Trying to understand the history behind Russell's Paradox

[GrandizerII]

[GrandizerII]

I think I get it now. The one-to-one mapping is done randomly, not based on any specific set of guidelines. When an entity is mapped to a subclass, it either is mapped to a subclass that contains it or it's mapped to a subclass that does not contain it. Sometimes it's possible that all entities are each mapped to a subclass that contains it, in which case it will not be possible to show that the one-to-one mapping f is impossible. Other times, however, some of the entities will be mapped to subclasses that don't respectively contain them, in which case it is then possible to show how the mapping f is impossible (because in this case one of the subclasses could not logically map back on to any of the entities, which means the number of the subclasses is greater than the number of the corresponding initial entities).

Did I get this right?

[BrianSoddingBoru4]

Can't help you.  My eye glazed over after 'considering'.

Boru

[GrandizerII]

Thanks for trying.

[Dr H]

Some entities in the domain would be mapped by f on to subclasses that contain them, whereas others may not.

Can anyone explain what the author means here, especially the last part of that sentence? What "others" may not be mapped by f on to subclasses that contain them? How in this context can an entity be mapped to a subclass not containing it? Or am I misunderstanding the term "map" here?

I want to try to understand this bit by bit so I can fully understand the whole paragraph. Thanks.

Russell's Paradox is a paradox of self-reference that leads to essentially a circular reasoning problem.

The "others" in the problematic statement are the "entities" which either belong to particular subclasses, or not.  
That classes/subclasses themselves may also be "entities" is where the problem arises.

For example:  "The set of all bowling balls" defines a limited set of entities -- bowling balls.  

"The set of all things that are not bowling balls" defines a much broader set of entities -- everything which is not a bowling ball.  This would necessarily include the set itself, since "the set of all things that are not bowling balls" is itself, clearly not a bowling ball.  (It would also include the set of all bowling balls -- just the set; not the balls --, since the set, as an entity, is not itself a bowling ball.)

So the second set includes itself as an element of itself, whereas the first set does not.

Don't know if that helps you or not.  As an engineer I often find that viewing things in concrete terms makes them more graspable, but the same can't always be said for others.

[GrandizerII]

Thanks, I understand the paradox just fine, just trying to understand that particular paragraph itself. Appreciate the post though.

[brewer]

So this has nothing to do with Goldie Hawn?

[Vicki Q]

[GrandizerII]