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.
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bleistift2@sopuli.xyz 3 weeks agoI find this axiomatization of the naturals quite neat:
- Zero is a natural number. 0∈ℕ
- For every natural number there exists a succeeding natural number. ∀_n_∈ℕ: s(n)∈ℕ (s denotes the successor function)
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.
unwarlikeExtortion@lemmy.ml 3 weeks ago
bleistift2@sopuli.xyz 3 weeks 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.
unwarlikeExtortion@lemmy.ml 2 weeks ago
Yeah, I assumed. No way 86 pages are needed for a proof of ‘1+1=2’.
That being said, it’d be nice for there to actually be a “proof” of 1+1=2, made as concise and simple as possible, while retaining all the precision required of such proof, including a complete set of axioms.
This, obviously isn’t is, nor does it try to. It’s not the “1+1=2” book, ot’s the theoretical fpindations of matheđatics book. Nothing wrong with that.
anton@lemmy.blahaj.zone 3 weeks ago
That axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
TeddE@lemmy.world 3 weeks 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 3 weeks 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 3 weeks 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).
Eq0@literature.cafe 3 weeks 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
kogasa@programming.dev 3 weeks 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.