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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/

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

jrh206 7y ago
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”]
- laqq3 7y ago
  It's an isomorphism in the category of Sets.
  - graphpapa 7y ago
    (The worst sort of isomorphism)
  - jrh206 7y ago
    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