} statement. Since it can be assumed that Π_{1}^{1} is absolute for N[G], this holds in V, too. Thus A is Ramsey.

Following this method of proof, we can show that

1. Every Σ_{1}^{1} subset of ^{ω}2 is completely Ramsey.

2. Every provably Δ_{2}^{1} subset of ^{ω}2 is completely Ramsey.

3. For a ⊆ ω, if ω_{1}^{L[a]} < ω_{1} then every Σ_{2}^{1}(a) subset of ^{ω}2 is Ramsey.

We prove the first result to demonstrate our method of proof and the difference between Silver’s and ours.

Then, by a similar proof we extend the first result to provably Δ_{2}^{1} subsets.

Finally, the simplicity of our forcing notion enables us to find the weakest cardinality hypothesis and to effectivize Silver’s result.

These results are optimal in the following sense: Since ZF+“V = L” implies the existence of a Δ_{2}^{1} non-Ramsey set, we cannot prove Δ_{2}^{1}’s Ramseyness from ZF only. Similarly, ZF+“V = L[μ]” (the universe is the inner model of measurability) implies the existence of a Δ_{3}^{1} non-Ramsey set. So we cannot prove Δ_{3}^{1}’s Ramseyness even from ZF+“∃measurable”.

Moreover note that our theorems are proved completely within ZF. Silver’s paper did not show that AC is not necessary.

References

[1] P. Erdös and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proceedings of the London Mathematical Society, vol. 2 (1952), pp. 417–439.

[2] J. H. Silver, Every analytic set is Ramsey, Journal of Symbolic Logic, vol. 35 (1970), pp. 60–64.

[3] F. Galvin and K. Prikry, Borel sets and Ramsey’s theorem, Journal of Symbolic Logic, vol. 38 (1973), pp. 193–198.

[4] A. R. D. Mathias, Happy families, Annals of Mathematical Logic, vol. 12 (1977), pp. 59– 111.

[5] A. Kanamori, The Higher Infinite, Springer, Berlin, 1994.