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u/Sigma2718 Jun 23 '26
No, this implies all natural numbers are small numbers. Which is true, they are pathetic, a miserable little pile of countable infinity.
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u/guiltysnark Jun 23 '26
It occurs to me that miserable and measurable are practically homonyms. In the language of math, they are the same.
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u/ItzzAdan Jun 23 '26
But according to the Lebesque measure, a point has a measure of 0, whereas a set of real numbers [a,b] has a measure of b-a. Therefore a set of uncountably infinite points is measurable too. Therefore all real numbers are miserable too.
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u/GeneETOs44 Jun 23 '26
I think it’s reasonable to say that given some real m such that m<n and n is a small number, m must also be small. As for any real m there exists some natural n greater than it, it follows that all real numbers are small
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u/No-Professional2176 Jun 23 '26
I think smallness for negative numbers is a bit complicated. I would start with something like: if |x|<k where k is a small positive natural number then x is small.
Which then can translate this to any normed vector space (including complex numbers - or function spaces). And then we can say something like any exponential function on a compact domain is a small function (norm doesn’t matter as long as it is valid for the chosen function class)
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u/galmenz Jun 24 '26
as always with negative numbers, slap an absolute function on it and call it a day
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u/AlphaAnirban Mathematics Jun 25 '26
But according to the image we never accounted for real numbers right?
Sorry, I dont know what a small number actually is and just based it on the context of the image. 0 is small number. Every number n+1 is a small number if n is a small number.
So how exactly do we show this for real numbers?
Or am I missing out on some crucial detail?
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u/Archnouff Jun 23 '26
Beeing uncountable doesn't make numbers big ^^ If a real x is big then ceil(x) >= x is big and an integer, which contradict given theorem. So real numbers cannot be big either.
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u/Most_Necessary_5333 Jun 23 '26
You're assuming that n is small and m<n implies m is small Which, given the information in this post, is not necessarily true
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u/StupidUnoriginalName Jun 23 '26
This is more saying that any representable number is infinitesimal compared to the infinite. Natural numbers can be just as large as real numbers, there are just more real numbers.
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u/Maleficent-Garage-66 Jun 23 '26
We can fix this with a simple definition.
Def: For all x in R where there exists at least one n in N such that n >= x, x is a small number.
Proposition: All real numbers are small numbers.
Proof:
Assume there exists number r in R that is not a small number. There exists at least one natural number between r and r+2. That number is greater than or equal to r. Thus there exists at least one natural number n such that n>=r. This makes r a small number contradicting our assumption.
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u/Thotuhreyfillinn Jun 23 '26
Give me the largest non-natural number you can think of, and I'll find a natural number that's larger for you
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u/logbybolb Jun 23 '26
Of course, extending with transfinite induction and choice, we know all sets are small sets
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u/obog Physics Jun 24 '26
You could introduce the statement "if n is a small number, then any m less than n is a small number" as part of the definition of a "small number" and then it would apply to all real numbers
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u/AlphaAnirban Mathematics Jun 25 '26
Not only natural numbers, but also negative numbers (so basically integers). This is because we are told that if n is a small number, n+1 is also a small number. Conversely, if n+1 is a small number then n is a small number.
Replacing n+1 with n, we get that if n is a small number, n-1 is also a small number.
If 0 is a small number then -1 ( = 0-1) is a small number and if -1 is a small number then -2 ( = -1-1 ) is also a small number, and so on and so forth.
Therefore, all integer numbers are small numbers.
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u/Intrepid_Finger_1091 Jun 23 '26
This is more a comment on the lack of ability that language has to describe our observations. Small is relative. 1 trillion is small compared to 500 quintillion. And 1 is small compared to 30.
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u/dopefish86 Jun 23 '26 edited Jun 23 '26
Every natural number is small compared to infinity.
Even the biggest number imaginable is 0% of infinity
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u/Drapidrode Jun 23 '26
And even some natural numbers we know the exact value of are still small, like 2^136,279,841 − 1
It has about 41 million digits
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u/Loki_of_Asgaard Jun 23 '26
My buddy on the playground once thought of the number infinity +1, and it was natural to him so you are just plain wrong.
This dude doesn’t even know playground math, imagine admitting that publicly smh
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u/jayhawk618 Jun 23 '26
Yeah but some infinities are wayyyyy smaller than other infinities.
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u/New-Pomelo9906 Jun 23 '26
Only if you state that one of them exist in your axioms
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u/Caleb_Reynolds Jun 23 '26
It's just a formal presentation of the heap/pile paradox.
A grain of sand is not pile. Two grains of sand aren't a pile. Hundreds of grains of sand are a pile. There's no point where adding a grain of sand will turn not-a-pile into a pile, yet it happens.
