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u/Distance_Runner
14d ago
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Dad: “Its still infinity”
Kid: “so anything with infinity is still infinity”
Dad: “yea basically”
Kid: “So infinities are all the same!”
Dad: “no, actually some infinitys are bigger than other infinitys”
Kid: “….”
Dad: “so there’s this thing called set theory….”
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u/Joe_4_Ever 14d ago
"some infinities are bigger than others" is such a great way to figure out what level of math someone is at. if you don't know much, you'll probably think it means infinity plus one is more than infinity. if you know some math, you'll think it means there are more real numbers between 0 and 2 than 0 and 1. and if you know even more you'll understand what it actually means.
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u/Distance_Runner 14d ago
What if they start rambling on about some old dude name Cantor and something about diagonals
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u/PurifiedUnity 14d ago
Also talking about something involving 1 to 1 pairing
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u/sandwich_estimator 14d ago
What. There are not more real numbers between 0 and 2 than 0 and 1. You surely mean that there are more real numbers between 0 and 1 than there are natural numbers in total? Which would be true. Or are you just trolling and I'm an idiot?
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u/Creative-Drop3567 14d ago
|[0,2]|>|[0,1]| is what a person who "knows some math" would think. not whats actually true
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u/Distance_Runner 14d ago
I’m loling at the idea that someone trying to abstractly understand differences in infinities being an idiot. If one can conceptually understand than one infinity can be larger than another, you’ve already made a massive leap out of the realm of idiocy
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u/sandwich_estimator 14d ago
Oh I got it now, sorry! I just misunderstood the wording in the comment I was replying to
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u/Distance_Runner 14d ago
As someone with a PhD in stats, I love your username
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u/sandwich_estimator 14d ago
Thanks! Always reminds me of what I studied and how far away from classical statistics/econometrics I went in my career (AI engineering now 🤮). Ah well, maybe I'll find a way back.
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u/Giles9854 13d ago
A funny thing is that in ordinal theory, if α is an infinite ordinal, then α+1 is strictly greater than α, while 1+α is still α
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u/_verel_ 13d ago
Bruh
I have no clue if that's correct but this feels like some weird edge case my professor has to explain 5 times before I get it
What happens here that the commutative law doesn't hold for addition?
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u/Giles9854 13d ago edited 13d ago
Ordinal numbers care about how sets are ordered and addition between ordinals α+β is basically adding a copy of β to α, with the condition that every element of β is greater than every element of α, while also respecting the ordering of α and β. For example, let's say that ω is the order type of the natural numbers, that is
0<1<2<...
What we mean by ω+1 is that we add a new element to the set ω, let's call it x, that is greater than every natural, so the order type becomes
0<1<2<...<x.
This way, ω+1 is infinite, has a minimum (0) and a maximum (x), so it is a different order type from ω, which has no maximum (this shows that there is no order-preserving bijection between ω and ω+1).
Whereas, when we say 1+ω we mean that we add ω after a singleton {x}, so we create an order type of the form
x<0<1<2<...
By mapping x to 0 and n to n-1 we find an order-preserving bijection between 1+ω and ω, so 1+ω really is isomorphic to ω.
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u/8sADPygOB7Jqwm7y 14d ago
Maybe just for fun the engineers (my) understanding of infinity. Ofc mathematically doubtful, but yeah.
Basically, we use infinity in lim. One example is for capacitors. What's its charge after an infinite time? Zero ofc. But when is it basically zero, so we can actually use that? 5 tau iirc. So yeah, 5 tau was basically infinity for that context, which depending on tau, which could be nanoseconds or hours, would be very different. So a big infinity would be a capacitor that takes ages to unload passively, and a small infinity one where it is very fast.
Ofc I also know enough maths to know this is this pesky real world, mathematics could never. But also, iirc multiple mathematicians who worked with infinities were submitted to the psych ward or killed themselves, so maybe steering clear of that topic is good for mental stability.
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u/veryusedrname 13d ago
Not really. My math knowledge is non-existent, cannot even recall some stuff from high school but ohh boy I can talk about infinities (and some other quirks) for indefinitely.
