I think you are missing some properties of successors (uniqueness and s(n) different than any m<= n)
That would avoid “branching” of two different successors to n and loops in which a successor is a smaller number than n
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anton@lemmy.blahaj.zone 1 day agoThat axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
Eq0@literature.cafe 19 hours ago
kogasa@programming.dev 23 hours ago
There are non-standard models of arithmetic. They follow the original first-order Peano axioms and any theorem about the naturals is true for them, but they have some wacky extra stuff in them like you mention.
TeddE@lemmy.world 1 day ago
Not sure what you mean by ‘loops’ - except perhaps modular arithmetic, but there are no natural numbers that are negative - you may be thinking of integers, which is constructed from the natural numbers. Similarly, rational numbers, real numbers, and complex numbers are also constructed from the naturals. Complex numbers are often expressed as though they’re two dimensional, since the imaginary part cannot be properly reduced, e.g. 3+2i.
I recommend this playlist by mathematician another roof: www.youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rm…
They build the whole modern number system ‘from scratch’
anton@lemmy.blahaj.zone 20 hours ago
I know how how natural numbers work, but the axioms in the comment i replied to are not enough to define them.
There could be a number n such that
m=s(n)andn=s(m). This would be precluded by taking the axiom of induction or the trichotomy axiom.If we only take the latter we can still make a second number line, that runs “parallel” to the “propper number line” like:
I know, but the given axioms don’t preclude it. Under the peano axioms it’s explicitly spelled out:
0 is not the successor of any natural number
TeddE@lemmy.world 8 hours ago
Ah! I see. Thanks for clarifying.
As to
m=s(n)andn=s(m), I think that is the motivation behind modular arithmetic and it gets used a lot with rotation, because 12 does loop back around to 1 in clocks, and a half turn to face backwards is the same position whether clockwise or counter. This is why we don’t use natural numbers for angles and use degrees and radians.I’m terms of parallels, I personally see that as a strength - instead of having successors (a term that intuitively embeds a concept of time/progression), I typically take the successor function as closer to the layman concept of ‘another’. Thus five bananas is
s(s(s(s(🍌))))and it does have a parallel to five carss(s(s(s(🚗)))). The fiveness doesn’t answer questions about the nature of the thing being counted (such as, "Are these cars: 🚓🚙🏎️🛵? "). Mathematicians like to use the size of the empty set as an abstract stand-in for when they don’t know what they’re talking about (in a literal sense, not broadly).As far as predecessors to 0 - undefined isn’t a problem for natural numbers, just for the people using them. And it makes a certain sense, too. You can’t actually have negative apples (regardless of how useful it may be to discuss a debt of apples).
anton@lemmy.blahaj.zone 6 hours ago
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.