russell's paradox
Very Low (Technical)Formal, Technical, Academic
Definition
Meaning
A contradiction in naive set theory, which asks whether the set of all sets that do not contain themselves, contains itself.
A foundational paradox in mathematical logic and set theory, discovered by Bertrand Russell (1901), demonstrating that a logically consistent concept (like 'the set of all sets') can lead to a logical contradiction, challenging the foundations of mathematics.
Linguistics
Semantic Notes
This is a proper noun referring to a specific, named paradox. It is used almost exclusively in discussions of logic, set theory, philosophy of mathematics, and foundational studies.
Dialectal Variation
British vs American Usage
Differences
No significant differences in usage or spelling; the term is identical in both variants.
Connotations
Identical academic and technical connotations.
Frequency
Equally rare and specialised in both varieties, confined to relevant academic disciplines.
Vocabulary
Collocations
Grammar
Valency Patterns
[Subject] illustrates Russell's paradox.The lecture covered Russell's paradox.A solution to Russell's paradox was proposed.Vocabulary
Synonyms
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Almost never used.
Academic
Core term in philosophy, mathematics, logic, and computer science (theory).
Everyday
Virtually never used in everyday conversation.
Technical
Precise term in formal logic, set theory, and foundations of mathematics.
Examples
By CEFR Level
- Russell's paradox is a famous problem in logic.
- The professor explained how Russell's paradox revealed a flaw in naive set theory.
- To circumvent Russell's paradox, mathematicians developed axiomatic set theories like Zermelo–Fraenkel set theory.
Learning
Memory Aids
Mnemonic
Think of a librarian making a catalogue of all catalogues that don't list themselves. Should this new catalogue list itself? If it does, it shouldn't. If it doesn't, it should. That's Russell's paradox.
Conceptual Metaphor
A Möbius strip of logic (a statement that twists back on itself to create an impossible loop).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Парадокс Рассела (correct). Avoid literal translations like 'противоречие Рассела' unless context is very informal.
Common Mistakes
- Incorrect possessive: 'Russel's paradox' (missing an 'l'), 'Russells paradox' (missing apostrophe).
- Using it as a general term for any paradox.
Practice
Quiz
Russell's paradox is primarily concerned with:
FAQ
Frequently Asked Questions
The British philosopher and logician Bertrand Russell in 1901.
A barber shaves all those, and only those, who do not shave themselves. Who shaves the barber? This leads to a contradiction similar to the set-theoretic one.
It triggered a crisis in the foundations of mathematics, showing that intuitive set theory could be inconsistent, leading to the development of more rigorous axiomatic systems.
Through the creation of axiomatic set theories (like ZF) which restrict the formation of sets (e.g., via the Axiom of Separation) to prevent the creation of 'the set of all sets' and similar problematic entities.