No, my comment was exactly correct: You haven't provided your definition, yet. You suggested what 'strict cardinality' might mean, but that was not a definition of cardinality.
Here is an expanded response:
If you think about it, you need the property of "size" of sets before you can define "size" of sets.
Huh?
(|A| is called strict iff for bijection f : A → { 0 , … , n } |A| is some n) OR (|A| is called non-strict iff for bijection f : A → { 0 , … , n } |A| is not any some n)
Ok, let's see if we can parse that. You seem to be defining something:
Yes, you are defining what it means for the cardinality of a set (which has not yet been defined) to be strict.
...iff for bijection f...
and that meaning is precisely equivalent to...um, er, something. "iff for bijection f"? Did you mean "iff there exists a bijection, f"? Let's assume so.
|A| is called strict iff there exists a bijection f : A → { 0 , … , n }
Well, that's more sensible. There is a problem with that set, {0, ..., n}, though. Shall we assume you mean a set composed of a sequence of whole numbers from 0 to n? That would require your set theory have considerably more added to it besides a simple concept of mapping. You need for the whole numbers to be defined and for them to be well-ordered, and maybe more things I'm missing. I will not ask you to define those things, however, since I know they can be defined, just not by you.
You still have a problem, though. You have n as a free variable. Shall we add a "there exists an n in N" (where N is the set of whole numbers)? Ok? If so, we'd be up to this:
|A| is called strict iff there exists n in N (there exists a bijection, f : A → { 0 , … , n })
If we are still good to this point, then you have an acceptable definition for whether the cardinality of some set is strict or not. Hurrah! Not sure why you need the term, strict, but Hurrah! nonetheless. (It is also not really what you meant; boundary examples can be a bear.)
Continuing:
Huh? What did you mean by this? It is not connected to all the stuff that precedes it. It has no meaning in predicate calculus sitting there where it does. It can be safely removed, and with that bit of nonsense removed, you are back to an acceptable definition for whether the cardinality of some set is strict or not.
Continuing with your statement, as corrected, we have:
(|A| is called strict iff there exists an n in N (there exists a bijection f : A → { 0 , … , n })) OR ...
And now we run smack into that OR. Odd thing, that. One does not usually find two definitions joined in a disjunction very often. Seems pointless, too, since definitions are universally true. The are definitions, after all.
No matter. Everything that follows the OR would require more correction than what was required for the left side, but it ends up being redundant and not worth any effort to correct. You've already defined what strict means and what not strict means. You don't need to define them again in the negation.
Still no sign of a definition for cardinality. Maybe there is something coming up in your post. Let's look:
Cardinality is a measure of the number of members of set A
Perfectly acceptable background information. Not a definition of cardinality, though.
...such that (A is finite iff |A| is strict) OR (A is non-finite iff |A| is non-strict).
Looks like you want to define something again. As before, let's fix it to be what you probably meant:
A is finite iff |A| is strict
Not very interesting. Why did you even bother with introducting the term, strict, in the first place? You could have just defined 'finite' directly without the smoke and mirrors.
And still no sign of a definition of cardinality.
