Hilbert's program

<h2 id="statement-of-hilberts-program">Statement of Hilbert's program</h2>
<p>The main goal of Hilbert's program was to provide secure foundations for all mathematics. In particular, this should include:
</p>
<ul><li>A formulation of all mathematics; in other words all mathematical statements should be written in a precise <a href="/facts/Formal_language/crDTyP8q">formal language</a>, and manipulated according to well defined rules.</li>
<li>Completeness: a proof that all true mathematical statements can be proved in the formalism.</li>
<li>Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.</li>
<li>Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a> sets) can be proved without using ideal objects.</li>
<li><a href="/facts/Decidability_(logic)/17SHPMo0">Decidability</a>: there should be an algorithm for deciding the truth or falsity of any mathematical statement.</li></ul>
<h2 id="gdels-incompleteness-theorems">Gödel's incompleteness theorems</h2>
<p class="note">Main article: <a href="/facts/G%25C3%25B6del%2527s_incompleteness_theorems/du8PRu8g">Gödel's incompleteness theorems</a></p>
<p><a href="/facts/Kurt_G%25C3%25B6del/P14DqKqK">Kurt Gödel</a> showed that most of the goals of Hilbert's program were impossible to achieve, at least if interpreted in the most obvious way. Gödel's second incompleteness theorem shows that any consistent theory powerful enough to encode addition and multiplication of integers cannot prove its own consistency. This presents a challenge to Hilbert's program:
</p>
<ul><li>It is not possible to formalize all mathematical true statements within a formal system, as any attempt at such a formalism will omit some true mathematical statements. There is no complete, consistent extension of even <a href="/facts/Peano_axioms/DhhDnNug">Peano arithmetic</a> based on a <a href="/facts/Computably_enumerable/lpzr8jHn">computably enumerable</a> set of axioms.</li>
<li>A theory such as Peano arithmetic cannot even prove its own consistency, so a restricted "finitistic" subset of it certainly cannot prove the consistency of more powerful theories such as set theory.</li>
<li>There is no algorithm to decide the truth (or provability) of statements in any consistent extension of Peano arithmetic. Strictly speaking, this negative solution to the <a href="/facts/Entscheidungsproblem/y8LDal2X">Entscheidungsproblem</a> appeared a few years after Gödel's theorem, because at the time the notion of an algorithm had not been precisely defined.</li></ul>
<h2 id="hilberts-program-after-gdel">Hilbert's program after Gödel</h2>
<p>Many current lines of research in <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>, such as <a href="/facts/Proof_theory/kopBdQn9">proof theory</a> and <a href="/facts/Reverse_mathematics/nHLm6JUQ">reverse mathematics</a>, can be viewed as natural continuations of Hilbert's original program. Much of it can be salvaged by changing its goals slightly (Zach 2005), and with the following modifications some of it was successfully completed:
</p>
<ul><li>Although it is not possible to formalize all mathematics, it is possible to formalize essentially all the mathematics that anyone uses. In particular <a href="/facts/Zermelo%25E2%2580%2593Fraenkel_set_theory/UYIsPbXw">Zermelo–Fraenkel set theory</a>, combined with <a href="/facts/First-order_logic/lcISqWIg">first-order logic</a>, gives a satisfactory and generally accepted formalism for almost all current mathematics.</li>
<li>Although it is not possible to prove completeness for systems that can express at least the Peano arithmetic (or, more generally, that have a computable set of axioms), it is possible to prove forms of completeness for many other interesting systems. An example of a non-trivial theory for which <a href="/facts/Complete_theory/TfBN8Fnb">completeness</a> has been proved is the theory of <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed fields</a> of given <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a>.</li>
<li>The question of whether there are finitary consistency proofs of strong theories is difficult to answer, mainly because there is no generally accepted definition of a "finitary proof". Most mathematicians in proof theory seem to regard finitary mathematics as being contained in Peano arithmetic, and in this case it is not possible to give finitary proofs of reasonably strong theories. On the other hand, Gödel himself suggested the possibility of giving finitary consistency proofs using finitary methods that cannot be formalized in Peano arithmetic, so he seems to have had a more liberal view of what finitary methods might be allowed. A few years later, <a href="/facts/Gerhard_Gentzen/9ujN0ssI">Gentzen</a> gave a <a href="/facts/Gentzen%2527s_consistency_proof/LTDpe0V8">consistency proof</a> for Peano arithmetic. The only part of this proof that was not clearly finitary was a certain <a href="/facts/Transfinite_induction/XlolgTDl">transfinite induction</a> up to the <a href="/facts/Ordinal_number/GelFLjJt">ordinal</a> <a href="/facts/Epsilon_number/4AX7o9ST">ε0</a>. If this transfinite induction is accepted as a finitary method, then one can assert that there is a finitary proof of the consistency of Peano arithmetic. More powerful subsets of <a href="/facts/Second-order_arithmetic/abBxQsaH">second-order arithmetic</a> have been given consistency proofs by <a href="/facts/Gaisi_Takeuti/ttdv6myL">Gaisi Takeuti</a> and others, and one can again debate about exactly how finitary or constructive these proofs are. (The theories that have been proved consistent by these methods are quite strong, and include most "ordinary" mathematics.)</li>
<li>Although there is no algorithm for deciding the truth of statements in Peano arithmetic, there are many interesting and non-trivial theories for which such algorithms have been found. For example, <a href="/facts/Alfred_Tarski/7h6NySAj"> Tarski</a> found an algorithm that can decide the truth of any statement in <a href="/facts/Analytic_geometry/3qWP0kM0">analytic geometry</a> (more precisely, he proved that the theory of <a href="/facts/Real_closed_field/Vj1Un2fw">real closed fields</a> is decidable). Given the <a href="/facts/Cantor%25E2%2580%2593Dedekind_axiom/0bJcuH6k">Cantor–Dedekind axiom</a>, this algorithm can be regarded as an algorithm to decide the truth of any statement in <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>. This is substantial as few people would consider Euclidean geometry a trivial theory.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Grundlagen_der_Mathematik/v5tuKokm">Grundlagen der Mathematik</a></li>
<li><a href="/facts/Foundational_crisis_of_mathematics/yjpbbUkt">Foundational crisis of mathematics</a></li></ul>

