But I am not taking about an amount of different things, but a parallel or branching number line being part of the set of natural numbers.
I am not talking about modular arithmetic on its own, but as part of the set of natural numbers.
Under the missing axioms those constructs would be part of the natural numbers, including an x in N such that s(x)=x and therefore x+1=x. While some might think this implies 0=1, it doesn’t, because we don’t have the axiom of induction, an thus can’t prove a+c=b+c=> a=b.
The usefulness of such a system questionable but it certainly doesn’t describe the natural numbers as we understand them.
I apologize. I went back and reread from the top and I see my error.
My mobile Lemmy client indicates replies with cycling colors, and I had the misunderstanding that your objection was to the axioms presented in Principia Mathematica. But your reply was fair in the context of the axioms you were actually replying to.
While it was probably not the best use of our time, it certainly made me think about relations and algebra in more interesting ways than the last uni course did.
I’ve had this asserted before, but I’m not sure it lives up to the mathematical rigor of our conversation to this point. I recommend substantially more investigation. 😉
But I am not taking about an amount of different things, but a parallel or branching number line being part of the set of natural numbers.
I am not talking about modular arithmetic on its own, but as part of the set of natural numbers.
Under the missing axioms those constructs would be part of the natural numbers, including an
xin N such thats(x)=xand thereforex+1=x. While some might think this implies0=1, it doesn’t, because we don’t have the axiom of induction, an thus can’t provea+c=b+c => a=b.The usefulness of such a system questionable but it certainly doesn’t describe the natural numbers as we understand them.
I apologize. I went back and reread from the top and I see my error.
My mobile Lemmy client indicates replies with cycling colors, and I had the misunderstanding that your objection was to the axioms presented in Principia Mathematica. But your reply was fair in the context of the axioms you were actually replying to.
While it was probably not the best use of our time, it certainly made me think about relations and algebra in more interesting ways than the last uni course did.
I’ve had this asserted before, but I’m not sure it lives up to the mathematical rigor of our conversation to this point. I recommend substantially more investigation. 😉