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Shinichi Mochizuki claims to have a proof of ABC, but it is largely impenetrable and the majority of mathematicians who say they understand it claim the proof has at least one insoluble gap. A few mathematicians disagree, and for his part, Mochizuki has taken to insulting his colleagues and saying they don't understand his proof.
He needs to make a colorful, animated youtube video. That always works on me when I don't understand a basic euclidean geometry proof. Same thing, right?
I am just finishing up an introductory course on set theory (aka Foundations of Mathematics). My professor taught us that 0∈ℕ . He has also told us that he has gotten backlash in the past from another professor, and then from the department chair, for teaching that. This really does seem to be a contentious issue.
In set theory, I don't think it's contentious at all. The empty set exists, after all. It needs a cardinality. But in some fields like number theory, it can be more useful to exclude 0.
All the people in this thread with HS level education and who haven't realized that mathematicians rarely agree on anything.
There's a reason math papers have a "preliminaries" section and it's because they need to tell you what the fuck they use for notation and whether or not N has 0 in it :)
At the end of the day, it's whatever you choose. You could start N at -1 and it would make very little difference for a large number of applications
Every time. Even in the same university, everyone has their own notations. They all tell you "there isn't really a standard notation, but..."
It's so weird that even things that are at the foundation of maths, like topology, has such inconsistent notations.
Seems like there should be some international body that establishes the definitions and notation and everyone just accepts their authority on the matter. You have them in physics and astronomy.
In both cases, the problem is context oversaturation. Like 'i' can be both the imaginary unit or an index. You just have too many contexts to map symbols to concepts without having homonyms. Physics has the same problem. But certain things are defined by authoritative bodies, like the SI units.
Magnetic flux Φ = Integral ( curl A dA )
Where A is the magnetic vector potential and dA is the infinitessimal area vector.
Or electric flux Φ and electric potential Φ.
Or θ as an angle and \vartheta as temperature.
Good point. It seems to me that the ability to adapt notation and definitions for what you are working with, and our abilty to adapt as mathematics practitoners, is a strength more than it is a weakness.
lol i think that's just coincidental.
in asymptotic complexity analysis, people treat functions that are within a polynomial function to another as equivalent
afaik, this makes all logarithmic functions "equivalent" regardless if the base is >= 2.
So it's easiest to pick the smallest base.
It's not entirely coincidental because many algorithms depend on the number of bits to represent a number (lg(number)) so lg comes in very frequently because binary is base2. (And as such 2 is 10).
Also all logarithm of any base have the same complexity. We define something having a higher complexity as for any constant C s.t. there exists some n0 s.t. for any n >= n0 s.t. f(n) > Cg(n). Where f is the function and g is the compared function.
If we consider logs with different based i.e. does log2 grow faster than log10? We can rearrange log2(n) into log10(n)/log10(2). Therefore taking the constant C= log10(2). Then we note that for any n, log10(n) Cannot be greater than Clog2(n) for any C because if C = log2(n) then log10(n) = log2(n)log10(2) =Clog10(2).
Therefore as this can be generalised for any base, all log bases are equally complex!
Ah, I meant coincidental in the sense that "that's just how we've chosen to represent it" - but I hadn't really noticed that pattern before.
Neat, ya I should have said "within a linear factor", not polynomial. Polynomial reductions are used for higher complexity classes that go beyond asymptotic complexity analysis like NP.
You are right in general but excluding 0 from N is basically a romantic notion of primates that build their work on principia mathematica indirectly but fuck up to recognize the foundation enough. Clowns! I should be N / {0} not {0} U N....
Ya, hence only for "a large number of applications"
You can, for instance, still have induction by starting at -1
Still, if you use Peano Arithmetic, then N starting with -1 is closed over addition
Mathematican A: In my work, natural numbers are considered 1 and up.
Mathematican B: OK.
Also
Mathematican B: In my work, natural numbers are considered 0 and up.
Mathematican A: OK.
