Inclusion

I am having trouble understanding the translation of the definition of subclasses in the text I am currently studying which covers First Order Predicate Calculus /w Equality.

 

 

The definition (inclusion)

 

[math] A \subseteq B \leftrightarrow \forall z \left( z \in A \rightarrow z \in B \right ) [/math]

 

is being translated as "A is a subclass of B, or a subset of B if A is a set, or B includes A" in the text.

 

 

The definition (proper inclusion)

 

[math] A \subset B \leftrightarrow A \subseteq B \land \exists z \left( z \in B \land z \notin A \right ) [/math]

 

is being translated as "A is a proper subclass of B, or a proper subset of B if A is a set, or B properly includes A" in the text..

 

 

Why in both of these cases is A a subset(or proper subset) of B if A is a set? I think my inability to understand the statement stems from my inability to discern between a Class and a Set. I'm not exactly sure how Russels Antinomy in refuting the Axiom of Comprehension creates the difference but from what I understand it is this conflict that brings rise to the distinction between the two where all Sets are Classes but not all Classes are Sets.

 

 

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I think I see the error of my own logic where I was tying the two statements "A is a subclass of B, or a subset of B" together.

 

What I should have read this as is "A is a subclass of B, or a subset of B if A is a set, or B includes A" breaks down to this:

 

- A is a subclass of B

 

or

 

- A is a subset of B if A is a set

 

or

 

- B includes A

 

Error in my interpretation of the grammar? The statement was just an English reinterpretation of the definition as opposed to a literal translation?

 

'cause I mean the literal translation is obviously A is a subclass of B if and only if for all z if z is in A then z is in B ...

Edited October 19, 201015 yr by buttacup