Yeah, that’s what meant with “2 is just the shortened representation of 1+1”.
Same with 1+1+1=3, really. We need to decide the value of 1,2,3,4… Before we can do anything. In hindsight if you think about it, for someone that doesn’t know the value of the symbols we use to represent numbers, any combination that mixes numbers requires the axiom of 1+1+1+1+… = X
I’d be surprised if someone proved that something equals 5 without any kind of axiom that already makes 5 equal to another thing.
bleistift2@sopuli.xyz 1 day ago
I find this axiomatization of the naturals quite neat:
Now the neat part: If 0 is a constant, then s(0) is also a constant. So we can invent a name for that constant and call it “1.” Now s(s(0)) is a constant, too. Call it “2” and proceed to invent the natural numbers.
anton@lemmy.blahaj.zone 19 hours ago
That axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
Eq0@literature.cafe 10 hours ago
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
TeddE@lemmy.world 17 hours 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 10 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
kogasa@programming.dev 14 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.
unwarlikeExtortion@lemmy.ml 13 hours ago
What’s missing here os the definition that we’re working in base 10. While it won’t be a proof, Fibbonaci has his nice little Liber Abbaci where he explains arabic numerals. A system of axioms for base 10, a definition of addition and your succession function would suffice. Probably what the originals were going for, but I can’t imagine how that would take 86 pages. Reading it’s been on my todo list, but I doubt I’ll manage 86 pages of modern math designed to be harder to read than egyptian hieroglyphs.
bleistift2@sopuli.xyz 51 minutes ago
That ‘86 pages’ factoid is misleading. They weren’t trying to prove that 1+1=2. They were trying to build a foundation for mathematics, and at some point along the way that prove fell out of the equations.