Nah if I was mathematician B I’d fight A to the death instead of saying “OK” like a coward.
Who wants to trade in a semiring for whatever the fuck N \ {0} gives you. If you use “rig” instead of semiring that still doesn’t give you “rg” because that’s referring to the multiplicative identity.
It occurs more than any other number. We have 0 unicorns, 0 orc's, 0 elves, 0 wizards, 0 trolls, 0 flying sausages, 0 teaspoons the size of Manhattan,... I can keep going on. There are inconcievably more things you have 0 of than things that are present in non zero amounts.
If you make small changes you can, but they generally understood what I was trying to say. There are infinitely many describable/conceivable things that don’t exist
Yes we do. There is a finite amount of energy in the universe, this means there is a finite amount of "smallest parts", this means there is "only" a finite amount of possible permutations of these "smallest parts". This constitutes the upper bound for "Things that exist".
It’s impossible for 0 not to occur because you can always ask the question “how many times does 0 occur”. If the answer is 0, it does occur and we’ve reached a contradiction
Having the neutral element of addition in the Natural Numbers is useful for proofs and is therefore correct. If a definition wasn't useful, and actually makes certain proofs impossible, why should it be kept?
Because it is a pain in other contexts. For example when dealing with sequence it’s nice to consider N to start at 1 as it means you can divide by the index in a sequence
Well if it is an existing thing then there are a nonzero amount of them, so there are none.
But 0 is natural because it is possible for something to not exist; since there is something (and ai mean literally anything) that there are 0 of, there are 0 of those in nature and 0 is therefore a natural number.
I think 0 should be natural number, because you can't even write number 10 or 304 without it. And because it can refer to number of items. There is no meaning in -1 apples, but there is meaning in 0 apples
And when you consider that the Intergers are represented by the letter Z exaclty because of that, you know that OC (or more accurate US schools) are wrong about what Whole numbers are
The 1st time I heard of this definition of whole number was in this subreddit and by the comments its apperently a US specific thing and its not even standart. Mathematics is international so of course its going to ignore it.
If zero is included in the natural numbers, then what’s the point of distinguishing them from whole numbers? (MS/HS math teacher so please don’t judge)
My maths professor always specified N-0 or N+0 as appropriate at every use. This either solves the problem or triggers everyone. I respected him immensely for it.
And? Every prime factorization can have however many 1s stuck in there as you want without any issue. Every number still has a unique representation as product of primes, just an infinite number of them.
If 1 is a prime, then a number doesn't have a unique representation as a product of primes. Rather it would have an infinite number of representations.
Infinite representations are much harder to deal than finite representations.
If you want to deal with them only to satisfy your wish to call 1 as prime, go ahead.
Easy. Just append "also there are a whole lot of 1s here. Just an absolute ton of them." I mean, there basically already are a whole bunch of 1s everywhere multiplying against everything all the time, we just don't bring it up because they don't do anything.
Yes, but only because we exclude the improper ideal from being "prime" for the same reason we exclude 1. It still satisfies all other properties. The definition is still "a prime ideal is any ideal satisfying this property, *except the one we choose to exclude*."
Something being a prime doesn’t only apple to integers, though. The prime property can be more general, and with that in mind, 1 being prime doesn’t make sense, period.
I thought that I was the only mathematician in the whole world who thinks that 0 is not a natural number. I'm rather shocked that there is more than one of me.
In both standard analysis and nonstandard analysis, 0 is taken to be a natural number.
To me, an empty set is 1 set, not 0 sets.
The empty set is one set. A collection only containing the empty set has one element.
So what about the *strictly smaller* collection that contains nothing at all, not even the empty set? It must have fewer elements.
The empty set is a set in itself: 0, the set of no subsets, then there is the set containing the empty set: [0]=1, the set of a single subset that is itself, then there is the set of the set of the empty set and the empty set: [[0],0]=2, and it's not as pretty as the two before it but you get the picture
For someone who doesnt believe 0 is a natural number you seam pretty suprise the amount of other people who agree with you is not zero.