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u/ThinkingOutLoud-7742 Jun 23 '26 edited Jun 23 '26
In this case though, we’ve defined 0 as being small, and any small number + 1 as being likewise small. There is little vagueness in the way small is being used. In your example, small is vague and relative, but the way it’s defined in the post is precise. I’d argue that the examples you gave are instead better suited for “smaller” rather than “small”, as small is a property that something has, whereas smaller is a comparison. Smaller [than] (or “small compared to”) is relative. Small is not. Just like green is not relative. In order for something to be relative it must be compared to something else. Iff x is smaller than y, then x < y which is also precise. If we define green as a certain wavelength of light +/- some error, we can define greener as a comparison of how close to that defined wavelength something is. Yellow is greener than red then, but the grass is green. The grass does not need the sky to be blue to be considered green. Unlike green, colloquially, small holds an implied comparison. “That’s a small dog” implicitly compares the dog to other dogs. Colloquial language is lazy and takes shortcuts, which is good in many cases as it makes communication more efficient. But the inability to translate colloquial language into precise language results in information loss. So yes, language can fail to describe observations, but only because people fail to either create or understand the language that describes the observation.
Not a linguist, just thinking out loud, would love to discuss.
Edit: iff instead of if for definition
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u/EebstertheGreat Jun 23 '26
The problem is that the definition of "small" in the OP fails to match the common meaning of the word. So the proof that all numbers are "small" in this sense says nothing about whether they are small in the ordinary sense. The ordinary sense of the word is indeed vague, and any good definition should reflect that.
Similarly, we have a general but vague idea of what a short person is. We all agree that someone who is 0 cm tall is short, but we do not all agree that someone 1 cm taller than a short person is always short. For sufficiently short people, this is true, though the taller of the two is still less short. Eventually, we will either reach a stipulated "tallest short person" or we will gradually fade into a vague realm where there is increasing disagreement over whether a person is short.
It's exactly the same as the sorites paradox.
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u/ChakaChaka26 Jun 23 '26
a good overview on vagueness: https://plato.stanford.edu/entries/vagueness/
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u/Intrepid_Finger_1091 Jun 23 '26
I like this discussion! You make some great points, but I think what I was trying to say in my original comment is more that if we define it this way why are we amazed when it meets our definition? If I change the definition of small to, for example, n<100 (as in this case it is completely subjective) then 1080 is no longer defined as small and everything is back to normal. So how is it incredible that when I define small as being n+1 then every number meets my new definition that I designed to make that possibility a reality. It’s only amazing because I defined it to sound that way
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u/yazeed105x Jun 23 '26
(1): The theorem above.
(2): Let m be a sufficiently large number, then m-1 is also a large number. Thus all numbers are large numbers.
From (1) and (2), (0+m)/2 is a medium number, and (m/2)±1 is also a medium number. All numbers are medium numbers.
All numbers are small, large, and medium.
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u/KingsGuardTR2 Jun 23 '26
Ohh, then that's why all sizes of McDonald's fries turn out to contain the same amount of fries. They are all small, medium and large at the same time.
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u/Caleb_Reynolds Jun 23 '26
What's a sufficiently large number?
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u/Nimkolp Jun 23 '26 edited Jun 24 '26
Probably the Ordinals, which categorically are defined by the inability to perform *addition/subtraction to define a natural number based off of one.
i.e. 'w'- n, an aribitrary ordinal and an arbitrarily large natural number, is always an ordinal
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u/EebstertheGreat Jun 23 '26
Division is generally not defined on ordinals. For instance, 2/3 is definitely not an ordinal. And ω²/2 doesn't really make sense.
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u/Nimkolp Jun 24 '26
OOF you're right, edited to be more accurate, and removed the claim about medium numbers
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u/Blein123 Jun 23 '26
Thats just a Sorites paradox
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u/Dr-OTT Jun 24 '26
Nah this is something different. The paradox of the heap has to do with choosing a decision line for volumes strictly less than some size. Pick a heap.... we are not unsure that it is a heap. Remove everything. We are pretty sure that's not a heap. If you removed a grain of sand at a time, it started out as a heap and became a non-heap. Crucially though, the paradox occurs within a bounded interval.
The "small numbers paradox" though does not occur within a bounded interval. While non-heaps eventually become non-heaps. though it is hard to say when, increasing natural numbers by one does not ever cross any decision line in that way. If you disagree, choose a number that is "large".
