6 ms·
For that matter, even with 1, how do you generate 0?
by ummwhat 7y ago
For that matter, even with 1, how do you generate 0?
- roywiggins 7y agoZero being a member of the natural numbers or not depends on convention.
- traderjane 7y agoI 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.
- deleted 7y ago[deleted]
- deleted 7y ago[deleted]
- 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.
- OscarCunningham 7y agoThere are actually many circumstances in which we do want to treat 0 as a prime. For example it generates a prime ideal in the integers. One way to state unique prime factorisation including 0 is to treat the set of primes as a pointed set with 0 as the point. The smash product makes the category of pointed sets into a symmetric monoidal category. Then unique prime factorisation can be stated by saying that the multiplicative monoid of natural numbers (with 0) is isomorphic to the free commutative monoid object on the set of primes.
- yakubin 7y agoPrime numbers are defined as elements of rings, so we need two operations, the other being addition (understood as a group operation). So you generate 0 by 1-1.