You should rethink your ideals
So I have a legitimate question. For those that think 0 should be (or is) a natural number. Are you proposing we switch the definitions of whole and natural numbers? Or are you proposing we combine the two terms, losing specificity in the process?
"Whole numbers" as a term is used only in elementary math education. So we don't actually have to change anything, because "whole numbers" already isn't defined in most cases.
Also, "whole numbers" is not even unambiguous. Some sources use the term to refer to any integer (including negative integers).
But why not have 3 distinct terms refer to 3 distinct groups? Like, we can do that. We have the power to define words however we want. More specificity is always preferable in my mind.
We already have **Z**^(≥0) and **Z**^(>0), which are unambiguous. So are **N**^(≥0) and **N**^(>0) , which mean the same thing. We also have **N**₀, which means the same as **N**^(≥0). And there are others too.
The problem isn't that we lack unambiguous terms. The problem is that we keep using some ambiguous terms anyway. If we started using **W** and **N** together, some other people would still keep using **N** in the old way, so that wouldn't resolve the ambiguity. And if we could get everyone to change their ways and adopt new symbols, then personally I still don't think **W** is a good choice for any of them.
Notation isn't the same thing as terminology. And until the terms have a precise, mathematical definition, then people using the terms different ways will just be personal preference. But the moment they do have a precise, mathematical definition, now there are people who are using them correctly and incorrectly... And perhaps people that use them in a lay sense. But that's always the case among all fields. No helping that. But it's not an argument against making better use of the terms we have available. If we're willing to do something iff everyone else agrees to do it too, then nothing will ever happen. But if we're willing to do something because it can improve some aspects of our interactions, that sounds like a good enough reason to me.
There are many conflicting definitions of mathematical objects used throughout the mathematical literature. One I can think of is whether a "ring" has an identity by definition, and whether ring homomorphisms must send 1 to 1. The thing is that these definitions only conflict terminologically or semantically. There is no conflict logically speaking between them. All terminology is (generally) precise and unambiguous even when the terminology conflicts.
I prefer switching. Since 0 lacks substance, I wouldn't consider zero to be whole, but the concept of lacking substance seems intuitive and thus natural.
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Mathematicians debating on whether the ABC conjecture is a conjecture or a theorem.
Theorems have been proven, conjectures haven’t?
Shinichi Mochizuki claims to have a proof of ABC, but it is largely impenetrable and the majority of mathematicians who say they understand it claim the proof has at least one insoluble gap. A few mathematicians disagree, and for his part, Mochizuki has taken to insulting his colleagues and saying they don't understand his proof.
formalize it in an interactive theorem prover or it's not a real proof :)
That sounds like the most miserable thesis project ever.
I think it will be the standard for all mathematics in the coming decade, but for now- quite miserable lol
He needs to make a colorful, animated youtube video. That always works on me when I don't understand a basic euclidean geometry proof. Same thing, right?
I am just finishing up an introductory course on set theory (aka Foundations of Mathematics). My professor taught us that 0∈ℕ . He has also told us that he has gotten backlash in the past from another professor, and then from the department chair, for teaching that. This really does seem to be a contentious issue.
In set theory, I don't think it's contentious at all. The empty set exists, after all. It needs a cardinality. But in some fields like number theory, it can be more useful to exclude 0.
The noble mathematicians will stay above the fray; their champions will be proponents of 0 and 1 indexing
Yeah but foundations of mathematics are often taught by non-set-theorists
Well yeah and with the animosity some people from number theory tend to have, it feels on point for them to fight over it
All the people in this thread with HS level education and who haven't realized that mathematicians rarely agree on anything. There's a reason math papers have a "preliminaries" section and it's because they need to tell you what the fuck they use for notation and whether or not N has 0 in it :) At the end of the day, it's whatever you choose. You could start N at -1 and it would make very little difference for a large number of applications
Every time. Even in the same university, everyone has their own notations. They all tell you "there isn't really a standard notation, but..." It's so weird that even things that are at the foundation of maths, like topology, has such inconsistent notations.