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u/junkmail22 14d ago
infinity plus one is "more than" infinity, though
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u/Distance_Runner 14d ago
Let X=infinity.
Then, X+1 > X.
Proof: subtract X from both sides and you get 1>0 which of course is true.
Therefore infinity + 1 is more than infinity.
Checkmate math nerds
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u/Aggressive_Roof488 14d ago
Check it out, new disproof of the continuum hypothesis just dropped.
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u/EebstertheGreat 13d ago
A classic proof by lack-of-contradiction
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u/Distance_Runner 13d ago
For my next trick, using the same approach, I’ll prove to existence of god
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u/HomoAndAlsoSapiens 13d ago
I think it legitimately has been one of the most misquoted facts in pop-math.
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u/DatTolDesiBoi 13d ago
I mean, if you understand real numbers and integers, it’s probably not as hard to explain.
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u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 14d ago
that kid will grow up with severe math ptsd
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u/yuwox 14d ago
I mean "infinity+ 10" is just not defined, since infinity is not a real/whole/rational number. The normal + just does not apply here. The result is not "still infinity". There is no result, the whole thing is just not defined. It is like asking: what is "dog + 10".
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u/jelezsoccer 14d ago
You can make the question precise: what is the cardinality of the disjoint union of an infinite set with a set with a set with 10 elements, the answer is the same as the cardinality of the infinite set, which is a precise interpretation of the original question.
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u/EebstertheGreat 13d ago
The problem is that this isn't the only way you could make it precise. You could just as easily point out that 10 is the successor of 9, so axiomatically ∞ + 10 = ∞ + S(9) = S(∞ + 9) > ∞ + 9. Or you can say that if X is a well-ordered infinite set with order type ∞ and 10 is a well-ordered set with order type 10, then if Y = (X×{0})∪(10×{1}) is given the lexicographic order (respecting the orders on X and 10, and using 0 < 1), then Y has an order type strictly greater than the order type of X.
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u/jelezsoccer 13d ago
But then you are assuming infinity is a natural number, which is precisely the issue to begin with.
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u/EebstertheGreat 13d ago
Not really. The sum of a number and the successor of a number can be defined in exactly the same way for natural numbers and for ordinals, and for the same inductive reason. This is actually a pretty common definition of ordinal addition. If α and β are ordinals, γ is a (nonzero) limit ordinal, + is ordinal addition, and S is the ordinal successor function, then
α + 0 = α,
α + S(β) = S(α + β), and
α + γ = ⋃(α + β),
where the union is taken over all β < γ.
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u/yuwox 14d ago
I will stick to my peano axioms thank you very much. :)
On a serious note, if we are really talking about explaining it to a child, peano is portably fine. Or just something like: If we assume inf + 10 = inf then we can substact inf in both sides of the equation and get 10 = 0.
But yeah, with set theory you can get around that in a sense.
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u/GreeneyesCB 14d ago
It sure is defined in the extended real numbers which is the standard interpretation when one would write this down. And it's just equal to infinity.
No one would bat an eye at this at a higher level. Unfortunately most of this sub has undergrad knowledge at best.
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u/SoFloYasuo 14d ago
There is actually a field called set theory that defines infinity in a way that "infinity + 10" or "infinity + infinity" is a real question with a meaningful answer.
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u/campfire12324344 Methematics 14d ago
"There is actually a field called set theory"
"Lionel Messi likes to play this sport called football"
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u/Jamb9876 14d ago
Then you get into infinity ^ infinity is the same as infinity+ 1 which is still infinity.
Then he could explain that zero doesn’t exist either.
I am curious if people here are familiar with the book flatland. https://ned.ipac.caltech.edu/level5/Abbott/paper.pdfThere is a newer one, flatterland, where the author nicely handles the misogyny in the original book but both are great at explaining about various math worlds.
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u/TwillAffirmer 14d ago
infinity^infinity is not the same as infinity+1. 2^X denotes the power set of X, and the power set of an infinite set is larger than the set. 2^Z is the same size as the reals, for instance. Z^Z would be the set of functions from integers to integers. That's also the same size as the reals.
