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! https://archive.org/details/Logicomix-Comic-EarlyLifeOfBertrandRussell/page/n7/mode/2up
Wow, I have to admit I wasnt expecting much and the first 10 pages arent that gripping, but it really does get good
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
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.
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).
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.
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.
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
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.
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: https://www.youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rmb34ZBYe5ohrkL3t
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.
Not sure what you mean by ‘loops’
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:
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 numberAh! 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).
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
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.
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.
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.
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…
It took a long while to reach division
and even longer to reach long division?
Lambda calculus be like
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.
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?
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.
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 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.
The proof might be somewhat lengthy, but it is quite rigorous.
Gödel has entered the chat.
One year to prove it, 82 years for that banger of a title
Pffh, Terrence Howard will disprove it in only 4 pages!
/s
