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SUMMARY:Leszek Kołodziejczyk (University of Warsaw)
DTSTART:20201013T200000Z
DTEND:20201013T210000Z
DTSTAMP:20260423T004819Z
UID:CTA/25
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CTA/25/">Rev
 erse mathematics of combinatorial principles over a weak base theory</a>\n
 by Leszek Kołodziejczyk (University of Warsaw) as part of Computability t
 heory and applications\n\n\nAbstract\nReverse mathematics studies the stre
 ngth of axioms needed to prove various\nmathematical theorems. Often\, the
  theorems have the form $\\forall X \\exists\nY \\psi(X\,Y)$ with $X\, Y$ 
 denoting subsets of $\\mathbb{N}$ and $\\psi$\narithmetical\, and the logi
 cal strength required to prove them is closely\nrelated to the difficulty 
 of computing $Y$ given $X$. In the early decades\nof reverse mathematics\,
  most of the theorems studied turned out to be\nequivalent\, over a relati
 vely weak base theory\, to one of just a few typical\naxioms\, which are t
 hemselves linearly ordered in terms of strength. More\nrecently\, however\
 , many statements from combinatorics\, especially Ramsey\ntheory\, have be
 en shown to be pairwise inequivalent or even logically\nincomparable.\n\nT
 he usual base theory used in reverse mathematics is $\\mathrm{RCA}_0$\, wh
 ich\nis intended to correspond roughly to the idea of "computable mathemat
 ics".\nThe main two axioms of $\\mathrm{RCA}_0$ are: comprehension for com
 putable\nproperties of natural numbers and mathematical induction for c.e.
 \nproperties. A weaker theory in which induction for c.e. properties is\nr
 eplaced by induction for computable properties has also been introduced\,\
 nbut it has received much less attention. In the reverse mathematics\nlite
 rature\, this weaker theory is known as $\\mathrm{RCA}^*_0$.\n\nIn this ta
 lk\, I will discuss some results concerning the reverse mathematics\nof co
 mbinatorial principles over $\\mathrm{RCA}^*_0$. We will focus mostly on\n
 Ramsey's theorem and some of its well-known special cases: the\nchain-anti
 chain principle CAC\, the ascending-descending chain principle ADS\,\nand 
 the cohesiveness principle COH.\n\nThe results I will talk about are part 
 of a larger project joint with Marta\nFiori Carones\, Katarzyna Kowalik\, 
 Tin Lok Wong\, and Keita Yokoyama.\n
LOCATION:https://researchseminars.org/talk/CTA/25/
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