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        <title type="main" level="a">Second incompleteness theorem: research developments and consequences</title>
        <author>
          <persName n="1" ref="https://orcid.org/0000-0001-9441-7226" type="ORCID">
            <forename>Duccio</forename>
            <surname>Pianigiani</surname>
            <placeName type="affiliation">University of Siena, Italy</placeName>
          </persName>
        </author>
        <respStmt>
          <resp>This is a section of <title>Lectures in Proof Theory and Complexity</title>(DOI: <idno type="DOI">10.36253/979-12-215-0778-2</idno>) by </resp>
          <name>Duccio Pianigiani</name>
        </respStmt>
      </titleStmt>
      <publicationStmt>
        <publisher>Firenze University Press, USiena Press</publisher>
        <pubPlace>Florence</pubPlace>
        <date when="2025">2025</date>
        <idno type="DOI">https://doi.org/10.36253/979-12-215-0778-2.09</idno>
        <availability>
          <p>Available for academic research purposes</p>
          <p>Open Access</p>
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            <p>Metadata licence CC0 1.0</p>
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        <p>This is original content, published for academic research purposes</p>
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      <abstract xml:lang="en">
        <p>We discuss the Second incompleteness theorem, some research developments and consequences. We introduce Turing’s progressions as attempt to attempt to overcome the problem of incompleteness. The we introduce the basics of classical and intuitionistic Provability Logic and some recent developments. Lastly we  provide a guide to consistency proofs of arithmetic developed by Gentzen and Schütte.</p>
      </abstract>
      <textClass>
        <keywords>
          <list>
            <item>Derivability conditions</item>
            <item>consisteny</item>
            <item>Turing’s progressions</item>
            <item>Provability Logic</item>
          </list>
        </keywords>
      </textClass>
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    <body>
      <p>It is available online at https://doi.org/10.36253/979-12-215-0778-2.09<ref target="https://doi.org/10.36253/979-12-215-0778-2.09" /></p>
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