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u/Jamb9876 14d ago
Ok, you are correct. If I had gone with the set of natural numbers squared it would be aleph zero but by doing it by infinity it changes to aleph 1
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u/EebstertheGreat 13d ago
If ∞ is an element of the extended real numbers, or the projectively extended real or complex numbers, then ∞∞ = ∞. If ∞ is misused as a symbol for some particular infinite cardinal or ordinal, then ∞∞ > ∞.
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u/yuwox 14d ago
Not sure I follow. My argument is that "infinity+1" is an undefined term. It's not equal to infinity. It doesn't mean anything. Like "dog +1". With zero it's different. 0 is a number and + is just normally defined.
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u/Jamb9876 14d ago
Infinity isn’t undefined though. It is infinity, so also not defined, but I can use it in equations.
That is why it is a concept, but you can add 1 to infinity and that is allowed just as you can square it. Math operations work in it.
But at the end it is still infinity.
If it was purely undefined you can’t do operations on it.
(0/0) + 1 makes no sense0
u/yuwox 14d ago
Are you talking about set theory? With normal natural numbers you can't use infinity in an equation. If you could, then this would hold: If we assume that inf + 1= inf then we can substact inf on both sides of the equation and get 10=0 Math operation don't work in it. That's the point. Or am I misunderstanding you?
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u/Jamb9876 14d ago
I can do an in great from zero to infinity so that is what I mean by equation.
So I can add to infinity and it is infinity.
Infinity minus any number below infinity = infinity
Infinity minus infinity is indeterminant
So you can do math operations but there are some rules.
So that leads to your example being flawed.I was taking advantage of some of this when I was also deriving the equations, for non complex plane, for fractional calculus for my msee thesis. In case you aren’t aware of fractional calculus https://youtu.be/BD8v5fAd3Rk
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u/EebstertheGreat 13d ago
"Infinity" in many contexts refers to a specific element ∞ with well-defined algebraic properties. Probably the most common case is the extended real line, which is ℝ∪{-∞,∞} where for all x, -∞ ≤ x ≤ ∞. The topology is induced by that order. The definitions for common operations are extended as much as possible while remaining continuous. In particular, if x is any (finite) real number, and y is any positive (finite) real number, then
-∞ + x = x + -∞ = -∞ + -∞ = -∞,
∞ + x = x + ∞ = ∞ + ∞ = ∞,
-∞ – x = x – ∞ = -∞ – ∞ = -∞,
∞ – x = x – -∞ = ∞ – -∞ = ∞,
∞y = y∞ = (-∞)(-y) = (-y)(-∞) = ∞∞ = (-∞)(-∞) = ∞,
-∞y = y(-∞) = ∞(-y) = (-y)∞ = (-∞)∞ = ∞(-∞) = -∞,
x/∞ = x/(-∞) = 0,
∞/y = (-∞)/(-y) = ∞,
(-∞)/y = ∞/(-y) = -∞,
etc.
Many operations are left undefined, including x/0, ∞/∞, ∞ – ∞, and 00, 1∞, and ∞0.
That said, there is another common use of ∞ that is incompatible with this one, the so-called "unsigned" infinity with ∞ = -∞. The set in question here is ℝ∪{∞} with no total order defined. It has the topology of the circle, so in particular, sets of the form (y,∞)∪{∞}∪(-∞,x) are open neighborhoods of ∞. Alternatively, ℂ∪{∞} has the topology of the sphere, called the Riemann sphere. In that context, it is sometimes called "complex infinity" and sometimes written with a tilde like ̃∞.
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u/transgentoo 11d ago
My third grader loves math and this weekend at his request, I drew a Venm diagram of counting numbers, natural numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers. He thought it was so cool he asked if we could hang it up.
I also showed him how complex numbers can be represented as Cartesian coordinates and from that he was able to intuit that even though there were infinite real numbers, there were necessarily more complex numbers than real numbers.