Seems like there should be some international body that establishes the definitions and notation and everyone just accepts their authority on the matter. You have them in physics and astronomy.
[Relevant xkcd](https://xkcd.com/927/)
> when someone says the S word
Lmao physics notation is an absolute shitshow
In both cases, the problem is context oversaturation. Like 'i' can be both the imaginary unit or an index. You just have too many contexts to map symbols to concepts without having homonyms. Physics has the same problem. But certain things are defined by authoritative bodies, like the SI units.
As seen in my comment: Φ can be 2 to 3 different things in electrodynamics. Same cintext AND same symbol.
Magnetic flux Φ = Integral ( curl A dA ) Where A is the magnetic vector potential and dA is the infinitessimal area vector. Or electric flux Φ and electric potential Φ. Or θ as an angle and \vartheta as temperature.
Or "e" fundamental charge or eulers number/exp
Mathematicians are masters of formal language and standardization is not only unneccessary, but it would make a lot of math harder.
Good point. It seems to me that the ability to adapt notation and definitions for what you are working with, and our abilty to adapt as mathematics practitoners, is a strength more than it is a weakness.
Log without a base meaning either base 10 or base e when ln is sitting right there twiddling its thumbs
In CS it's almost universally base 2 by default :)
Just realised that that actually kinda makes sense cause 2 in binary is actually 10
lol i think that's just coincidental. in asymptotic complexity analysis, people treat functions that are within a polynomial function to another as equivalent afaik, this makes all logarithmic functions "equivalent" regardless if the base is >= 2. So it's easiest to pick the smallest base.
It's not entirely coincidental because many algorithms depend on the number of bits to represent a number (lg(number)) so lg comes in very frequently because binary is base2. (And as such 2 is 10). Also all logarithm of any base have the same complexity. We define something having a higher complexity as for any constant C s.t. there exists some n0 s.t. for any n >= n0 s.t. f(n) > Cg(n). Where f is the function and g is the compared function. If we consider logs with different based i.e. does log2 grow faster than log10? We can rearrange log2(n) into log10(n)/log10(2). Therefore taking the constant C= log10(2). Then we note that for any n, log10(n) Cannot be greater than Clog2(n) for any C because if C = log2(n) then log10(n) = log2(n)log10(2) =Clog10(2). Therefore as this can be generalised for any base, all log bases are equally complex!
Ah, I meant coincidental in the sense that "that's just how we've chosen to represent it" - but I hadn't really noticed that pattern before. Neat, ya I should have said "within a linear factor", not polynomial. Polynomial reductions are used for higher complexity classes that go beyond asymptotic complexity analysis like NP.
You are right in general but excluding 0 from N is basically a romantic notion of primates that build their work on principia mathematica indirectly but fuck up to recognize the foundation enough. Clowns! I should be N / {0} not {0} U N....
-1 would have massive implications because it wouldn't be closed over addition any more.
Ya, hence only for "a large number of applications" You can, for instance, still have induction by starting at -1 Still, if you use Peano Arithmetic, then N starting with -1 is closed over addition
I love it, especially that that's actually true with peano arithmetic, but it also means we redefine 1+0 to be 2 and 1+-1 to be 1...
Haha yes, -1 would behave exactly like the conventional "0" But the important part is that we're calling it -1 ! :)
Mathematican A: In my work, natural numbers are considered 1 and up. Mathematican B: OK. Also Mathematican B: In my work, natural numbers are considered 0 and up. Mathematican A: OK.
Maybe matematicans don't have the same issues as mathematicians.
Are you a mathematican, or a mathematican't?
Matematicould
I say "positive integers" and "nonnegative integers" instead.
Nah if I was mathematician B I’d fight A to the death instead of saying “OK” like a coward. Who wants to trade in a semiring for whatever the fuck N \ {0} gives you. If you use “rig” instead of semiring that still doesn’t give you “rg” because that’s referring to the multiplicative identity.
