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murshid_islam
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well, i read about the russell's paradox recently. but what i was wondering is "how is the paradox resolved?"
can you please give an example of what cannot be allowed to be called a set (to avoid the paradox)?matt grime said:By disallowing unrestricted comprehension. I.e. restricting what is allowed to be called a set.
It's an algebraic sort of thing: ZFC explicitly posits the existence of only two sets: the empty set, and the set of natural numbers... but it provides us with a lot of ways to build new sets out of old sets.all of those went above my head. can you give some easier examples?
Russell's paradox is a logical paradox discovered by philosopher and mathematician Bertrand Russell in the early 20th century. It questions the foundations of set theory and highlights the limitations of formal systems. The paradox revolves around a set that contains all sets that do not contain themselves. It is significant because it challenged the widely accepted principles of set theory and required a resolution to avoid contradictions.
Russell's paradox creates a contradiction by questioning the existence of a set that contains all sets that do not contain themselves. If this set exists, then it must both be a member of itself and not be a member of itself, which is a logical contradiction.
There have been several proposed solutions to Russell's paradox, including axiomatic set theories that restrict the formation of sets and deny the existence of the set that leads to the paradox. Another solution is the theory of types, which organizes sets into different levels or types to avoid self-referential sets. The most widely accepted solution is the Zermelo-Fraenkel set theory with the axiom of choice (ZFC), which avoids the paradox through the axiom of regularity.
The ZFC set theory resolves Russell's paradox by introducing the axiom of regularity, which states that every non-empty set must have an element that is disjoint from the set. This means that the set that leads to the paradox cannot exist in the ZFC set theory, as it must contain itself as an element. This solution also avoids other paradoxes, such as the Burali-Forti paradox and Cantor's paradox.
While the ZFC set theory with the axiom of choice is widely accepted as the solution to Russell's paradox, there are still some criticisms and challenges. One criticism is that the axiom of regularity is ad hoc and not logically necessary. There are also ongoing debates about the consistency and completeness of ZFC, and some mathematicians have proposed alternative set theories that avoid the paradox without relying on the axiom of regularity. Overall, Russell's paradox remains a topic of interest and discussion in the fields of logic and mathematics.