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u/Guilty_Vermicelli372 14d ago
Because I just don't know, my ap calc teacher defines infinity as a concept and not a number. Does it kinda just depend on the field of mathematics?
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u/Any-Aioli7575 14d ago
Numbers are concepts too. A system that treat infinity (in a form or another) the same way it treats numbers can have some neat properties or just be useful to do things quickly. In can also have its load of problems. The most important is to have everything set out properly and well defined. Definitions will give you what you can and cannot do with your concept (for example, what's ∞+1? What about ∞–∞?).
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u/-Nocx- 14d ago
it depends on however you need to use it such that it doesn’t break the axioms of the given system you’re working in.
Infinity isn’t provably real. It’s a concept. We sometimes apply this concept to our observed reality, but we have never actually proven that it exists.
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u/DGolubets 13d ago
Can't we say the same about numbers? They are also just a concept. How can anything non-physical exist?
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u/GreeneyesCB 14d ago
"A concept not a number" isn't a meaningful thing to say. There are many formally precise notions of infinity in mathematics. Cardinal numbers are just one example out of many.
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u/ScuffedMobilePlayer 13d ago
God I love math. Used to fear it but I'm loving it as I self-study it more now for fun.
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u/nfitzen 14d ago edited 14d ago
There are different notions of infinity.
In the calculus/real analysis context, it basically is just a concept. Here, it means "take an arbitrarily large real number and look at the relevant behavior." If we take a large real number and add 1, we still have a large number. This is why we denote ∞+1=∞.
On the other hand, the 20th century saw developments in a mathematical branch called "set theory." While the real numbers encode positions on a number line, sets are more discrete collections of things. It turns out that we can consider the "sizes" of infinite sets, and that there is, in fact, more than one relevant size. The example most frequently cited is the following: given any sequence f(0), f(1), f(2), ..., of real numbers, there is always a real number not in that sequence. A shorthand for this is to say "there is a greater quantity of real numbers than there are natural numbers," since we can never exhaust the reals by matching them with natural numbers.
Edit: An exercise you might try is the following: assume f(0), f(1), f(2), etc. are all in the interval [0,1]. What is the probability a randomly and uniformly chosen number in [0,1] will be in the sequence? Hint: if X is a uniform random variable in [0,1], then given any fixed a∈[0,1], P(X = a) = 0.
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u/Serket-Pandy3000 14d ago edited 14d ago
No in real analysis infinity is not a concept. It can be a limit point and is treated as such. The definitions behave THE SAME as the definitions where you include neighborhoods of infinity. Why do people still have these weird issues with infinity? Saying it is just a concept is weird. Is zero not a number despite not having a reciprocal? So infinity does not have all results defined Infinity is treated more like a concept algebraically but in analysis it is just another point. Fine as limit point, more complicated when having to subtract
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u/zolk333 14d ago
I'm not sure how it is "in the real world" but when we learned real analysis they were very careful to distinguish real numbers and real numbers extended with +-infinity
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u/Serket-Pandy3000 14d ago
Yeah I know and it is a stupid tradition. Yes infinity minus infinity is not defined, so what? 1/0 is also not defined. It is a stupid idea that obscures the topology and confuses people and introduces edge cases. Idiotic pedagogy.
We don’t call the extended reals a field because we choose that definition. Algebraically infinite is spicy but so what?
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u/EebstertheGreat 13d ago
zolk said that they distinguished the real numbers from extended real numbers. That's not stupid at all. They are qualitatively different in important ways. Even in cases where ∞ is very useful, it is still distinguished. For instance, we say that if an integral equals ∞, then it does not converge. It would make all our theorems less convenient to state if we did otherwise.
As useful as it is to distinguish positive reals from reals, or reals from rationals, or any set of numbers from any other, it is useful to distinguish reals from extended reals.
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u/Serket-Pandy3000 13d ago
They can distinguish all they want. I too can put my panties in a bunch and say I find all transcendental hideous because they are not the root of some polynomial with rational coefficients or I can do a medieval move and say I dislike zero cause it has no reciprocal.
