+ is a map from N×N to N where a + 0 = a and a + S(b) = S(a + b) (S is the successor function that gives the next number).
Then 1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2.
No, it’s correct. You define the operation by it’s properties. It’s not saying that “a plus 0 = a” but “the result of applying the binary operation ‘+’ to any number with 0 should give the original number.”
is just a symbol. You could instead write it as +(a,0)=a and +(a,S(b))=S(+(a,b)).
You have to have previously defined 1=S(0), 2=S(1), 3=S(2), and so on.
I really recommend the YouTube channel “Another Roof”. His first few videos were building up exactly this idea, as well as building up all the real numbers (possibly complex too if I’m remembering correctly). Sounds like a dry topic but he uses humour really well throughout. https://youtube.com/@anotherroof
ok, I define 1 as {∅} and 2 as {∅, {∅}}
proving the addition holds is slightly more complicated
Now define “+”
I love the comment that it’s “occasionally useful”
Hmm yes… set theory… I don’t understand anything happening here
There is actually a really good explanation for us math-curious non-mathematicians here:
https://blog.plover.com/math/PM.html
That’s some good read, thank you so much.
+is a map fromN×NtoNwherea + 0 = aanda + S(b) = S(a + b)(Sis the successor function that gives the next number).Then
1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2.seems a little sus to use + to define +
No, it’s correct. You define the operation by it’s properties. It’s not saying that “a plus 0 = a” but “the result of applying the binary operation ‘+’ to any number with 0 should give the original number.”
You have to have previously defined 1=S(0), 2=S(1), 3=S(2), and so on.
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I really recommend the YouTube channel “Another Roof”. His first few videos were building up exactly this idea, as well as building up all the real numbers (possibly complex too if I’m remembering correctly). Sounds like a dry topic but he uses humour really well throughout. https://youtube.com/@anotherroof
Here is a playlist of the topic: https://youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rmb34ZBYe5ohrkL3t
ooh, that looks interesting!