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The proof-theoretic strength of a theory is measured by the $\mathsf{\Pi}_{1}^{1}$-ordinal of the theory (indeed, there are other ordinal analyses, like the $\Pi_{2}^{0}$-ordinal of the theory).

Consider the following statements:

Finite Ramsey Theorem ($\mathrm{FRT}$): For any $r,k,p\in\mathbb{N}$, there exists $N\in\mathbb{N}$ such that for any set $S$ of cardinality $N$ and any function $c:[S]^{p}\rightarrow [r]$ there exists a subset $H\subseteq S$ of cardinality $k$ such that $c|_{[H]^{p}}$ is constant.

By $[r]$ we mean the set $\{1,\ldots r\}$, and the function $c$ is often called an $r$-coloring of the $p$-element subsets of $S$. The subset $H$ is called a homogeneous $k$-subset of $S$.

What is the proof-theoretic strength of $\mathrm{FRT}$? My understanding is that it should not be more than $\omega^{\omega}$ because $\mathsf{RCA}_{0}+\mathsf{IΣ}_{1}$ proves $\mathrm{FRT}$ since we are essentially applying induction to the formula: "there exists an $N$ such that every $r$-coloring of the $p$-element subsets of $\{1,\ldots,N\}$ has a homogeneous $k$-subset". But $\mathsf{RCA}_{0}+\mathsf{IΣ}_{1}$ seems more than we really need, and it may very well be that the proof-theoretic strength is $\omega^{n}$ for some low $n\ge 1$.

Infinite Ramsey Theorem ($\mathrm{IRT}$): For any $r,p\in\mathbb{N}$, and for any function $c:[\mathbb{N}]^{p}\rightarrow [r]$, there exists an infinite subset $H\subseteq \mathbb{N}$ such that $c|_{[H]^{p}}$ is constant.

Let $\mathrm{IRT}(p)$ denote the instance of $\mathrm{IRT}$ for exponent $p$. That is, $\mathrm{IRT}$ is simply the statement $\forall p\,\,\mathrm{IRT}(p)$.

First, we consider the case of $A$ countable:

  1. Over $\mathsf{RCA}_{0}$, $\mathsf{ACA}_{0}$ proves $\mathrm{IRT}(P)$, $p\geq 0$ (Lemma III.7.4, p.123 of 1). Therefore, the proof-theoretic strength of $\mathrm{IRT}(p)$, $p\ge 0$ is at most $\varepsilon_{0}$. Moreover, Theorem III.7.6, p.124 of 1 shows that over $\mathsf{RCA}_{0}$, $\mathsf{ACA}_{0}$ is equivalent to $\mathrm{IRT}(p)$, $p\ge 3$. Therefore, the proof-theoretic ordinal of $\mathrm{IRT}(p)$, $p\geq 3$ is indeed $\varepsilon_{0}$. The case $\mathrm{IRT}(0)$ is trivial. Does the proof-theoretic strength of $\mathrm{IRT}(1)$ (i.e. the infinite pigeonhole principle) equal $\omega^{\omega}$? What is the proof-theoretic strength of $\mathrm{IRT}(2)$?
  2. Remark III.7.7, p.124 in 1 indicates that the statement $\forall p\,\,\mathrm{IRT}(p)$ is provable from $\mathsf{ACA}_{0}$ plus $\mathsf{\Pi}_{2}^{1}$-induction over $\mathsf{RCA}_{0}$. From this I gather that the proof-theoretic strength of $\mathrm{IRT}$ could be $\varepsilon_{\alpha}$ for some ordinal $\alpha$ (possibly $\alpha=\omega$?) But I am not sure. What is the proof-theoretic strength of IRT?

Second, we consider $A$ arbitrary infinite: Then we must consider $\mathrm{IRT}$ as a statement of $\mathsf{ZFC}$, for it has been shown in [2] that $\mathrm{IRT}$ is not a theorem of $\mathsf{ZF}$ and that some form of the Axiom of Choice is necessary to prove it. What is the proof-theoretic strength of $\mathrm{IRT}$ in this context?

