Why do we assume the square root of two is irrational because it's proven to be not rational?

the square root of 2 is a real number
real numbers include the set of rational numbers
any real number not in the set of rational numbers is called irrational

We don't assume we know. We know becuase it is definitionally true. An "irrational number" means a specific thing, the square root of 2 meets that definition. You seem to fail to understand that words communicate ideas, they are not themselves that idea.

>but there are statements that are neither true or false.

Your statement is actually both true and false.

the definition of rational is that the number can be expressed as the ratio of two integers.

since the sqrt(2) can not be expressed as the ratio of two integers, it is not rational. You can call it whatever you want, but it is not rational.

Who are the "we" that "know" what "definitionally true" means? And just because "you" "know" something does not mean it is not a valid question. And anyways, you are just shifting the question to what that definition of "irrational number" is, which probably whenever I have seen the proof of "the irrationality of the root of two" has not been mentioned (perhaps it has but that is another question).

So I can call it something like "not a number" or "something that does not exist"? So the "Real" number line is full of "not a number" and only some "numbers" in them, but just very few of them.
Why does everybody make it seem like it is an easy question when it is not. Well not true, No.9287367, actually agrees it is a long discourse and should do some "research on Youtube?" well I like watching Norman Wildberger and I think he says in a video, "all we have proven is that no number is the square root of 2".
Which actually is the first time I got some meaning out of that "proof". But then again there has to be "more" to it. And I don't know what the right question is.

Is that not like "circular reasoning". A number is a number that ...

"This statement is false" can be true as long as you interpret "this" as referring to some other, untrue statement.

I don't know if it is a "stupid" word game, but I do entertain the idea that "irrational numbers" are not numbers and in my head it makes more sense to think of them as concepts as abstract as say the "gold at the end of the rainbow" than as anything remotely similar to a number.

No, it's something very clear and plain that you apparently just don't have the equipment to understand. That's probably why rhetorical word games appeal to you as a coverup.

>real numbers include the set of rational numbers
holy fuck
and N is a subset of Z right?
then please construct (Z,+) via (N,+), where ≤ and ≥ are orders defined on.. yes N only so far.
Hf doing this without equivalence classes.
And then tell me that N is a subset of Z is a subset of Q is a subset of R is a subset of C
After you constructed Q a similar way, try the same with R
there are different ways doing this (one of the most introductions is done in intuitionistic logic via something similar to cauchy series).
Then you realize that all those '+' and '*' are indeed different operations on similar looking fields where one led to the construction of the other, but are NOT THE SAME and thus not a subfield.

Ok, how about the coninuums hypothesis then?
is it either true or false
a quick google search gives you some hits that it is neither
so statements can be undecidable for example

Nice, though I don't think I have the patience to make all those constructions.

Now that is a result I find interesting, hope someone would make an exposition for an Engineer somewhat interested in these topics.
Do you know of any such works for not so layman people?

well math is supposed to have a logic foundation, for mathematical logic we use a specific kind of logic (there are many) that you probably know as "first order logic", this type of logic has a specific property known as the law of the excluded middle, which says a proposition can either be true or false, propositions that can be both are called paradoxes, and in mathematical logic we have plenty of ways to deal with them, we constructed math from the "beggining" excluding paradoxes which lead us from basic propositions to construct rings and fields to give out definition or Irrational numbers, because of this we can't have something that's at the same time both true and false, and we defined rational and irrational in such a way in which if we know some number is not rational it has to be irational.

Get a logic book if you are interested in the topic, I recommend "Laws of truth" by Smith.

What the fuck
"Q is a subset of R" here just means we have a canonical injection of Q in R.
Nobody cares about equivalence classes. Everything in maths is up to isomorphism.

Because it's a useful convention to call all numbers that are not rational as irrational. We could have agreed to call these numbers "cocks" for example. Your argument is one of semantics.

The difference between classification of irrationals and your paradoxic statement is that we can prove the sqrt (2) is rational is not true. So it can't be simultaneously true and false. It's just not true.

Wild berger says no number is the sqrt 2 because he beleives only naturals should exist up to a finite amount. Which is retarded. If 1 atom can represent 1 number, 1 aTom can represent n numbers.

You might appreciate constructivism / intuitionism. The basic idea afaik is that statement A can't be proven true just by proving that not A is false.

>they just taught me the construction of real numbers and I know that you need Q before R!
don't sperg out before you learned more math. Q is canonically included (aka a subset, sub topological space, sub Q-algebra) in R

The first thing I found for the construction of Z:
homepages.warwick.ac.uk/~masgar/Teach/2006_361/2006_10_18lec_ZZ.pdf

There you can see that Z consists of the equivalence class [(m, n)] with m,n element N and [(m, n)] beind defined as m+q=p+n with q,p element of N.
and + in Z defined as
[(m,n)] + [(q,p)] = [(m+q,n+p)]
so techically, N is not a subset of Z, but embedded in the construction.

