bjoern_tantau@swg-empire.de
on 01 Oct 13:16
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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.
SaharaMaleikuhm@feddit.org
on 01 Oct 13:45
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Yeah, but how many pages did it take?
InternetCitizen2@lemmy.world
on 01 Oct 13:59
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As many as needed.
SaharaMaleikuhm@feddit.org
on 01 Oct 14:43
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But if it’s less than 83 do we really know if it’s better than whatever the initial 1+1 guy wrote?
pebbles@sh.itjust.works
on 01 Oct 13:50
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Yk thats something some religious folks gotta understand.
Diplomjodler3@lemmy.world
on 01 Oct 15:04
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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
on 01 Oct 18:31
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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
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.
emergencyfood@sh.itjust.works
on 01 Oct 13:56
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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.
bjoern_tantau@swg-empire.de
on 01 Oct 14:19
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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
on 02 Oct 22:28
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Propositional logic as a system is both complete and consistent.
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.
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.
fushuan@lemmy.blahaj.zone
on 01 Oct 17:33
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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
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
fushuan@lemmy.blahaj.zone
on 01 Oct 17:54
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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.
bleistift2@sopuli.xyz
on 01 Oct 18:21
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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.
anton@lemmy.blahaj.zone
on 01 Oct 23:46
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That axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
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.
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).
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.
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.
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.
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
on 02 Oct 04:52
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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.
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
unwarlikeExtortion@lemmy.ml
on 02 Oct 04:57
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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.
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
on 02 Oct 21:58
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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.
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).
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”?
threaded - newest
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.
Yeah, but how many pages did it take?
As many as needed.
But if it’s less than 83 do we really know if it’s better than whatever the initial 1+1 guy wrote?
Yk thats something some religious folks gotta understand.
What are you talking about, filthy infidel? My holy book contains the single, eternal truth! It says so right here in my holy book!
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
Dumbfuckery at its finest…
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.
Doesn’t that only apply for sufficiently complicated systems? Very simple systems could be provably self-consistent.
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.
Propositional logic as a system is both complete and consistent.
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.
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.
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
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
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.
I find this axiomatization of the naturals quite neat:
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.
That axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
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’
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
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).
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
xin N such thats(x)=xand thereforex+1=x. While some might think this implies0=1, it doesn’t, because we don’t have the axiom of induction, an thus can’t provea+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.
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.
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.
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. 😉
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.
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
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.
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.
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.
It’s only difficult to prove if you somehow aren’t able to observe objects in real world.
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
None of that sounded fun…
Lambda calculus be like
and even longer to reach long division?
That’s a misinterpretation of the incompleteness theorem: you should reread it. They did prove 1+1=2 from axioms with their methods.
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).
One year to prove it, 82 years for that banger of a title
Pffh, Terrence Howard will disprove it in only 4 pages!
/s
<img alt="" src="https://lemmy.world/pictrs/image/1b38b539-b407-4c23-a40d-2d4f701cd6ed.jpeg">
Gödel has entered the chat.
The proof might be somewhat lengthy, but it is quite rigorous.
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”?
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
If you want a nice evening, I highly recommend a visual novel called Logicomix.
It covers Bertrand Russel’s life in a popular way and brings a lot of understanding of his struggle.
I have a hard copy, but it’s available on archive.org too! archive.org/details/…/2up
Wow, I have to admit I wasnt expecting much and the first 10 pages arent that gripping, but it really does get good