Is it or nah?
>>7936060
I feel that it is very natural to include it.
>>7936064
lel, no.
N is created only to have The principle of induction, which is the most dubious inference rule of all.
>>7936060
return Does 0 fuck up shit ? 0∉ℕ : 0∈ℕ;
It's gets pretty real on a dmt trip.
>>7936060
>lowest non-finite ordinal
>set of all finite ordinals
>set of all finite ordinals except the smallest one
Yeah no, out of these three, the last one definitely sounds way more natural to me.
>>7936060
[math] \mathbb{Z}^{\geq 0} [/math] includes it
[math] \mathbb{Z}^{>0} [/math] does not
problem solved
>>7936452
You didn't solve shit, buckaroo. The question is which of these sets is N?
Just look at the Peano Axioms.
If you are considering the Von Neumann ordinals as your set theoretic basis for the natural numbers, then yes. If you are a working mathematician then probably no, at least the mathematicians I've met. It doesn't affect anything though it is just convention.
>>7936060
Find an example of 5 in nature. How about 30? 54297?
Now find an example of zero in nature.
Zero is a concept; it's is the idea of "nothing". That's why it's a whole number, not a natural number. It represents a possible quantity of something, but is not a natural thing in itself.
>>7936450
Why arbitrarily exclude the smallest one?
>>7936915
I think he was being ironic
>>7936908
every number is a formalization. zero is just a formalization, as a number, of something .
>>7937556
But it's not a formalization that is observable in nature.
You can see, count, organize, reference, and use 4 coconuts. Try doing that with 0.
I didn't invent the definition of natural numbers, this is just always how I've visualized the discussion of how it came to be called natural, and why it's used.
That argument falls apart when you think of the term "whole" numbers, though... Since... How can zero of something constitute a whole.. But anyways.
Really, only people discussing mathematics, who need to reference all positive integers above 0, use the term Natural numbers. It has very little purpose other than to say the above in a short and recognizable way.
>>7936908
I haven't found 3, 4 or 2137 in nature
I may have saw coconuts, which quantity I can describe as 3, but haven't seen 3 in nature on its own
>>7937572
the alpha male that's fucking your girflfriend steals your 4 coconuts and shoves them up your mother's cunt, how many do you have now? oh thats right you cant say how many because youre a fag
>>7937589
Jokes on you, I don't have a girlfriend.
>>7936098
Really?
You literally don't believe there's a set of natural numbers.
Wow.
>>7937744
just like any mathematician taking seriously finite mathematics or even ultrafinite mathematics
... as though numeration is in any way "natural".
>>7937995
Ok, so, I guess that that sort of cancer works in ZF minus the axiom of infinity, plus the assertion that everything is dedekind finite.
So finitary algebraic structures and graphs. That's all they can do. And they don't have induction.
Just sad.
>>7937585
What's your first reactionary thing to say when someone asks "how many apple are in that (empty) basket?".
"None".
You don't say zero. You mention the absence of apples, but don't refer to it as a count/number of apples equal to zero.
Again, didn't make the rule, this is just how I see it making sense. If you have a better argument for why they're separate and called Natural, please share.
>>7938044
>>Just sad.
but less fantasmatic
>>7936473
Don't use N because it a shit
people think that the equality on N is decidable
help me /sci what method should i use to solve it?
>>7939781
trig sub, x=sqrt(6)*sin(theta)
>>7939781
[eqn]\displaystyle u = \sqrt{6} \sin{\theta}[/eqn] Now go away.
>>7939808
What's "u"?
>>7939824
Oh shit nigger, thanks for catching that.
That should actually be [math]x[/math], idk why the fuck I said [math]u[/math].
N or Q ?
I actually think it is a part of the natural numbers
>>7936450
[math]\omega[/math] is just a model for [math]\mathbb{N}[/math]. When not doing set theory nobody gives a shit if the naturals are ordinals, they just have to be a Peano structure. Or do you construct [math]\mathbb{R}[/math] such that [math]\omega \subset \mathbb{R}[/math]? Doing so is a fucking mess and serves no practical purpose.
>>7940455
[math] \omega [/math] is a set, which viewed as a model endowed with the obvious interpretation of [math] \in [/math] is a linear order. But strictly speaking I would not say it's a model over the language of arithmetic, though the natural interpretations of [math] + [/math] and [math] \cdot [/math] are definable over it.
how is ω defined ?
what changes between ω and N ?
>>7936060
It is natural if I get dubs
>>7940567
0 confirmed for not being a natural number.
>>7936098
>what is a real number
the set of the real numbers is the the smallest complete set containing the rational numbers.
So a real number is a limit of a rational cauchy sequence.
>describe precisely the operations:
sum/product of the limits is the limit of the sum/product
>prove the laws of arithmetic for these.
trivial limits manipulation.
>provide plenty of illustrative examples
e is the sum of of the 1/n!
pi/4 is the sum of the (-1)^n/(2n-1)
e+pi/4 is the sum of the 1/n! + (-1)^n/(2n-1)
>>7940549
b) [math] \omega [/math] is defined as the least infinite ordinal. [math] \mathbb{N} [/math] is defined as the structure that everyone intuitively knows -- the set of natural numbers endowed with binary operations [math] + [/math] and [math] \cdot [/math] interpreted in the canonical way, often also with constant symbols [math] 0 [/math] and [math] 1 [/math].
