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[-] fushuan 13 points 1 day ago

In logic class we kinda did prove most of the integer operations, but it was more like (extremely shortened and not properly written)

If 1+1=2 and 1+1+1=3 then prove that 1+2=3

2 was just a shortened representation of 1+1 so technically you were proving that 1+1 plus 1 equals 1+1+1.

Really fun stuff. It took a long while to reach division

[-] Taldan@lemmy.world 17 points 1 day ago

Presumably you were starting with a fundamental axiom such as 1 + 1 = 2, which is the difficult one to prove because it's so fundamental

[-] bleistift2@sopuli.xyz 15 points 1 day ago

I find this axiomatization of the naturals quite neat:

  1. Zero is a natural number. 0∈ℕ
  2. 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.

[-] anton 3 points 1 day 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 2 points 19 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 5 points 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: https://www.youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rmb34ZBYe5ohrkL3t

They build the whole modern number system 'from scratch'

[-] anton 1 points 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.

Not sure what you mean by 'loops'

There could be a number n such that m=s(n) and n=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:

n,s(n),s(s(n)),s(s(s(n))),...
0,s(0),s(s(0)),s(s(s(0))),...

there are no natural numbers that are negative

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 1 points 7 hours ago

Ah! I see. Thanks for clarifying.

As to m=s(n) and n=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 cars s(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 1 points 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 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.

[-] TeddE@lemmy.world 2 points 5 hours ago

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.

[-] anton 1 points 5 hours ago

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.

[-] TeddE@lemmy.world 1 points 4 hours ago

Time you enjoy wasting is not wasted time.

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. 😉

[-] kogasa@programming.dev 2 points 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.

[-] unwarlikeExtortion@lemmy.ml 1 points 23 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 1 points 10 hours 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 points 6 hours 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.

[-] fushuan 4 points 1 day ago

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.

[-] MeThisGuy@feddit.nl 3 points 1 day ago

It took a long while to reach division

and even longer to reach long division?

[-] titanicx@lemmy.zip 3 points 1 day ago

None of that sounded fun....

[-] JackbyDev@programming.dev 2 points 1 day ago

Lambda calculus be like

this post was submitted on 01 Oct 2025
402 points (100.0% liked)

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