• MBM@lemmings.world
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    2 hours 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”?

  • Björn@swg-empire.de
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    19 hours 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
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      5 hours 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.

      • Diplomjodler@lemmy.world
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        18 hours 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
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          14 hours 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

      • TaterTot@piefed.social
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        16 hours 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.

    • Fushuan [he/him]@lemmy.blahaj.zone
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      15 hours 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
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        15 hours 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

          • captainlezbian@lemmy.world
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            4 hours 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
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          14 hours 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.

          • unwarlikeExtortion@lemmy.ml
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            4 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.

          • anton@lemmy.blahaj.zone
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            9 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
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              6 minutes 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
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              4 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.

            • TeddE@lemmy.world
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              7 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: https://www.youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rmb34ZBYe5ohrkL3t

              They build the whole modern number system ‘from scratch’

              • anton@lemmy.blahaj.zone
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                26 minutes 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

        • Fushuan [he/him]@lemmy.blahaj.zone
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          15 hours 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
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      17 hours 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.

    • emergencyfood@sh.itjust.works
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      19 hours 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
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        18 hours 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.

      • Björn@swg-empire.de
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        18 hours 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.