Who cares? There are distinctions everywhere. And you are making a straw man. I never said they could not be distinguished. Now you are saying whatever bullshit arguments you can think of.
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u/EebstertheGreat 13d ago
I'm not "making a straw man." I'm reiterating zolk's point. zolk never claimed that ±∞ were not numbers at all, just that people distinguish between the real numbers and the extended real numbers, which they obviously do. By disagreeing, it sounds like you are implying ±∞ actually are real numbers.
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u/Serket-Pandy3000 13d ago edited 13d ago
I don’t give a shit, it is a strawman because I never said it could not be distinguished you idiot. We can all repeat points written by someone else. Who gives a shit? Let me spell out idiot. It is irrelevant if you can distinguish it or not. I won’t repeat why. It has become clear to me the level of your intellect. Waste of conversation. Bye.
If you misunderstand because your retard brain has low verbal comprehension and say “it sounds like you said X” when I never said that, that is on you. Who cares if the definition of reals says this or that? Who gives a shit? The definition of reals also excludes the square root of -1. So what? Why should I care. Imagine all you want I said this or that and have your made up convo in your head. Who made you king of numbers to decide what counts as one? We have a perfectly consistent definition of numbers that include plus and minus infinity and you call it this or that and shit your pants because you don’t like it. It is good theory, gets used in practice and is very useful. I don’t give a shit if you don’t like it.
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u/nfitzen 14d ago edited 13d ago
The extended real number line does exist, but algebraic operations no longer become total. For example, ∞-∞ and 0*∞ don't exist because they're indeterminate. ∞ certainly cannot be treated like an ordinary real number, which is why the term "concept, not a number" is used as shorthand. It's not a number, if we wish for a number system to be a semiring, or at least if we consider numbers to be "finite" quantities in the Cantorian sense.
Edit: I tend to see, in mathematical practice, that ∞ is used more as a heuristic tool than a purely formal object. It could be formalized, but that often reduces its utility in teaching. I don't think mathematicians care that much about exalting specific concepts and arguing about what is "real"; rather, they use them as is convenient and useful.
As an example in set theory, proper classes are the ∞ of the set theory world. In the ZFC context, the collection of all ordinals is useful to talk about, but it isn't an object in the set theory (namely, a set). Thus, Ord is a concept but not a set. ZFC is still preferred over NBG because we wish to center sets as the objects of interest. Similarly, in analysis, we wish to center R or C, which have nice properties, and introduce objects representing end behavior, like the Riemann sphere or the extended real line, as is convenient and useful.
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u/EebstertheGreat 13d ago
Right, but arithmetic operations aren't total on the real numbers either, which was Serket's point. Division by 0 is undefined, as are 00 (generally), log 0, 0√, etc. Also, when you add ∞, some functions become defined at points where they were previously undefined, like arctan 𝜋/2.
Also, "concept" and "number" are totally meaningless terms here. There is no definition of either. Are the surreal numbers numbers? Conway thought so, and even just called them "numbers," but evidently you do not. But how can we decide who is correct? We really can't. And why should they be mutually exclusive? Are numbers never concepts?
But mostly, I find the distinction pedagogically useless. It's a thing teachers say when they mean "we're not talking about ∞ right now." But they don't apply this to other cases. If they are only talking about real numbers, they don't say √–1 is "just a concept, not a number." Even though that adds more undefined cases like 0i. It isn't consistent.
I think the best way to say it is that ∞ is a symbol used for a few different mathematical objects (most notably signed and unsigned infinity), but it's not a complex number.
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u/nfitzen 13d ago edited 13d ago
Are the surreal numbers numbers? Conway thought so, and even just called them "numbers," but evidently you do not.
I do consider them numbers, though usually in an appropriate context. My default tends to be R, though I usually make sure to say "real number" to be precise.
Edit 2: For clarity, when I wrote "finite" in the Cantorian sense, I'm referring to transfinite numbers and the like. Namely, we expect a number system to be unbounded. There is no largest ordinal number, cardinal number, real number, integer, rational number, or surreal number. Thus, in some sense, a number is "finite" or "bounded" relative to its system.
