# Ordinal Analysis of Theories

PGR-P-1203

## Key facts

Type of research degree
PhD
Country eligibility
International (open to all nationalities, including the UK)
Funding
Competition funded
Supervisors
Professor Michael Rathjen
Schools
School of Mathematics
Research groups/institutes
Logic, Pure Mathematics

A central theme running through all the main areas of modern Mathematical<br /> Logic is the classification of sets, functions, theories or models, by means of<br /> transfinite hierarchies whose ordinal levels measure their rank or complexity in<br /> some sense appropriate to the underlying context. Such hierarchy classifications<br /> differ widely of course, in their modes of construction and their intended application,<br /> but they often provide the means to discover deep connections between<br /> areas which may on the surface seem quite unrelated.<br /> In Proof Theory, from the work of Gentzen in the 1930s up to the present time,<br /> this central theme is manifest in the assignment of proof theoretic ordinals to<br /> theories, measuring their consistency strength and computational power, and<br /> providing a scale against which those theories may be compared and classified.<br /> There is a process (not yet fully understood in the abstract, but emerging clearly in practice) by means of which the proof theoretic ordinal of a theory is computed:<br /> one first unravels the induction and comprehension principles of the given theory<br /> into infinitary rules which reflect their intended meaning. A proof which was<br /> finite in the original theory thus becomes an infinite well-founded derivation-tree<br /> whose height is measured by some ordinal. The problem is then to transform<br /> this tree with its complex logical structure and comprehension rules, into another<br /> derivation-tree in which the premises of any rule are less complex (logically)<br /> than is the conclusion. For then it is easy to see, by induction through the<br /> derivation, that no inconsistency can be proven. This, generally speaking, is<br /> Cut Elimination or Normalization. If one can estimate the operational-cost of<br /> Cut Elimination, in terms of the transformation in the sizes of derivation-trees<br /> as they get normalized, then the least ordinal closed under that operation will<br /> measure the length of transfinite induction needed to prove the theory consistent,<br /> i.e. consistency strength. In fact more explicit computational information is<br /> gained as well: this ordinal also turns out to be the least upper bound on the<br /> termination orderings of all functions which can be provably computed in the<br /> given theory. Thus if a program can be proved to terminate in the theory, the<br /> cut-elimination transformation provides a complexity bound in terms of the socalled<br /> Fast Growing Hierarchy and directly leads to combinatorial independence results for specific theories.

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University&#39;s website</a>. Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PHD Pure Mathematics FT</strong></em> and in the research information section&nbsp;that the research degree you wish to be considered for is <em><strong>Ordinal Analysis of Theories</strong></em>&nbsp;as well as Professor <a href="https://eps.leeds.ac.uk/maths/staff/4073/professor-michael-rathjen">Michael RATHJEN </a>as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;s minimum English language requirements (below).</p> <p style="margin-bottom:11px"><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p class="MsoNoSpacing">Applications will be considered on an ongoing basis. &nbsp;Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. &nbsp;We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p>Please note that you must provide the following documents at the point you submit your application:</p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> <li>Funding information including any alternative sources of funding that you are applying for or if you are able to pay your own fees and maintenance</li> </ul>

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