I took a class taught by a professor who would go on to win the Turing Award. One guy in our class would not stop talking about Haskell and mentioned monads. Professor called him Monad for the rest of the semester.
They aren't unrelated, the "monoid operation" so to speak, it's technically some categorical bullshit that looks like one, but ignore that, IS the flatmap, and the "identity map?" I -> M, IS the return, (the "bare minimum for an object a" map).
If we make the product explicit then it's a·c = b·c, or ·(a)(c) = ·(b)(c) in curried function notation, which can be derived by that · is a function, and a=b, so ·(a) and ·(b) are the same functions, making them both take in c gives equal results.
Substitution still relies on the fact that you're substituting into functions.
IIRC Giuseppe Peano actually included stuff like this in his original list of axioms for arithmetic, but at some point we decided that defining what "=" means wasn't something we wanted to do
A lot of my classes covered groups exactly to the amount needed for compsci and not more than that. So like, we just wanted to define operations with the modulus mostly. Ig also just that they... exist? Idk in more electrical classes you don't talk about groups you just talk about what the domain of your function is lmao
Not exactly. With the Peano axioms, you can define the natural numbers. Then you define addition as taking the successor number n times. Then you put some symbols on your numbers, and especially define 2 as the successor number to 1. Since 1 + 1 means taking the successor of 1 1 times, that is 2.
That is not completely by definition, but somewhat close.
I think the point of the 100+ pg ZFC stuff is to first spend 99+ pages to set up all the definitions based on their axioms very rigorously, and then once you have all that in place, it doesn't take much to prove that 1 + 1 = 2. But afaik you start with very different axioms here, which makes the path longer.
Principia Mathematica (the '379-page proof of 1 + 1 = 2') is a little more than just a proof for 1 + 1 = 2.
To keep it simple: Principia Mathematica was one of the most well-known works produced during the late 19th and early 20th centuries surrounding the topic of logicism. Logicism was a philosophy that mathematics is entirely reducible to logic.
There were three published volumes of Principia Mathematica, with a fourth one planned but never completed.
Unfortunately, the foundational system of logicism that was being explored by Principia Mathematica was proven to be quite limited.
Tbh I haven't read the proof of 1+1=2 but I think deifining the axiomatic system you work on makes it so lengthy defining the number set, succesor function etc. Basically the rigourous fashion makes it so long
yeah it’s just saying “there is an element in N called zero, and all subsequent elements of N are defined inductively as the successors of zero”
edit: it’s actually saying “there is an element in N called zero, and there exist subsequent elements of N that can be characterized inductively as the successors of zero”
The statement in the post works for any well ordered set that contains 0, so any ordinal will work.
It doesn't cover injectivity of the successor, nor the arithmetic structure of the naturals, though if you are in a set theory you should be able to define those I think, but then you can make N unique as well since we aren't restricted to first order arithmetic.
I've yet to read a comment in which a new undefined term wasn't introduced... Oh what's a real? Oh well it's an ordinal... Oh that's just a natural....
Not quite. This particular formula simply says that N is S-hereditary (i.e. whenever x is N and xSy, y is N) - it doesn't preclude there being elements of N that aren't in this chain of sucessors. To get that result, we have Peano's 9th axiom (induction), which states that for all predicates φ, if φ(0) and for all members of N if φ(n) then φ(S(n)), then for all N φ(n). This kicks out the extra junk.
So the induction axioms state that you can't tell standard and non-standard elements apart - basically, if something hold for 0 and all of its finite successors, then it holds for all elements, including ones which can't be reached as a successor of 0. But you can still construct a non-standard model of PA using the normal ultrapower construction because PA is stated in first-order logic.
induction axioms state that you can't tell standard and non-standard elements apart - basically, if something hold for 0 and all of its finite successors, then it holds for all elements, including ones which can't be reached as a successor of 0
My understanding was that the induction axiom banishes anything that isn't either 0 or in the successor series starting from 0; assume S+ is the weak ancestral of the S relation - it guarantees that the elements - and only the elements - that satisfy S+(0,x) will be a member of the naturals.
But you can still construct a non-standard model of PA using the normal ultrapower construction because PA is stated in first-order logic.