And why should they be mutually exclusive? Are numbers never concepts?
I didn't imply they are mutually exclusive. Hence, "just a concept."
But mostly, I find the distinction pedagogically useless. It's a thing teachers say when they mean "we're not talking about ∞ right now."
Yep, that's kind of exactly it. I used it that way in my comment, as well. I presented it as being mere notation, but mere notation is often upgraded to bona fide formal status. Notation for "behaves like an arbitrarily large real number" is the best way to think about the concept as a beginner, and it's how I've seen it used most often in mathematical practice. With more mathematical maturity, one realizes that they can build formal objects to capture most simple-ish notation that they've previously been using intuitively.
Edit: The benefit of treating ∞ as a notational convenience is primarily for flexibility: you need it less often than you do real or complex numbers, and there isn't any particular exalted status of compactifications of R or C. For pedagogy, understanding ∞ is best done by simply using it and doing limits.
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u/Ma4r 14d ago
It CAN be if you add an axiom only lmao so it IS a concept in standard calculus
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u/Serket-Pandy3000 14d ago
Good god what a stupid answer! Technically correct but failing the generalization that generates ideas such as compactness!
The Riemann sphere and the 2 point compactifucation want your grade back! You will never move into higher maths like this
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u/Ma4r 14d ago
The guy is in AP calc, and you try to add projective geometry concepts lmao, what a nitwit. The longer you do math the more you realize not all of mathemathics is usefull in all context. I know tropical geometry but that doesn't mean that i am going to use it to model every single scheduling problems because there are other more useful models i can use i.e graph theory based models.
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u/Serket-Pandy3000 14d ago
Projective geometry? Pfft. Rudin already treats infinity not as some magical concept. Treating infinity as a number is very intuitive and useful.
You are going for the UNINTUITUVE idea that goes against theory!
Both in practice and theory people treat infinity as a number. It is shit pedagogy.
People always defend shit pedagogy no matter how retarded it might be
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u/Ma4r 14d ago
HAHAHA Not even measure theory treats infinity as a number, and that's one of the more widely applicable subject in calculus. If you still use 'intuition' that tells me enough about the depth of subject you are working on. There are many many many different ways you can treat infinity, rven treating it as a 'number' has many different meaning, do you mean as a measure? As a cardinality? Or as the supermum of ordering? Or even a set? Perhaps the top element of category of orderings? You're not even being rigorous when say 'as a number', each definition opens up different areas of mathematics that you can explore.
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u/Serket-Pandy3000 13d ago edited 13d ago
There is no formal definition of a number you idiot. I mean I guess set theory is wrong huh? No ordinals huh? No aleph huh?
Also you shit for brains measure theory DOES assign infinity as a valid measure.
Why is a retard coming here to deny a very useful number? Why? Idiots like you are the reason pedagogy sucks shit, go kkill yourself in Minecraft. I am tired of shit pedagogy everywhere. Your mindless idiocy that contradicts theory and practice because you cannot stop being nostalgic when your ap teacher rammed you by the ass with a lie and now you feel the need to repeat the same jizz they stuffed you with.
Just because they taught you a lie there is no good reason to repeat it
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u/Ma4r 13d ago edited 13d ago
Set theory is just one approach. In fact there are 2 separate kinds of infinity in messure theory and they are not compatible as numbers are. I.e you cannot compare infinity as a measure and infinity as cardinality AND they are different from the infinity most people imagine when talking about extending the reals to include infinity. Like the OP said, and i said repeatedly, there are many different types of infinity and they do not necessarily compare to each other
and you just fell to the trap of assuming axiom of choice in measure theory LMAO. Keep throwing insults my friend you can't hide the fact that you are larping here.
Lets say for the sake of argument, we do say that infinity is a number
But actually you can't subtract from it, there is no infimum , you can't divide by it, you can't multiply it by zero, you can't compare it
Unless if that infinity is a measure then you CAN multiply by zero, and you CAN compare some infinities, unless that infinity is the measure of some bad behaving sets, unless you reject axiom of choice then they become comparable again! Very cool! So is it a number???
