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intersection

Is proof the same as truth?

Work through Russell’s paradox, inspect the evidence behind five historical moments, and then reconsider your first judgment about proof and truth.

  1. Make an initial judgment
  2. Work through the paradox
  3. Explore five historical moments
  4. Reconsider your first judgment
  5. Crossroad voyage complete

Make an initial judgment

  1. Every true statement can eventually be proved.
  2. Only provable statements can be called true.
  3. With enough axioms and rules, every mathematical question can be decided.
Do these three statements look like the same claim?

This judgment is stored only on this device.

Work through the paradox

Does the set R of all sets that do not contain themselves contain itself?

Under unrestricted comprehension, defining R={x | x∉x} makes both R∈R and R∉R imply their opposites, yielding a contradiction.

R ∈ R

Assume R∈R. By the definition of R, it must follow that R∉R.

R ∉ R

Assume R∉R. By the definition of R, it must follow that R∈R.

Both assumptions lead to their opposites. Now follow how this contradiction reshaped questions about proof and truth.

View evidence and sources

Check the boundary between the source language, translation, and paraphrase.

  1. Russell's Paradox

    Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

    Section 1, unrestricted comprehension, definition of R, and the two-case derivation

Explore every moment in this step before continuing.

Explore five historical moments

  1. 11879Writing logic in a formal language

    Gottlob Frege

    While Frege taught at Jena, Begriffsschrift was published by Louis Nebert in Halle in 1879.

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    1. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens

      Gottlob Frege · 1879 · Halle a/S: Louis Nebert, 1879, first edition · Primary work · de

      title page and imprint

    2. Gottlob Frege

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      Bibliography, 1879 Begriffsschrift entry

    The work formalized quantification and function–argument analysis, laying foundations for modern predicate logic.

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    1. Gottlob Frege

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      Logic and philosophy of language sections

    1. Begriffsschrift is published in Halle

      1879

    2. Frege teaches at Jena

      In 1879

      • Jena · Teaching and appointment · Location confidence: high

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        1. Gottlob Frege

          Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

          Life and academic career at Jena

    Is being able to state something precisely in a formal language the same as being able to settle everything within it?

  2. 21901–1902A paradox and a letter

    Bertrand Russell · Russell's Paradox

    Russell discovered the paradox around 1901 and communicated it to Frege in a letter dated 16 June 1902.

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    1. The Russell–Frege Correspondence

      Bertrand Russell and Gottlob Frege · 1902 · Critical correspondence excerpt, Cambridge University Press · Critical edition · de

      Russell to Frege, 16 June 1902

    2. Russell's Paradox

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      History, discovery and correspondence paragraphs

    The paradox exposed an inconsistency tied to Basic Law V while volume II of Grundgesetze was in press.

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    1. Russell's Paradox

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      Significance for Frege and Basic Law V

    1. Russell discovers the paradox

      Around 1901

      • Cambridge · Discovery · Location confidence: high

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        1. Russell's Paradox

          Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

          History of the 1901 discovery

    2. Russell writes to Frege

      16 June 1902

      • Cambridge · Letter sent · Location confidence: high

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        1. The Russell–Frege Correspondence

          Bertrand Russell and Gottlob Frege · 1902 · Critical correspondence excerpt, Cambridge University Press · Critical edition · de

          Russell to Frege, 16 June 1902

      • Jena · Letter received · Location confidence: high

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        1. The Russell–Frege Correspondence

          Bertrand Russell and Gottlob Frege · 1902 · Critical correspondence excerpt, Cambridge University Press · Critical edition · de

          Frege to Russell, 22 June 1902

    When a contradiction is found, should we abandon the system or rebuild it with stricter limits?

  3. 3The 1920sA program that reexamined proof

    David Hilbert

    Hilbert formulated the program in the early 1920s and developed proof theory with collaborators through the decade.

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    1. Hilbert's Program

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      Historical development of Hilbert’s program

    1. Hilbert and collaborators develop proof theory

      The 1920s

      • Göttingen · Teaching and appointment · Location confidence: high

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        1. Hilbert's Program

          Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

          1917–1920s development at Göttingen

    What must be made explicit if mathematics is to examine its own security mathematically?

  4. 41930–1931Marking the limits of formal systems

    Kurt Gödel · Incompleteness Theorems

    Gödel informally announced the first incompleteness result in Königsberg on 7 September 1930; the paper appeared in 1931.

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    1. Gödel's Incompleteness Theorems

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      History, 7 September 1930 announcement and January 1931 publication

    2. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I

      Kurt Gödel · 1931 · Monatshefte für Mathematik und Physik 38, 173–198 · Primary work · de

      journal metadata, volume 38, pages 173–198

    The theorem applies, under stated consistency assumptions, to effectively axiomatized formal systems strong enough for arithmetic.

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    1. Gödel's Incompleteness Theorems

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      First and second incompleteness theorem statements

    1. Gödel announces the first incompleteness result

      7 September 1930

      • Königsberg · Announcement · Location confidence: high

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        1. Gödel's Incompleteness Theorems

          Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

          7 September 1930 Königsberg discussion

    2. The incompleteness paper is published

      1931

      • Vienna · Teaching and appointment · Location confidence: medium

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        1. Kurt Gödel: Life, Work, and Legacy

          Institute for Advanced Study · Institutional record · en

          Vienna career before emigration

    When we respect a theorem’s assumptions and scope, what can we say about what remains unproved?

  5. 5From 1933Distinguishing the language used to speak of truth

    Alfred Tarski

    Tarski’s Polish work on truth in deductive languages appeared in Warsaw in 1933; an expanded German version followed in 1935.

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    1. Pojęcie prawdy w językach nauk dedukcyjnych

      Alfred Tarski · 1933 · Warsaw: Towarzystwo Naukowe Warszawskie, 1933 · Primary work · pl

      catalog record and title page

    2. Tarski's Truth Definitions

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      1933 program and bibliography

    The English translation of that work appeared in 1956; Tarski’s 1944 article is a different publication.

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    1. Tarski's Truth Definitions

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      Bibliography and publication history

    Convention T is a material-adequacy requirement; one snow biconditional is an instance, not the full definition.

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    1. Tarski's Truth Definitions

      Stanford Encyclopedia of Philosophy · Scholarly encyclopedia · en

      Material adequacy and Convention T

    2. The Semantic Theory of Truth

      Internet Encyclopedia of Philosophy · Scholarly encyclopedia · en

      T-schema and semantic theory sections

    1. Tarski’s Polish work on truth is published

      1933

      • Warsaw · Publication · Location confidence: high

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        1. Pojęcie prawdy w językach nauk dedukcyjnych

          Alfred Tarski · 1933 · Warsaw: Towarzystwo Naukowe Warszawskie, 1933 · Primary work · pl

          catalog record and Warsaw imprint

    To explain truth for a language, what distinctions must be made outside that language?

Explore every moment in this step before continuing.

Reconsider your first judgment

Look again at the three opening statements. Do they look like the same claim now?

Your position does not need to change. Notice which assumptions and kinds of evidence you now distinguish more clearly.

Explore every moment in this step before continuing.

Crossroad voyage complete

You followed both sides of the paradox and five historical moments, examining when proof, decidability, and truth cease to mean the same thing.

You can end the voyage here or return to the mathematics home.

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