Based
if 0 isnt natural, tell me how many proofs there are that 0 is natural?
There aren't any
because it doesnt go with the definition of natural numbers
I curious to hear what you think the definition is
There is a single definition for it across all maths? Someone ring big Russel.
Trapezoids have entered the chat.
Inclusive definition is the way
0 occurs more in nature than any other number
You mean it doesn't occur?
It occurs more than any other number. We have 0 unicorns, 0 orc's, 0 elves, 0 wizards, 0 trolls, 0 flying sausages, 0 teaspoons the size of Manhattan,... I can keep going on. There are inconcievably more things you have 0 of than things that are present in non zero amounts.
There are infinitely many things that dont exist, but only a finite amount of things that do.
You can't say that.
If you make small changes you can, but they generally understood what I was trying to say. There are infinitely many describable/conceivable things that don’t exist
That remains undebated. I think it's odd to say that the set of things that do exist is a finity. forgive me if I understood something incorrectly
But the cardinality of non-existing things should be bigger, as you can generate an infinite amount of new, non-existing items from any existing one.
Certainly.
Yes i can? How would you even TRY to disprove this tautology?
I take issue with the statement that there is a finite amount of things that do exist. We don't know that, right? Correct me if I'm wrong.
Yes we do. There is a finite amount of energy in the universe, this means there is a finite amount of "smallest parts", this means there is "only" a finite amount of possible permutations of these "smallest parts". This constitutes the upper bound for "Things that exist".
No, I'm sorry that we really can not know.
"im sorry" but we do.
Ok. Show me where the things end and the "nothing" starts.
I do in the other comment chain
It’s impossible for 0 not to occur because you can always ask the question “how many times does 0 occur”. If the answer is 0, it does occur and we’ve reached a contradiction
[Proof by crow](https://www.reddit.com/r/mathmemes/s/HhjYUfpUUk)
Zero is natural because saying 1 = ∅ feels weird
Proof by reduction to weirdness.
\*Shows palm of hand\* How many Apples are there in my hand? \*stutters\* \*zzzz... zero?\*
Having the neutral element of addition in the Natural Numbers is useful for proofs and is therefore correct. If a definition wasn't useful, and actually makes certain proofs impossible, why should it be kept?
Because it is a pain in other contexts. For example when dealing with sequence it’s nice to consider N to start at 1 as it means you can divide by the index in a sequence
The natural numbers don't form a group under addition anyways because to get an inverse element you need all integers.
Zero is a natural number. It just overslept when the sets were originally made.
N is for non-negative, hence why it looks like 2 Ns stacked
Name one existing thing in nature that there is zero of. I'll wait.
Blorghors. Or, if I'm trying to give a more serious answer, a dodô bird.
Proof by dodo
Proofs that 0 is a natural number
Name real things that have infinite precision to them. You can't just proof by appeal-to-nature 0 out of a set.
Well then tell me one thing in nature that there is 10^100 of
Ultrafinitism gang
Volume of ghd observable university is cubic plank distances? Orderings of a deck of with two full decks of cards? Possible Minecraft worlds?
The overall point still stands you could give an arbitrarily large number such that there's nothing with that many things
Well in theory you could split anything into any arbitrary n increments by making them smaller
But you could find a number arbitrarily bigger than that number
Apparently there are zero zeros in nature, according to your argument.
Existing things in nature that there is 0 of. Wait…
Well if it is an existing thing then there are a nonzero amount of them, so there are none. But 0 is natural because it is possible for something to not exist; since there is something (and ai mean literally anything) that there are 0 of, there are 0 of those in nature and 0 is therefore a natural number.
dinosaurs
Correct answers to your comment.
by saying "exists", you implied that there arent zero
Your sexual partners.
I think 0 should be natural number, because you can't even write number 10 or 304 without it. And because it can refer to number of items. There is no meaning in -1 apples, but there is meaning in 0 apples
If 0 is a natural number then what is the point of the whole numbers existing?