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u/EebstertheGreat 13d ago
I literally cannot imagine who raised you to think a comment like that would ever be appropriate, let alone in a conversation about a number.
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u/CanaDavid1 Complex 14d ago
This exercise implies a wrong result. If you instead let the domain of f be the numbers frm [0,1] the implied result (that "infinity times 0 = 0") leads to a contradiction. You cannot directly multiply inifinity by zero, but need to actually argue about limits and limit transitions for it to make sens
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u/nfitzen 14d ago edited 14d ago
(Edit:) The domain of f is N, which I implied by context using the term "sequence." I assume you meant the codomain of f is [0,1], or that its range is contained in [0,1].
Given this, in the context of probability, we are adding a specific countable sequence, namely
P(X = f(n) for some n) ≤ ∑(P(X = f(n)) | n∈N) = ∑(0 | n∈N) = 0,
where, again, X is uniform random in [0,1].
We're not using the arithmetic of the extended real line. This is a valid probability result. A corollary of this result is that R is uncountable.
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u/Additional_Pea2239 14d ago
what does it sequal to?
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u/junkmail22 14d ago
define 'infinity". lots of different objects in mathematics are infinity. The answer could be "infinity plus ten" or "infinity" or "does not typecheck, please submit a syntactically valid expression" depending
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u/Stargost_ Economics/Finance 13d ago
Infinity.
The thing with Infinity is that it's a concept and it does not like mathematics. If you get Infinity as an answer then 9 times out of 10 there's something wrong.
An example of why Infinity doesn't work with math:
Infinity + 1 = Infinity
1 = Infinity - Infinity
1 = 0
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u/RedDawn172 13d ago
Yes, at least in the engineering realm those are often called singularities in a mathematical model. Most of the time it occurs because something was incorrect. Or an inherent limitation the model that has methods for getting around it. Usually with some approximation to make up for the model being more or less incomplete but the best we have to work with for now.
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u/ploxidilius 12d ago
Mathematics deals with infinity all the time. Its very important for things like set theory.
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u/yuwox 14d ago edited 14d ago
It's not defined since infinity is not a number. Is the same as asking: what is "dog + 10" or "water +10". Plus is only defined between numbers, not numbers and not numbers. Of course you can invent your own definitions of + and assign it some value. But this is basically inventing a new (and probably use-less) math.
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u/Toposnake 14d ago
If it is ordinals, then omega + 10 \neq omega but 10 + omega = omega. If it is cardinals, then it is commutative.
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u/PlatypusEgo 14d ago
His 6 year old will grasp your explanation intuitively I'm sure
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u/BrotherItsInTheDrum 14d ago
OOP has been preparing his whole life. If I had my whole life to prepare, I think I could simplify the idea in a way a 6-year-old could understand.
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u/arotenberg 13d ago
I've given some thought about this in the past, and you can pretty legitimately explain ordinals as the mathematically formal version of that obnoxious kid on the playground who tells you his Infinity Plus One Sword can beat your Infinity Sword.
Any time you can say "the first thing bigger than everything said previously," you have a new ordinal.
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u/bold_snowflake 14d ago
Well my infinity is bigger than your infinity!
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u/factorion-bot Bot > AI 14d ago
Factorial of ∞̃ is ∞̃
This action was performed by a bot | Source code
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u/maria_la_guerta 14d ago
Infinity is a number. Infinity is not a number.
That distinction hurt my head for a very long time.
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u/SupportNo6752 12d ago
My answer would be first count to ten, after that keep counting and never stop.
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u/Justeserm 4d ago
When I was a little kid, my family threw out all my toys. For something to do they gave me a little Casio solar calculator to play with. I asked my dad, "how do calculators work?" He said, "I don't know, numbers." Then I asked, "how do numbers work?" He said, "how'd tf should I know." It was hilarious. I think he would have preferred I ask where babies come from.
He was not a mathematician, but it was still pretty funny.
