The Russell–Frege Correspondence
Telling Frege his life's work is inconsistent
Venue: Letter from Russell to Frege, 16 June 1902; Frege's reply, 22 June 1902; appendix to *Grundgesetze* II (1903).
In a brief letter, Russell shows Frege that his foundational axioms generate a contradiction. The gracious reply is one of the most poignant moments in modern philosophy.
Russell, while finishing *The Principles of Mathematics*, discovered that Frege's Basic Law V — the principle that the extension of any concept is itself an object — generated a contradiction, namely Russell's paradox (see Experiments #112). Russell wrote to Frege on 16 June 1902, just as the second volume of Frege's *Grundgesetze* was at the printer. Frege replied within days, acknowledging that his system was indeed inconsistent and adding a hasty appendix to the volume. In the appendix Frege wrote: "Hardly anything more unwelcome can befall a scientific writer than that one of the foundations of his edifice be shaken after the work is finished." The correspondence is the founding document of 20th-century foundations of mathematics: it forced the development of Russell's type theory, Zermelo's axiomatic set theory, and ultimately ZFC.
Historical Context
Frege had spent more than two decades developing his logical foundations for arithmetic; the second volume of *Grundgesetze* was his final synthesis. Russell was two decades his junior and largely unknown.
Parties
Naive comprehension of sets (or extensions of concepts) is inconsistent; foundations of mathematics must restrict comprehension by some principled method (type theory).
Key arguments
- Construct R = {x : x ∉ x}. Then R ∈ R iff R ∉ R. Contradiction.
- The contradiction is independent of any contingent feature of Frege's system; it threatens any system that allows unrestricted comprehension.
- Type theory (later: ZFC) restricts comprehension to avoid the paradox while preserving most useful set-theoretic reasoning.
Acknowledged the inconsistency immediately; attempted a patch (Frege's "Way Out") in the appendix to *Grundgesetze* II that was itself inadequate.
Key arguments
- Acknowledged that Basic Law V, as stated, is untenable in light of Russell's paradox.
- Attempted a "Way Out" restriction that turned out to be only marginally consistent (Quine later showed it permits a one-element universe at most).
- Eventually concluded that the logicist programme had failed and abandoned the position late in life.
Allied schools
Dimensions Engaged
Information
Foundational for the structural integrity of formal informational systems: comprehension must be restricted to avoid paradox.
Verdict in retrospect
Russell was decisively right about the inconsistency. Frege's logicist programme failed; mathematics found its foundations in type theory (Russell-Whitehead) and axiomatic set theory (Zermelo, Fraenkel). Russell's admiration for Frege's graciousness in defeat is one of the most generous moments in philosophy: "as I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege's dedication to truth."
Related Debates
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Related Experiments
Experiments that share dimensions and/or aligned schools with this debate.
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Related Contemporary Dilemmas
Dilemmas that engage the same dimensions as this debate.
Further reading
- Russell & Frege, letters in van Heijenoort (ed.), *From Frege to Gödel* (1967)
- Burge, *Truth, Thought, Reason: Essays on Frege* (2005)
- Russell, *Autobiography*, vol. 1 (1967)