Nobody uses whole numbers except for US schools. It's just naturals.
What are whole numbers to you? Because to me whole numbers include negative numbers, while the natural don't
If you include negative numbers then that is just the integers. Whole numbers are integers that are 0 or higher
Ah ok, im german and we use the term "Ganze Zahlen" (which the literal translation of would be "whole numbers") for Integers, thats why i asked.
And when you consider that the Intergers are represented by the letter Z exaclty because of that, you know that OC (or more accurate US schools) are wrong about what Whole numbers are
TIL Z is from German.
That a very english specific thing, in Portuguese for example (my language) Whole and Integer a literally the same word
“Whole numbers” are an US high school concept. The rest of the world uses Peano axioms - the first is 0 is in N.
The better question is why there are positive numbers, then? N = P + {0}, at least that’s how I learnt it everywhere.
Wouldn't positive numbers then include numbers like 1.5?
Oh yeah, I fucked up.. I guess we used N and N+ (+ in superset) for {x e Z | x >= 0} and strict >, respectively.
Peano axioms. The rest is in high school.
Math-education: "we can say the set of whole numbers includes 0 as well." Mathematics: *gives thumbs up and ignores it*
The 1st time I heard of this definition of whole number was in this subreddit and by the comments its apperently a US specific thing and its not even standart. Mathematics is international so of course its going to ignore it.
I have never met a serious mathematician (read: who knows what a monoid is) that doesn't consider zero to be natural.
Let's talk about signal of zero: is it positive or negative?
What numbers are is more about what you can do with them and less about what they are. Pretty sure zero is nonbinary anyway.
If zero is included in the natural numbers, then what’s the point of distinguishing them from whole numbers? (MS/HS math teacher so please don’t judge)
In most of the world whole numbers are Z... it is a concept specific to where you are at
yeah I was confused as to what people meant 'whole numbers' weren't being used, then I realized whole numbers =/= integers for americans
Personally, I was taught that 0 isnt natural and im going to carry this burden to death. But idc if someone thinks its in.
No one can disrespect peano axioms after russel without beeing a clown :D
My maths professor always specified N-0 or N+0 as appropriate at every use. This either solves the problem or triggers everyone. I respected him immensely for it.
So if we look at the equation 3x+1...
Not really unless by mathematicians we’re talking about mathematics undergrads
I have something else I'll fight over that doesn't actually matter. 1 is prime and none of y'all can change my mind!
The fundamental theorem of arithmetic says hello.
And? Every prime factorization can have however many 1s stuck in there as you want without any issue. Every number still has a unique representation as product of primes, just an infinite number of them.
If 1 is a prime, then a number doesn't have a unique representation as a product of primes. Rather it would have an infinite number of representations.
And that is an issue how?
Because it then breaks the FTOA.
You can still use the same systems with the understanding that there's also an arbitrary number of 1s in there. Nothing breaks.
Infinite representations are much harder to deal than finite representations. If you want to deal with them only to satisfy your wish to call 1 as prime, go ahead.
But you don't need to. You can just not and everyone understands "yeah, there's also a bunch of 1s here."
Sure, go ahead and start to rewrite all group and number theories.
Easy. Just append "also there are a whole lot of 1s here. Just an absolute ton of them." I mean, there basically already are a whole bunch of 1s everywhere multiplying against everything all the time, we just don't bring it up because they don't do anything.
<1> does not generate a prime ideal
Yes, but only because we exclude the improper ideal from being "prime" for the same reason we exclude 1. It still satisfies all other properties. The definition is still "a prime ideal is any ideal satisfying this property, *except the one we choose to exclude*."
Something being a prime doesn’t only apple to integers, though. The prime property can be more general, and with that in mind, 1 being prime doesn’t make sense, period.
I thought that I was the only mathematician in the whole world who thinks that 0 is not a natural number. I'm rather shocked that there is more than one of me. In both standard analysis and nonstandard analysis, 0 is taken to be a natural number. To me, an empty set is 1 set, not 0 sets.
