bjoern_tantau 111 points a year 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.

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pebbles 52 points a year ago

Yk thats something some religious folks gotta understand.

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Diplomjodler3 29 points a year 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!

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GandalftheBlack 13 points a year 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... and yet people still tell you it unanimously supports their single eternal truth

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Stonewyvvern 5 points a year ago

Dumbfuckery at its finest...

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TaterTot 3 points a year 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.

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Klear 17 points a year 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.

Both "specifically" and "generally" would work.

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SaharaMaleikuhm 16 points a year ago

Yeah, but how many pages did it take?

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InternetCitizen2 14 points a year ago

As many as needed.

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SaharaMaleikuhm 5 points a year ago

But if it's less than 83 do we really know if it's better than whatever the initial 1+1 guy wrote?

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fushuan 15 points a year 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

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Taldan 19 points a year 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

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bleistift2 15 points a year 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.

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anton 2 points a year ago

That axiomisation is incomplete as it doesn't preclude stuff like loops, a predecessor to zero or a second number line.

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unwarlikeExtortion 1 point a year 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.

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fushuan 4 points a year 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.

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Matriks404 1 point a year ago

It's only difficult to prove if you somehow aren't able to observe objects in real world.

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captainlezbian 5 points a year 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

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MeThisGuy 4 points a year ago

It took a long while to reach division

and even longer to reach long division?

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titanicx 4 points a year ago

None of that sounded fun....

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JackbyDev 3 points a year ago

Lambda calculus be like

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HexesofVexes 15 points a year ago

Ehh...

So, it's more a case that the system cannot prove it's own consistency (a 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. Fortunately both are equiconsistent (each can be used to imply the consistency of the other).

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lmmarsano 11 points a year ago

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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emergencyfood 9 points a year 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.

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Shelena 15 points a year 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.

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bjoern_tantau 4 points a year 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.

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CompassRed 1 point a year ago

Propositional logic as a system is both complete and consistent.

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MBM 18 points a year ago

Is this where I go "actually it took 83 pages to set up an extremely rigorous system and then a couple of lines to show you could use it to prove 1+1=2"?

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Donkter 5 points a year ago

If those couple of lines couldn't be determined to be true without the 83 page setup then it took all 83 pages to prove 1+1=2

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nop 12 points a year ago

Gödel has entered the chat.

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Batman 10 points a year ago

One year to prove it, 82 years for that banger of a title