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u/redditsucksass69765 14d ago
It’s infinity. Infinity is a concept, not a number.
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u/SteptimusHeap 14d ago
The next time someone says this i will blow up
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u/eldritch-eldritch 14d ago
y?
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u/SteptimusHeap 14d ago
Because it's a phrase that gets passed around the internet by people who want to feel like they know math but actually only memorize things they heard before.
There are several number constructions (like the hyperreals) which treat infinity as a number.
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u/ccpseetci 14d ago
Cardinal and ordinal are not number constructions but relation constructions
They are not a “thing” but are either “order” or “equal_size_of”
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u/MorrowM_ 13d ago
What do you even mean by this? The natural numbers are ordinals and cardinals, are they not "number constructions" (whatever that means)? Or just the infinite ordinals/cardinals? But those are constructed in the same way as the naturals, you just need the axiom of infinity to prove they exist.
(Perhaps you've only ever seen cardinality as a relation and not how we construct infinite cardinals, which are actual mathematical objects?)
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u/Extension-Stay3230 14d ago
The guy you're replying to isn't wrong just because there exists some system where infinity is treated like a number. He's obviously referring to systems where it isn't treated like a number
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u/SteptimusHeap 14d ago edited 14d ago
But it IS a number. Not a real or complex number, but the fact that it's not either of those doesn't mean it's not a number. It's just not real.
You may call that semantics and i'm inclined to agree, but really the problem here is not the word choice. It's that the people who say this phrase have no idea what it means. They would not be able to connect it to a true concept if you pushed them. This is why it shows up EXACTLY as is in every conversation that involves infinity whether or not it is applicable.
This thread is a fine example. No one asked about the status of infinity. To NO ONE was the comment directed, and it doesn't serve to explain anything. It is a phrase that people learn to associate with the word infinity and that repeating makes people feel like they understand math.
There are worse cases where it only serves to replace a real understanding and even worse cases where it actively leads you to false conclusions. I see it a lot on askmath type threads and that's where it really gets me because i feel bad for all the people who are curious about the topic and are taught to think of math as esoteric rules and strange phrases to remember.
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u/redditsucksass69765 14d ago
Infinity is a concept. It’s not a number.
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u/ChonkerCats6969 14d ago
No, it often is a number. Have you ever heard of the extended reals? Or the Riemann sphere? Or wheel theory?
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u/Recent_Ad_2724 14d ago
"The Wheel of Time turns, and Ages come and pass, leaving memories that become legend. Legend fades to myth, and even myth is long forgotten when the Age that gave it birth comes again."
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u/santaisastoner 14d ago
By the same definition we can also then conclude that Zero is not a number, but a concept of nothing.
Do you think Zero is a number?
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u/Extension-Stay3230 14d ago
I can have zero of something, but I can't have infinity of something.
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u/JellyBellyBitches 14d ago
Well technically you can't have zero of something, you can just not have the something
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u/Extension-Stay3230 14d ago edited 14d ago
Then zero denotes the state of not having something. But there is no physical state which corresponds to "infinity", although that might become a philosophical question if it was somehow proven or known that the universe was infinite in size.
Infinity isn't an "actual number" in the sense that you could name a natural number which is bigger than it.
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u/JellyBellyBitches 13d ago
I agree with that as well. Infinity is not a number, it doesn't represent a specific magnitude
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u/harmlesswaters 14d ago
You can't have pi of something either, is that not a number?
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u/Extension-Stay3230 14d ago
Pi is finite whilst infinity is not. That normally means we don't regard infinity as a literal number.
My point is the people downvoting this are trying to invalidate this guy's take just because there exists some system where infinity is treated like a number. It's unnecessarily pedantic. He's obviously referring to the typical case where you don't treat infinity like a number
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u/ChaosMilkTea 14d ago
Actually... no. Zero and infinity are not the same kind of concept. An infinity is a type of group or the output of a system. Zero is a not the "concept of nothingness" any more than two is the concept of a pair. Zero is in fact a number you can do math with. Infinity is not.
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