Trying to understand the history behind Russell's Paradox

[GrandizerII]

So I'm reading about Russell's Paradox on IEP, and I'm trying to fully grasp the History section as best I can but currently struggling.

Link to article:
https://www.iep.utm.edu/par-russ/

The second paragraph is what I'm currently stuck at. Here's the paragraph in full:

Russell discovered the contradiction from considering Cantor's power class theorem: the mathematical result that the number of entities in a certain domain is always smaller than the number of subclasses of those entities. Certainly, there must be at least as many subclasses of entities in the domain as there are entities in the domain given that for each entity, one subclass will be the class containing only that entity. However, Cantor proved that there also cannot be the same number of entities as there are subclasses. If there were the same number, there would have to be a 1-1 function f mapping entities in the domain on to subclasses of entities in the domain. However, this can be proven to be impossible. Some entities in the domain would be mapped by f on to subclasses that contain them, whereas others may not. However, consider the subclass of entities in the domain that are not in the subclasses on to which f maps them. This is itself a subclass of entities of the domain, and thus, f would have to map it on to some particular entity in the domain. The problem is that then the question arises as to whether this entity is in the subclass on to which f maps it. Given the subclass in question, it does just in case it does not. The Russell paradox of classes can in effect be seen as an instance of this line of reasoning, only simplified. Are there more classes or subclasses of classes? It would seem that there would have to be more classes, since all subclasses of classes are themselves classes. But if Cantor's theorem is correct, there would have to be more subclasses. Russell considered the simple mapping of classes onto themselves, and invoked the Cantorian approach of considering the class of all those entities that are not in the classes onto which they are mapped. Given Russell's mapping, this becomes the class of all classes not in themselves.

I understand the first few sentences of the above paragraph just fine. The first sentence I'm stuck at trying to understand is the following:

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.

[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]

As an example, let's look at the set of numbers from 1 to 5. Let's suppose there were a function f which maps each member onto a subset. It might map 1 onto {1,3,4}. This contains 1. It might map 2 onto {1,5}. This doesn't. That's the gist of the line you're looking at.


The next step is to say 'let's look at the numbers like 2 in the example which don't go to subsets with them in'. Make a subset of those numbers. What number could map to it? It can't be one in the subset because of the definition of the subset, but it can't not be in the subset because then it fulfils the criteria for being in it.


This is a lengthier version: here


I hope this helps.

[GrandizerII]

As an example, let's look at the set of numbers from 1 to 5. Let's suppose there were a function f which maps each member onto a subset. It might map 1 onto {1,3,4}. This contains 1. It might map 2 onto {1,5}. This doesn't. That's the gist of the line you're looking at.


The next step is to say 'let's look at the numbers like 2 in the example which don't go to subsets with them in'. Make a subset of those numbers. What number could map to it? It can't be one in the subset because of the definition of the subset, but it can't not be in the subset because then it fulfils the criteria for being in it.


This is a lengthier version: here


I hope this helps.

Thank you. Exactly how I interpreted later.