# How Is Russell's Paradox Resolved?

• murshid_islam
In summary: In ZFC, sets are objects that are all of the same type. For instance, in ZFC there is only one type of set: the set of sets. So if we have a set S, then S is a set. But there are also other types of sets: for example, the set of natural numbers is a set, but the set of Integers is not a set. The set of Integers is a subset of the set of natural numbers. In other words, the set of Integers is a set, but it's not the same type of set as the set of natural numbers. Russell called these types of sets "types". So in ZFC there are three types of sets: the set of sets, the
murshid_islam

By disallowing unrestricted comprehension. I.e. restricting what is allowed to be called a set. Things that are too large to be called sets are called (proper) classes.

matt grime said:
By disallowing unrestricted comprehension. I.e. restricting what is allowed to be called a set.
can you please give an example of what cannot be allowed to be called a set (to avoid the paradox)?

You are always allowed to construct sets like

$$\{ x \in X \, | \, P(x) \}$$

(where X is some set, and P is some logical predicate) and

$$\{ f(x) \, | \, x \in X\}$$

(where f is some logical function) but you aren't generally allowed to construct

$$\{ x \, | \, P(x) \}$$

(And, of course, you can always do pairs, unions, and power sets)

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all of those went above my head. can you give some easier examples?

No. Nothing, a priori, is a set. All we know is that we have a notion of what we want a set to be, and the rules the collection of sets should obey. We also know what rules we can't allow sets to obey (Russell's paradox). Thus we have to sit down and write out a collection of axioms that we want sets to obey, and then think about what things can be sets *in some model* of the axioms. Thus some collection of objects might be a set in one theory and not in another. What all set theories do agree on is that there is an empty set, that is an object that satisfies x in X is false for all x. Apart from that you're on your own as to what is and what isn't a set.

Some things cannot be sets in any model. For instance if M is some model of some set theory, the collection of sets in M is not a set in M. The class of all cardinal numbers is not a set in any theory. Indeed that is one way to show something is not a set: show it has 'cardinality' greater than kappa for any cardinal kappa.

There are models of set theory where the real numbers as you know them are not a set.

In short think of 'set' as a label that we can apply to some objects and not to others, depending on the axioms we have chosen.

(Actually, this wasn't how Russell got round the issue, but his notion of higher types did not catch on as it is far too wieldy).

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all of those went above my head. can you give some easier examples?
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.

For example, if X and Y are sets, then we can apply the { , } operator to create the "pair set": {X, Y}.

suppose we construct a set $$A = \{ S \ | \ S \not\in S \}$$. so to avoid Russell's paradox, this kind of sets are NOT allowed. am i right?

Correct; that construction is in neither of the two forms I mentioned earlier, so it is not expressly permitted. (And the paradox proves that it cannot be permitted)

Russel and Zermelo

Two people actually solved the problem independently. Zermelo introduced an axiom called the axiom of separation and Russell introduced his theory of types.

## 1. What is Russell's paradox and why is it significant?

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.

## 2. How does Russell's paradox create a contradiction in set theory?

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.

## 3. What are the different proposed solutions to Russell's paradox?

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.

## 4. How does the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) resolve Russell's paradox?

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.

## 5. Are there any remaining criticisms or challenges to the resolution of Russell'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.

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