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I would argue that Zero is generally considered to be a natural. The naturals are often constructed via an initial object, and you can call that initial object
by traderjane 7y ago
I would argue that Zero is generally considered to be a natural. The naturals are often constructed via an initial object, and you can call that initial object whatever you want (some people write Z or Zed), but it's metaphorically a zero.
- rocqua 7y agoThe pushback I have gotten om my tongue-in-cheek rants defending 0 as a natural number would suggest not everyone agrees. Some people really care about 1/n being possible for every natural number N.
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- yakubin 7y agoAnd other people care about natural numbers being the set of cardinal numbers for finite sets. Others want the set of natural numbers with addition to be a monoid. And actually most-known series in analysis are usually indexed starting from zero, because that's when the formulas are usually simpler.
- traderjane 7y agoBut if you asked the same people to build the naturals, they'd likely include an initial object that behaves like 0.
- rocqua 7y agoI fully agree. However, there is also pushback on ordinals vs cardinals, and monoids vs semigroups.
- OscarCunningham 7y agoYou could just use 1 as the initial element though. Peano's axioms would become 1 != Sn Sn = Sm implies n=m n + 1 = Sn n + Sm = S(n + m) n × 1 = n n × Sm = (n × m) + n Induction That's not any more complex than the usual set of axioms.