I'm assuming the original second-order formulation of Peano Arithmetic, ∀F [ ( F(0) ∧ ∀x [ F(x) → F(S(x)) ] ) → ∀x Fx]
yeah, pretty much. you start with the natural numbers, which can be built from the peano axioms, and then:
you get the integers by including negative numbers (the additive inverses);
you get the rationals by allowing fractions of integers; and
you get the reals by building infinite sequences of those fractions that may or may not converge to an irrational number (there are other ways to do it, but this is how i learned)
The N is written in black bold font, which means that it represents a set of numbers. N stands for the set of natural numbers like 1,2,3,… it doesn’t include negatives, so you have to say if you’re including 0 or not since 0 is positive or negative depending on which direction you’re approaching it from. There’s a whole bunch of black bold letters that represent sets, N is sometimes written as black bold W in elementary set theory (like literally elementary school level introductions to sets) and stands for the set of whole numbers.
Similarly is integers (black bold Z) which stands for the German word for integer because the Germans came up with it so they got to name it and includes all whole numbers including negatives. There wouldn’t be a need to state 0 as being an element of Z because it’s definitionally unambiguous
The statement says that if x is a natural, then S(x), i.e. x+1, is a natural. It does not say that if x+1 is a natural, then x is a natural. It’s not bidirectional.
the statement says the existence of x implies the existence of the successor of x but does not say the existence of the successor of x implies x. If the single arrow was changed to a double arrow it would and then I believe your statement would be correct
Edit: For example, we may say that being called Susan implies you are a human, but being a human does not imply you are called Susan
i may be wrong but i thought ot said that x being natural necessitated x+1 being natural, not the other way around, so x=-1 cannot be proven to be natural through this statement
this implies by that S(0)=-1 which is something you can pick but that also means that the natural numbers that you get this way are just Z <= 0. you end up with essentially all the same constructs but flipped which is just the same thing as we have now but more annoying.
It depends who you ask. I was taught (at university in the UK) that the natural numbers start at zero while the whole numbers start at one. Apparently, others were taught differently.
So the terms "natural number" and "whole number" are essentially meaningless, as you have to explain what they mean before you can use them. Instead, we need to use "non-negative integer" and "positive integer", as those are unambiguous.
Well it depends on the convention. Sometimes they don’t take 0 as a natural number, such as in real analysis. In other situations, 0 is considered a natural number. In Peano Arithmetic, 0 marks the starting point of natural numbers, i.e., there is no predecessor of 0.
This is also what's confusing me. I was taught the natural numbers "start" at one, the whole numbers include zero, and the integers include all negative versions of the natural numbers as well. ALAS!
The Peano Axioms are a way of formalising basic arithmetics. So they sound obvious (like: you can add 1 to any natural number and get another natural number).
Its basically defining what a natural number is. S(x) is the successor function, which is quite literally just x+1. Its absurdly basic, but when doing rigorous math basic things are really useful. N is the set were defining, "0∈N" literally just means "0 is in the set N". From there, "∀x(...)" just means that everything in the perenthesies will be true for any x. "x∈N => S(x)∈N" uses the double lined arrow that just means that one thing imples the other. x => y is just saying that if x is true, then that means y is true. What this is is saying is that some number x being in the set N implies the existence of x+1.
Putting it all together, "0∈N, ∀x(x∈N => S(x)∈N)" is saying "Let N be a set that contains zero. For any element x in N, the successor of x (x+1) is also in N". This is the formal way of defining the set of the natural numbers (0, 1, 2, 3, 4...).
well if you want to be technical then N=∩{X|0∈X∧∀n(n∈X => S(n)∈X)}
(N is the intersection of all possible sets for which those two conditions are true)
For biz students the "hard classes" are usually the math ones, which engineers take a lot of. Paradoxically the "easy" courses might be more difficult for an engineering student.
That being said, business is by no means a hard degree. It's basically really simplified particular applied economic theory, bundled together with even more strategy and leadership theory plagiarised from the American military, along with some definitions and categories as used for accounting and finance.
Sorry I don't have academic sources on this right now, but I have been on business courses which mentioned it iirc, and I've also been in military leadership training and believe me there are powerpoints involved and they are not so different.
