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SUMMARY:Victoria Gitman (CUNY Graduate Center)
DTSTART:20250206T190000Z
DTEND:20250206T200000Z
DTSTAMP:20260423T035747Z
UID:OLS/165
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/165/">Pa
 rameter-free schemes in second-order arithmetic</a>\nby Victoria Gitman (C
 UNY Graduate Center) as part of Online logic seminar\n\n\nAbstract\nSecond
 -order arithmetic has two types of objects: numbers and set of numbers\, w
 hich we think of as the reals. Which sets (reals) have to exist in a model
  of second-order arithmetic is determined by the various set-existence axi
 oms. These usually come in the form of schemes\, of which the most common 
 are the comprehension scheme\, the choice scheme\, and the collection sche
 me. The \\emph{comprehension scheme} $\\Sigma^1_n$-${\\rm CA}$ asserts for
  a $\\Sigma^1_n$-formula $\\varphi(n\,A)$\, with a set parameter $A$\, tha
 t the collection it determines is a set. The \\emph{choice scheme} $\\Sigm
 a^1_n$-${\\rm AC}$ asserts for a $\\Sigma^1_n$-formula $\\varphi(n\,X\,A)$
  that if for every number $n$ there is a set $X$ such that $\\varphi(n\,X\
 ,A)$ holds\, then there is a single set $Y$ such that its slice $Y_n$ is a
  witness for $n$. The \\emph{collection scheme} $\\Sigma^1_n$-${\\rm Coll}
 $  asserts more generally that among the slices of $Y$\, there is a witnes
 s for every $n$. The full comprehension scheme for all second-order assert
 ions is denoted by ${\\rm Z}_2$\, the full choice scheme by ${\\rm AC}$\, 
 and the full collection scheme by ${\\rm Coll}$. Although the theories ${\
 \rm Z}_2$+${\\rm AC}$ and ${\\rm Z}_2$ are equiconsistent\, Feferman and L
 \\' evy showed that ${\\rm AC}$ is independent of ${\\rm Z}_2$. It is also
  not difficult to see that ${\\rm Coll}$ implies ${\\rm Z}_2$ over $\\Sigm
 a^1_0$-${\\rm CA}$\, and hence that ${\\rm Coll}$ implies ${\\rm AC}$ over
  $\\Sigma^1_0$-${\\rm CA}$.\n\nIn this talk\, I will explore how significa
 nt the inclusion of set parameters is in the second-order set-existence sc
 hemes. Let ${\\rm Z}_2^{-p}$\, ${\\rm AC}^{-p}$\, and ${\\rm Coll}^{-p}$ d
 enote the respective parameter-free schemes. H. Friedman showed that the t
 heories ${\\rm Z}_2$ and ${\\rm Z}_2^{-p}$ are equiconsistent and recently
  Kanovei and Lyubetsky showed that the theory ${\\rm Z}_2^{-p}$ can have e
 xtremely badly behaved models in which the sets aren't even closed under c
 omplement. They also constructed a more ``nice" model of ${\\rm Z}_2^{-p}$
  in which $\\Sigma^1_2$-${\\rm CA}$ holds\, but $\\Sigma^1_4$-${\\rm CA}$ 
 fails. They asked whether one can construct a model of ${\\rm Z}_2^{-p}$ i
 n which $\\Sigma^1_2$-${\\rm CA}$ holds\, but there is an optimal failure 
 of $\\Sigma^1_3$-${\\rm CA}$. I will answer their question by constructing
  such a model. I will also construct a model of ${\\rm Z}_2^{-p}+{\\rm Col
 l}^{-p}$ in which $\\Sigma^1_2$-${\\rm CA}$ holds\, but ${\\rm AC}^{-p}$ f
 ails\, thus showing that ${\\rm Coll}^{-p}$ does not imply ${\\rm AC}^{-p}
 $ even over $\\Sigma^1_2$-${\\rm CA}$.\n
LOCATION:https://researchseminars.org/talk/OLS/165/
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