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I remember my mind being blown when in Discrete Math 101 we learned that natural numbers are isomorphic with whole numbers. Was it, you map 0 to 0, even n to n/2, odd n to -(n+1/2) and all of us were like whoa and the TA is like "wait till I tell you about diffrent cardinalities" :D

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This is just a bijection - for an isomorphism you need more structure. Roughly, having an isomorphism (which is a function) says:

0. I have two sets of things, and I can perform operations on each set

1. These two things are the same size (this is your example)

2. If I do operation A on elements (a, b) in set A, then perform the isomorphism on the result, this gives me the same thing as if I performed the isomorphism on a and b independently then performed operation B on their result

3. The isomorphism has an inverse with the same properties

[There are various types of isomorphism depending on what part of maths you’re thinking about. But they all have roughly this idea of “same size” and “preserves something cool that I can do in each of the spaces”]

It's an isomorphism in the category of Sets.

(The worst sort of isomorphism)

Yes, I probably ought to have mentioned that. Though ime people are more likely to talk about isomorphisms in other categories, since the only “structure” a set has is its size

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