<ul><li>G. Gentzen, 1936/1969. Die Widerspruchfreiheit der reinen Zahlentheorie. <i><a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a></i> 112:493–565. Translated as 'The consistency of arithmetic', in <i>The collected papers of Gerhard Gentzen</i>, M. E. Szabo (ed.), 1969.</li>
<li>D. Hilbert. 'Die Grundlegung der elementaren Zahlenlehre'. <i>Mathematische Annalen</i> 104:485–94. Translated by W. Ewald as 'The Grounding of Elementary Number Theory', pp. 266–273 in Mancosu (ed., 1998) <i>From Brouwer to Hilbert: The debate on the foundations of mathematics in the 1920s</i>, Oxford University Press. New York.</li>
<li><a href="/facts/Steve_Simpson_(mathematician)/33WuyqGs">S.G. Simpson</a>, 1988. <a href="https://www.personal.psu.edu/t20/papers/hilbert-jsl-1988.pdf">Partial realizations of Hilbert's program (pdf)</a>. <i><a href="/facts/Journal_of_Symbolic_Logic/lfXVXcoy">Journal of Symbolic Logic</a></i> 53:349–363.</li>
<li><a href="/facts/Richard_Zach/fRUw8eB8">R. Zach</a>, 2006. Hilbert's Program Then and Now. <i>Philosophy of Logic</i> 5:411–447, <a href="https://arxiv.org/abs/math/0508572">arXiv:math/0508572</a> [math.LO].</li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Richard Zach. <a href="https://plato.stanford.edu/entries/hilbert-program/">"Hilbert's Program"</a>. In <a href="/facts/Edward_N._Zalta/U8JxHaaA">Zalta, Edward N.</a> (ed.). <i><a href="/facts/Stanford_Encyclopedia_of_Philosophy/LDuL5HW4">Stanford Encyclopedia of Philosophy</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Zach, Richard (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Hilbert's Program", The Stanford Encyclopedia of Philosophy (Spring 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2023-07-05 <a href="https://plato.stanford.edu/archives/spr2023/entries/hilbert-program/" target="_blank">https://plato.stanford.edu/archives/spr2023/entries/hilbert-program/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Hilbert's program open-in-new

Hilbert's program