Is there a deeper reason for why the rational numbers don't

>>7807022

excellent question! after thinking about it for a while, i'd say it's because you can define some bounded subsets of Q (:= the rationals) via properties that are too "strong" in some sense for Q, i.e. using operations that Q isn't closed under.

what i mean by that is: if you characterized sets only via properties like "x being small than q" with fixed q in Q, then you wouldn't feel the need for those irrationals inbetween. but the moment you permit properties like "x*x being smaller than 2" [which is essentially the same as saying x<sqrt(2)] characterizing sets, you use language that Q wasn't constructed for. Q, as a field, is closed under addition, multiplication, and respective inverses (save 0, obviously) - but it is not closed under sqrt or limits. yet math allows to define subsets via such properties.

well, that's the best insight/perspective i can come up with, anyway.

>>7807243
This would be my train of thought, but [math]\mathbb{Z}[/math] is also not closed under operations of that sort (e.g. sqrt) yet satisfies least upper bound property.

>>7807022
>deeper reason
gtfo

>>7807311

true that, but it comes at the price of [math]\mathbb{Z}[/math] not having multiplicative inverses.
[math]\mathbb{Z}[/math] by itself is coarse enough so that when you split it into a lower/upper part, there's always one integer to identify this particular splitting operation with - unlike in [math]\mathbb{Q}[/math].

another way to look at it: any bound subset of [math]\mathbb{Z}[/math] only has finitely many elements, unlike [math]\mathbb{Q}[/math]. this makes it easy to extract the smallest upper bound to a bounded set of integers. in the rationals, a bounded set has infinitely many upper bounds in every interval to its right side, which complicates finding a least upper bound.

>>7807318

i (not OP) think there's a big difference between having a verifiable proof for some statement and understanding/grasping why it is true, gaining an intuition to the underlying structures.
don't dismiss curiosity to understand things better this easily!

>>7807481
* any bounded subset

>>7807022
while we're at the subject of "why are things the way they are?":

why are [math]L_{2}[/math] norms so neat? why do we use least squares optimization/orthogonal projection? why are spheres the balls in [math]L_{2}[/math]-norm? why is Euclidean geometry the right one? what's so special about power 2? is it all because of the Pythagorean theorem? is there a deeper truth behind it, or is it just the way things are?

>>7808296
no one? don't you like to scrape deep when trying to understand things?

>>7808296
Going bigger makes things computationally expensive.
Going lower isn't possible because the cases for 0 and 1 are trivial.

>>7807022
>Is there a deeper reason
If formal mathematics is involved, no. Math maps onto its own axioms; nothing more, nothing less.

>>7809744
Don't you like to know the truth rather than confuse yourself with aberrant connotations?

Seriously. It's like you don't know where delusion comes from.

>>7808296
>is there a deeper truth behind it,
YES there is and it is called personal choices. literally nothing makes necessary to choose to formalize your experiences as traditional mathematics.

the deeper question is why do think that it is worth to formalize your abstraction living in natural languages, especially if you know that you will die.

The least upper bound property can be taken as an axiom.

If you don't take it as an axiom, instead taking the Archimedean property as an axiom, you can prove that there is only one ordered field, up to isomorphism, having the LUBP.

So if you want to know "why", check the proof of that.

But the real reason is essentially that R is constructed with a property like that in mind.