Thus, [math] \omega [/math] is a simple set, while [math] \mathbb{N} [/math] is a model over the first-order language of arithmetic [math] \mathcal{L} = (0,1,+,\cdot ) [/math].
bump for N
in the end, only finite sets are coherent
>>7941942
[eqn]\mathbf{N}[/eqn]
>>7940455
I do give a shit. I use it to write stuff like R^n and let the objects of the simplicial category be positive integers. Seems to be incredibly inconvenient to have an integer with infinite cardinality.
>>7936060
N does not include zero but it has an 'identity element'. I don't know if that was the exact name but it was 'something element'.
Basically a + e = a where e is the element so obviously e = 0 but 0 does not exist there, it is just a concept.
>>7942770
yeah e=1, +=*
>>7936098
Who is this guy?
>>7942790
We are talking about addition here.
>>7942770
The fancy math-degree words for what you're saying are: "Algebraic structures without identity can be embedded in algebraic structures with identity." When you enlarge your algebraic structure like that though, (from a semigroup to a monoid in this case) you have to start making the distinction of whether properties hold with respect to the smaller structure or larger structure, and that contributes to confusion and general annoyance.
>set of natural numbers
implying we need this for anything
literally why do u need the numbers to be in a set
simply replace in every textbook
>for any n in N
with
>for any natural number n
and we got rid of this shitty set
>>7942894
g2b youtube Norman
>>7942894
Mathematics would be so fucking dull if only finitary objects were considered.
Also, there goes calculus. Any function [math] \mathbb{N} \rightarrow \mathbb{N} [/math] would now be a class, and we cannot quantify over classes. So even in a Wildburger-style approach to calculus, you could prove theorems only on an individual basis, but you couldn't state, for instance, the fundamental theorem of calculus, as that would involve quantification over all classes.
>>7942894
how do you define a function without sets
Hard mode: more efficiently
>>7942791
wildberger
>>7942958
Okay this has got me confused.
What's the difference between saying
Every rational has a greater positive integer
But you can't prove that
Every real number has a greater positive integer
You can't do the first one exhaustively either.
I thought wildbergers point was that you can't take properties of the full set like cardinality or length
the nlab has a good article on N
https://ncatlab.org/nlab/show/natural+numbers+object#relation_to_object_of_integers
and also
https://ncatlab.org/nlab/show/natural+numbers+type
Oh good, another thread pointlessly arguing over terminology that has literally zero effect on anything in the real world.
>>7944905
>guys please stop talking about things I don't understand I feel left out
>>7942791
>he doesn't know about the based mathematician of finitism
Isn't any number times 0 a copy of something into/over nothing?
>>7945156
Existentially you can't divide nothing into something. Nihilistically, nothing/nothing is nihilism, beyond reality= not a real number. 0*x is abyssal. To be in oblivion is to be anything but oblivion, for existence IS NOT oblivion, how ever being a subset of oblivion. Vacuously we can say that something is illogical in the existential sense that something is anything or nothing. When existentially there is nothing, we have a real representation of the symbol 0.
Discussions like these are just bike-shedding. It's easy to have opinions on worthless shit, you're not supposed to take stuff like this seriously.
It's a natural number if it makes things simpler when you're working with something, it's not if it doesn't.
[eqn]0 \in \mathbb{N}_0[/eqn]
>>7945193
tl;dr
0 is the most natural number.
>>7936060
The only time people use [math]\mathbb N[/math] is when it makes no difference. Those among us who are intelligent just say [math]\mathbb Z_{> 0}[/math] or [math]\mathbb Z_{\ge 0}[/math] to avoid any possible confusion.
>>7945210
negative numbers are not eal
>>7936098
>provide plenty of illustrative examples
>last vague, desperate point in case someone is actually bored enough to prove him wrong on the first three
kek
>>7945210
when you use N it's usually pretty clear from the context what you mean. Z>0 charges the notation too much imo unless it's a one-off thing in which case sure
>>7936117
I think it's nicer to have zero in the natural numbers
Then at least it's a monoid
> At a foundational level, it's completely irrelevant whether 0 counts as a natural number or not; as sets (and even as natural numbers objects), the two options are equivalent, so we are really talking about the choice of rig structure (or inclusion map into the set of integers, etc).
>>7936060
If it's not a prime and not composite, it's not natural.
>>7946737
>1 isn't a natural number
>>7946743
Of course not. You don't "count" something if there is only one of it.
>>7946746
I think that the point of formalizing is to go as far from empiricism as you can. so it is alright to consider infinite sets and non predicative mathematics
>>7936098
to be clear, the deductive principle of induction is ''if you have property stable under the succession function and your property holds for one number, then your property holds for any number''.
N is the set stable under the succession function, but from this you do not have the deductive principle of induction.This principle is an extra choice of an inference rule.
What I meant is that it does not make sense to refuse one of the two constructions, once you choose to have the other one.
You either accept N+ the deductive principle of induction or you reject both.
>>7948496
It is possible to believe the Peano axioms to be inconsistent, but only if you believe all models of Robinson arithmetic to be nonstandard.
In other words, if you believe every model of arithmetic has infinite descending chains.
Of course, even if you believe the latter, you can still believe PA to be consistent. There exist nonstandard models of PA.
Also, even if all models of arithmetic are nonstandard, if you believe in the consistency of ZF you believe in the consistency of PA.
So basically you have to be really daft to reject PA. You have to both reject set theory and visualize the natural numbers as having an infinite descending chain.