Finally, if we modify $\mathrm{IRT}$ above by exchanging the sentence "Let $A$ be an infinite set" with "For any infinite set $A$"- does this change any of the results discussed in the first case above (i.e. when infinite means countably infinite)? What about the second case (i.e. when infinite means arbitray infinite)

1 Simpson, S. S., Subsystems of Second Order Arithmetic 2nd. ed. Cambridge University Press. Cambridge-New York (2009)

[2] Kleinberg, E. M., The independence of Ramsey's theorem. The journal of symbolic logic, vol. 34 (1969), pp. 205–206.

John
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1 Answers1

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In order to remove the question from the unanswered list, I will post as an answer the edit to the original question where I had been adding what I found out.

  1. Note that $\mathsf{FRT}$ is both expressible and provable in $\mathsf{PA}$ (In fact, it is expressible as a $\mathsf{I\Delta}_{0}(\exp_{2})$-formula but it is independent of $\mathsf{I\Delta}_{0}(\exp_{2})$ - see [*] p.188). However, in Chapter II.1 of Metamathematics of first-order arithmetic, by Petr Hájek and Pavel Pudlák, it is shown that $\mathsf{PA}^{-}+\mathsf{I\Sigma}_{1}^{0}$ proves $\mathrm{FRT}$. Furthermore, it appears that $\mathsf{EFA}$ (i.e. $\mathsf{I\Sigma}_{0}^{0}+\mathsf{exp}$, exponentiation is totat) is what is minimally required to prove $\mathrm{FRT}$. Therefore, the proof-theoretic strength of $\mathrm{FRT}$ is $\omega^{3}$.
  2. K. McAloon showed that over $\mathsf{RCA}_{0}$, $\mathrm{IRT}$ and $\mathsf{ACA}_{0}^{'}$ are equivalent (see K. MacAloon, Paris-Harrington incompleteness and progressions of theories, Proceedings of Symposia in Pure Mathematics 42(1985), 447-460). The proof theoretic strength of $\mathrm{IRT}$ is thus $\varepsilon_{\omega}$.
  3. J.L. Hirst showed in his Ph.D. thesis that over $\mathsf{RCA}_{0}$, $\mathrm{IRT}(1)$ and $\mathsf{B\Pi}_{1}^{0}$(equivalently $\mathsf{B\Sigma}_{2}^{0})$ are equivalent. It is also shown that $\mathsf{WKL}_{0}$ does not prove either $\mathrm{IRT}(1)$ or $\mathrm{IRT}(2)$. This seems to suggest that the proof-theoretic stregths of $\mathrm{IRT}(1)$ and $\mathrm{IRT}(2)$ correspond to ordinals at least $\omega^{\omega}$. But as I stated above, for $p\ge 3$ we know that the proof-theoretic ordinal of $\mathrm{IRT}(p)$ is $\varepsilon_{0}$. Hence, the proof-theoretic ordinals of $\mathrm{IRT}(1)$ and $\mathrm{IRT}(2)$ appear to lie between $\omega^{\omega}$ and $\varepsilon_{0}$. From this we have that the proof-theoretic ordinal of $\mathrm{IRT}(1)$ is between $\omega^{\omega}$ and $\omega^{\omega^{\omega}}$ and it may very well be precisely one of them. Regarding $\mathrm{IRT}(2)$, we have the following result.
  4. It has been shown that over $\mathsf{RCA}_{0}$, $\mathrm{IRT}(2)$ does not imply $\mathsf{ACA}_{0}$ (see David Seetapun and Theodore A. Slaman, On the strength of Ramsey’s theorem, Notre Dame J. Formal Logic 36 (1995), no. 4, 570–582).
  5. In The proof-theoretic strength of Ramsey's theorem for pairs and two colors, L. Patey and K. Yokoyama show that the case of $\mathrm{IRT}(2)$ when $r=2$ is simple enough that any consequence of its existence can be obtained purely finitistically (thus improving a result of Chong, Slaman and Yang that it does not imply $\mathsf{I\Sigma}_{2}^{0}$). This is rather interesting, since the case of $\mathrm{IRT}(3)$ when $r=2$ is already not finitistically reducible.

[*] Schwichtenberg, H. and Wainer, S.S., Proofs and Computations, Cambridge University Press, Cambridge-New York (2012)

John
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