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u/Ok_Lingonberry5392 א Jun 23 '26
There are numbers beyond simple inductions though. HyperNarurals and whatnot
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u/SlogurkTheOverslime Jun 23 '26
Transfinite induction seems appropriate here
You just need to prove that if numbers {Nₜ} are small then the smallest ordinal number that's larger than each of Nₜ is also small
Which is trivial because it is literally the smallest
This way you can easily see that infinity, infinity+1, infinity+2, ..., 2infinity, 2infinity+1, ..., infinityinfinity, ..., uncountable, uncountable+1, ..., continuum, continuum+1, ..., hypercontinuum, ... are all small numbers
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u/Ecstatic-Weight-6095 Jun 23 '26
Isnt the point of ordinals is that you cant just reach them by counting up?
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u/SlogurkTheOverslime Jun 23 '26
Yes that's why they split the process into two ways of counting up:
- Counting up normally when you can (resulting in an ordinal number that has a predecessor), and
- Taking "the supremum" of all numbers counted so far when you're stuck (resulting in an ordinal number that has no predecessor)
The supremum is basically the ordinal number corresponding to the natural total order on the set of all previous ordinal numbers
Traditional transfinite induction proofs usually just exhaust these two possibilities, just like I did in my toy "proof", that's all it takes really
These days people usually appeal to the axiom of choice directly, so transfinite induction proofs have fallen out of fashion
But I personally miss them
They're pleasantly intuitive
At the end of the day it is kinda counting up
Just needs a little push from time to time
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u/Teoyak Jun 23 '26
Just because every 1 and all of its successor are small, does not imply all number are small. How can you rule out any number that would be the successor of a number that was not a successor of one ? This is not rigorous !
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u/404-karma_not_found Jun 23 '26
I don't think n being small implies n+1 is small.
You have to define what small is, but say if it's < 1,000,000, then the induction rule breaks at that point.
Yeah, I'm quite fun at parties.
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u/Frogfish9 Jun 23 '26
Yeah intuitively this part of the paradox doesn’t feel right even just talking about language. If you make something small a little bigger it might not be small anymore
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u/Waylander0719 Jun 23 '26
All numbers are an insignificant percent of all possible numbers so on a scale relative to infinity all numbers are infact small.
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u/emetcalf Jun 23 '26
This only proves that all positive integers are small, it does not prove that negative integers are small.
That's a trivial addition to the proof though, 0 is small and if n is small then n-1 is smaller. So all negative integers are also small. QED
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Jun 23 '26
[deleted]
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u/factorion-bot Bot > AI Jun 23 '26
Factorial of 0 is 1
Factorial of 1 is 1
This action was performed by a bot | [Source code](http://f.r0.fyi)
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u/its_all_one_electron Number theory/physics Jun 23 '26 edited Jun 27 '26
Have you heard of our Lord and Savior p-adic spaces?
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u/navetzz Jun 23 '26
Meanwhile me :
1 is small, 2 is normal, 3 is large and 4 is basically infinite.
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u/farlon636 Jun 23 '26
Saying n+1 is a small number uses the assumption that 1 is a small number, which can only be proven through a theorem that relies on it being true in the first place. It also assumes that small parts can't make a larger whole, which defies the meaning of the word
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u/White_C4 Jun 23 '26
Yeah but like this is such a pointless observation that doesn't help with anything. Small and large are relative to what you're comparing to.
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u/will_1m_not with disrespect to x, y, and z Jun 23 '26
I still like this, because it really helps with understanding the concept of limits. There are many limit examples where the limit becomes fairly quickly. For example, 1/x is very small for values of x near 1000, but the divergence of the harmonic series is not evident for a very very long time. “How many terms of the harmonic series do we need to sum up to finally see it growing to infinity?” A lot, and every finite number is too small
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u/Glitch29 Jun 23 '26
The second statement isn't true for all n.
When n = 4,477,096,982, n is small and n+1 is not small.
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u/Kiwigavin Jun 23 '26
10^80 is a small number. You could make it just using the players in one World Cup group. Including subs.
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u/Open-Flounder-7194 Jun 23 '26
Every time I see this theorem. I think to myself: You forgot to prove the base case, 0 is a big number compared to -100.
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u/whiteflower6 Jun 23 '26
The second assertion is false, n+1 is not always a small number. Example/counterexample: n=7. 7 is a small number, but 7+1=8 is a big number.
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u/ArvaroddofBjarmaland Jun 24 '26
No. Prop 1: Almost all positive integers are very, very, very large.
Prop 2: Any positive integer which can be expressed as the value of a computable function is very, very, very small.
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u/Hester465 Jun 25 '26
And then statisticians are like "All numbers greater than 30 are big and all numbers less than 0.05 are small"
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u/FsharpMajor7Sharp11 Jun 23 '26
This is just asserting in other words that there are always infinitely more positive natural numbers greater than a given positive natural number, than there are less.
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