Metric spaces and enriched categories 1

[6665726e]

Science is fun when you know the secret!

[mediteight]

Total beginner's question (please be gentle) :) Is there any difference between a collection and a set? Is there a simple example of a category which is a collection of Ob(C)... but not a set of Ob(C)...?

[TheCatsters]

The difference between a "collection" and a "set" comes from the need for set theory to avoid Russell's paradox. This paradox arises if you try to construct a set X "containing every set that is not a member of itself". Is X a member of itself or not?? Set theory deals with this by saying that not all collections of things are allowed the status of "set". Eg the collection of all sets is "too big" to be a set. So the category of sets and functions has a collection of objects, not a set.

[chocojian]

Ummm shouldnt it be 0 <= d(a,a)?

[adudzik]

What he wrote was correct.

We're already given that distances are nonnegative. Therefore writing 0 >= d(a,a) is the same as writing 0=d(a,a).

The reason for this trick is: we want to say that d(a,a) is zero, because the distance between a point and itself should always be zero, but we'd like to do it without introducing an additional symbol, =.

[TheCatsters]

Yes, thanks for that. I tried responding myself about 10 times, but YouTube was not letting me do it!