More specifically he proved that you cannot prove that 1+1=2
That’s a misinterpretation of the incompleteness theorem: you should reread it. They did prove 1+1=2 from axioms with their methods.
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bjoern_tantau@swg-empire.de 6 months ago
And shortly after that some other guy proved that he was wrong. More specifically he proved that you cannot prove that 1+1=2. More more specifically he proved that you cannot prove a system using the system.
lmmarsano@lemmynsfw.com 6 months ago
fushuan@lemmy.blahaj.zone 6 months 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
MeThisGuy@feddit.nl 6 months ago
It took a long while to reach division
and even longer to reach long division?
JackbyDev@programming.dev 6 months ago
Lambda calculus be like
titanicx@lemmy.zip 6 months ago
None of that sounded fun…
Taldan@lemmy.world 6 months 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
Matriks404@lemmy.world 6 months ago
It’s only difficult to prove if you somehow aren’t able to observe objects in real world.
captainlezbian@lemmy.world 6 months ago
That’s just empirical data, not a mathematical axiom. I know it’s true, you know it’s true but this is math as philosophy not math as a tool
bleistift2@sopuli.xyz 6 months ago
I 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 6 months 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.
anton@lemmy.blahaj.zone 6 months ago
That axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
fushuan@lemmy.blahaj.zone 6 months 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.
Klear@quokk.au 6 months ago
I like how it’s valid to use “more specifically” as you’re specifying what exactly he did, but in both cases those are more general claims rather than more specific ones.
emergencyfood@sh.itjust.works 6 months ago
you cannot prove a system using the system.
Doesn’t that only apply for sufficiently complicated systems? Very simple systems could be provably self-consistent.
Shelena@feddit.nl 6 months ago
It applies to systems that are complex enough to formulate the Godel sentence, i.e. “I am unprovable”. Gödel did this using basic arithmetic. So, any system containing basic arithmetic is either incomplete or inconsistent. I believe it is still an open question in what other systems you could express the Gödel sentence.
bjoern_tantau@swg-empire.de 6 months ago
I think it’s true for any system. And I’d say mathematics or just logic are simple enough. Every system stems from unprovable core assumptions.
CompassRed@discuss.tchncs.de 6 months ago
Propositional logic as a system is both complete and consistent.
pebbles@sh.itjust.works 6 months ago
Yk thats something some religious folks gotta understand.
TaterTot@piefed.social 6 months ago
Sure, but I can hear em now. “If you can’t prove a system using the system, then this universe (i.e. this “system") can not create (i.e. “prove") itself! It implies the existance of a greater system outside this system! And that system is MY GOD!”
Torturing language a bit of a speciality for the charlatan.
Diplomjodler3@lemmy.world 6 months ago
What are you talking about, filthy infidel? My holy book contains the single, eternal truth! It says so right here in my holy book!
GandalftheBlack@feddit.org 6 months ago
The best thing is when the holy book *doesn’t * claim to contain the single, eternal truth, because it contains hundreds of contradicting truths of varying eternality due to being written by countless authors over more than a thousand years
Stonewyvvern@lemmy.world 6 months ago
Dumbfuckery at its finest…
SaharaMaleikuhm@feddit.org 6 months ago
Yeah, but how many pages did it take?
InternetCitizen2@lemmy.world 6 months ago
As many as needed.
SaharaMaleikuhm@feddit.org 6 months ago
But if it’s less than 83 do we really know if it’s better than whatever the initial 1+1 guy wrote?
HexesofVexes@lemmy.world 6 months ago
Ehh…
So, it’s more a case that the system cannot prove it’s own consistency (an system cannot prove it won’t lead to a contradiction). So the proof is valid within the system, but the validity of the system is what was considered suspect (i.e. we cannot prove it won’t produce a contradiction from that system alone).
These days we use relative consistency proofs - that is we assume system A is consistent and model system B in it thus giving “If A is consistent, then so too must B”.
As much as I hate to admit it, classical set theory has been fairly robust - though intuitionistic logic makes better philosophical sense.