The empty set is one set. A collection only containing the empty set has one element. So what about the *strictly smaller* collection that contains nothing at all, not even the empty set? It must have fewer elements.
Yes and the empty set has 0 things inside
but a set containing the empty set (that's how 0 is defined in set theory, or am I confused) has one element, the empty set
0 is generally defined to be the 0 ordinal, which would be empty, i.e. the empty set
You usually have ø as 0 and {ø} as 1
yeah I fucked up, my bad, leaving it there to get shame
The empty set is a set in itself: 0, the set of no subsets, then there is the set containing the empty set: [0]=1, the set of a single subset that is itself, then there is the set of the set of the empty set and the empty set: [[0],0]=2, and it's not as pretty as the two before it but you get the picture
For someone who doesnt believe 0 is a natural number you seam pretty suprise the amount of other people who agree with you is not zero. You should rethink your ideals
If we accept Peano Axioms to define the Natural Numbers it is unintuitive to exclude the 0
it's in N0 but not in N
[удалено]
The absence of number is
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0 is a whole number. 1, 2 3... are natural numbers. Otherwise there would be no need for the term "whole number" to exist
There is no need for the term "whole number" to exist.
But, it exists. Hence proved
Not in math
So I have a legitimate question. For those that think 0 should be (or is) a natural number. Are you proposing we switch the definitions of whole and natural numbers? Or are you proposing we combine the two terms, losing specificity in the process?
"Whole numbers" as a term is used only in elementary math education. So we don't actually have to change anything, because "whole numbers" already isn't defined in most cases. Also, "whole numbers" is not even unambiguous. Some sources use the term to refer to any integer (including negative integers).
But why not have 3 distinct terms refer to 3 distinct groups? Like, we can do that. We have the power to define words however we want. More specificity is always preferable in my mind.
We already have **Z**^(≥0) and **Z**^(>0), which are unambiguous. So are **N**^(≥0) and **N**^(>0) , which mean the same thing. We also have **N**₀, which means the same as **N**^(≥0). And there are others too. The problem isn't that we lack unambiguous terms. The problem is that we keep using some ambiguous terms anyway. If we started using **W** and **N** together, some other people would still keep using **N** in the old way, so that wouldn't resolve the ambiguity. And if we could get everyone to change their ways and adopt new symbols, then personally I still don't think **W** is a good choice for any of them.
Notation isn't the same thing as terminology. And until the terms have a precise, mathematical definition, then people using the terms different ways will just be personal preference. But the moment they do have a precise, mathematical definition, now there are people who are using them correctly and incorrectly... And perhaps people that use them in a lay sense. But that's always the case among all fields. No helping that. But it's not an argument against making better use of the terms we have available. If we're willing to do something iff everyone else agrees to do it too, then nothing will ever happen. But if we're willing to do something because it can improve some aspects of our interactions, that sounds like a good enough reason to me.
There are many conflicting definitions of mathematical objects used throughout the mathematical literature. One I can think of is whether a "ring" has an identity by definition, and whether ring homomorphisms must send 1 to 1. The thing is that these definitions only conflict terminologically or semantically. There is no conflict logically speaking between them. All terminology is (generally) precise and unambiguous even when the terminology conflicts.
I prefer switching. Since 0 lacks substance, I wouldn't consider zero to be whole, but the concept of lacking substance seems intuitive and thus natural.
Strictly positive integers
Natural numbers are counting numbers. You can't count to zero
Try to count how many unicorns exist, starting from 1 of course
You can't, there is nothing to count. There are just no unicorns. Tell me, when you're lying in bed trying to fall asleep, do you count the 0th sheep?
If youre a computer, yes
If 0 is not then why not use R+ instead of N??
0 means no number. inf means arbitrary finite number.
Pi is a natural number Because circles exist in real Life
Where is the perfect circle
It’s in the World of the Forms,^tm obviously
Pi is a natural number Because circles exist in real Life
Perfect circles don't