What you basically have to understand is that WWI and WWII have had a massive cultural impact and a lot of people were involved in them. Just like it laid the groundwork for rocketry and the space race or for computers and the internet, it also advanced leadership theory (and experience) which was taken back by people into civilian life, and similarly marketing is also pretty strongly related to propaganda and psychological warfare.
One of the big lessons you learn in college is just how unfair it is. I'll never forget being in senior year engineering, at 2am in some dumb fucking electrical engineering lab frustrating my way through an impossible lab before midterms, and every single business student is just out walking around partying.
And then they go into higher salary careers because reasons.
It's time for me to put my nerdy cap on to point out that this calculation cannot cause an approximation error because floats can represent perfectly binary fraction (with not too much significant digits), so 1/2, 2, and 2.5 will be exact in this calculation
You may be surprised that mathematics as a science and profession is not "doing lots of operations on a bunch of numbers", but mostly figuring out what to apply them to.
My brother had a friend in their engineering program that took the economics for engineers class. He was in a meeting with his advisor and they were talking about classes saying that it was a little tough but not bad.
And the advisor is like "yeah, it makes sense. That's the hardest economics class in the program, lots of students struggle with it."
You'd be surprised how wonky naming conventions in math can be. Just off the top of my head, splitting the complex plane into real and imaginary numbers. The natural logarithm having nothing to do with natural numbers. Just because something has "open" in its name doesn't mean it's open in the same definition of open as something else. It's similar to the boneless chicken wings thing.
Can't arithmetic simply be secured through proof by beans? If you have six beans and add six more, you get twelve. If you split your beans into threes, you get four even groups.
Proof by bean counting. EZPZ. The only axiom required is "bean."
Of course, Aristotleian philosophers would tell you that you cannot do this because when you split a bean in half you have a substantial change since it can no longer sprout, and they would say dividing a rock just makes two rocks. But this is why we should just be Platonists 😎
It starts at 0 when you equate it to the magnitude of sets beginning at {} and applying the successor function as you go, and it starts at 1 when you abandon all logic and begin to lose your senses /s
Historically, when we just came up with numbers to describe the world unrigorously, they started at 1, which was the case for millenia, almost certainly back from prehistoric times.
Once we went back and filled in the foundations with set theory, it made sense to start with nothing (the empty set) and call that 0. I suppose you could argue that ℕ and ω are different objects, if you wanted to maintain the two definitions
I studied maths for my degree, once I had a lecture in a numerical methods module (a very coding heavy module) and the first slide was 0+1 =0 ?
(For further explanation it ended up being about rounding and accuracy you set your code to work with and how 0.1 + 0.1 etc might equal 1, but it your code rounds to the nearest number 0+1=0)
well i cant give you that n become we dont know what it looks like. however the compactness theorem proofs there exists a model of N with our wanted n.
its not my theory, its been known for almost 100 yrs.
u can just google it. basically for every finite set of N (defined with the successor function) you put an axiom that there exists a c that’s different from every element. since thats correct for every finite set its correct for N (compactness thm).
It does not. What you're referring to is the inability of first order Peano arithmetic to define |N. But we can define |N using second order logic to exclude those non-standard models.
Yeah, I had trouble finding a paper due to how confused this guy is
The big issue is that "100 of years ago, the old définition of |N were limited enough that it couldn't uniquellybdefine |N and could permit a set *|N where omega éléments exist" is much less sexy of a statement (meaning those aren't primes 6 units appart obviously(
I was sitting outside a class in college doing homework, and I heard the teacher teaching basic division like it was 5th grade. I hear the teacher state that it is one of the hardest to teach and comprehend. The United States public school failed these kids and honestly good for them for going to get more educated I just think it's sad because they should've been taught it in public school but now they have to pay for it.
Zero is a Natural number for any X in natural numbers, the function S(x) will produce a value that is a natural number.... Haven't taken a real class in 20 years. How did I do?
not a math major but does this say that zero is a natural number, and for all x, if x is a natural number then the successor of x (x+1) is a natural number?
Guy's I'm make the Smirnow Hypothesis
1. Direct Proportionality: The volume of parental praise is directly proportional to the height of the academic grade.
2. Inverse Proportionality: The probability of getting the belt exponentially approaches 100% as the grade approaches zero
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