Or less mathematical (especially since "-" in N is not defined)
the equivalent class [(m, n)] is just every pair m,n that have the same result on the # operatior, for example.
[(1, 5)] is the class of {(0, 4),(1, 5),(2, 6),...}
since 1-5 = 0-4 = 2-6 = -4.
So [(1, 5)] is a representation of every pair of natural numbers that resemble/stand for one (and only one) whole number in Z.

So: m, n element N
but [(m, n)] element Z and can never be element N since N consists of "one tuples", or plain elements (which again is technically incorret, for example peano successors are the most done construction of N).

If and only if you have "-" defined on N, which is kinda dangerous to stupid, cause you have to live with waaay more restrictions than for example / and {0}, since not only one sepcial number is forbidden to operate to, but infinite of those.

That's not what's happening here, we're proving a statement is false by proving that it is not true.

to sperg out even more:
it depends how you see the numbers constructed.
let's say you have any construction for N and definition for + on N and heck, even - on N.
Then, if you construct your way up N->Z->Q you have a problem with R and probably end up with construction R via cauchy sequences.
Now, some dude can come and say:
"Hey, but then every Set of number is embedded into the other superset but cannot be operated within! that sux!"
then some non intuitionistic logic sperg says:
you know what, fuck this shit, we only take (N,+,-) and construct every other set of numbers via cauchy sequences, then we don't have a problem!
So N, Z, Q and R can be defined as subfields of C, like you are doing.

so yes, practically, they are embedded and algebraic operations are allowed between those sets, but technically it depends on the chain of construction (and, lol, on your belive).

you're shortsighted. we understand N to be included in Z and you can make this inclusion explicit as follows.

first construct the "fake" natural numbers (((N))). Then use it to construct Z. then define the natural numbers N as {Z >= 0}.

don't go around yelling about stuff you just learned

>don't go around yelling about stuff you just learned
better keep shut and let others do that, right m8y m8?

>homepages.warwick.ac.uk/~masgar/Teach/2006_361/2006_10_18lec_ZZ.pdf
This is a nice read, I am happy with myself that I now know enough math to actually make "some" sense out of such a pdf.
Now I would like to read the history of how such a "construction" was deemed necessary. I mean did Euclid think of Z and N in those terms?
What problem made "mathematicians" think, hey you know what, I have to construct Z because ... .
I mean after I put in the effort to understand and sort of be able to replicate some, if not most, of the steps in this construction, what mistakes/errors/misconceptions will I not fall anymore thanks to this knowledge.
The current motivation is to "know" that N,Z & Q are not subfields of R. But I think I will get much more than that.

lol

and the question still lingers, even if you went down that rabbit hole, do you get that square root of 2 is a number.

Or is square root of 2 not a number that is a member of the "irrational numbers" notwithstanding the name "number" in there?

It's a definition you dickHEAD

I don't think you understand my statement, and why is it paradoxical? I am never saying we can prove sqrt (2) is rational only that no "number" is the sqrt (2).

And yes I do believe it would make more sense to call them "cocks" for example, than numbers, given that they are "not numbers".
From what I understand from Wildberger, you cannot sum two irrational numbers, you cannot multiply two irrational numbers, at least not in a finite way. You can only give a representation of such an operation without ever performing it. (What can be learned from just stating operations without ever seeing the result?)

What about your "evidence" 1 aTom can represent n numbers" ? what is that supposed to mean?
Are you trying to say one aTom can represent infinity? Well okay, you do, and then? Can you perform any operation you want to it? Or are you incredibly restricted? What can you do with such representation? And if you do "do" something with it, how can you have any confidence you are doing something legitimate and not just playing with "cocks", and end up being completely lost in your delusions?

And returning to your first statement, what "proof" is there that it is useful to call "something that is not a number" an "irrational number"? And under your definition, are complex rational numbers, rational numbers? What about rational quaternions?

>14 year old "philospoher's" first thread on sci

>Why do we assume the square root of two is irrational because it's proven to be not rational?
Why do we assume the sky to to be blue if we all know that the sky is not not blue?

Completely and utterly wrong.
It is UNPROVABLE, which does not say anything about its truth value.
It is VERY different then saying it is neither false nor true.

The CH might be true, but we just can nor prove it in ZFC

If its unprovable how is it a hypothesis?

Why the fuck did that guy post a bunch of south park pics? I'm completely thrown here

This thread should have ended right here.

Any real number that can't be represented as the quotient of two integers is irrational.
Any number that can be represented as a Dedekind cut on the rationals is a real number.
The square root of 2 can be represented as a Dedekind cut on the rationals but not as a quotient of integers, therefore the square root of two is irrational.
>b-b-but Wilderberg said irrationals are not numbers!
Wilderberg has his head so deep up his ass it sticks out of his neck. If you want to claim irrationals are not numbers, you're gonna have to start by defining what a number is but you clearly won't because you're a troll or as stupid as Wilderberg, either way I'm not wasting my time with you.

To any poor soul that happens to stumble upon this thread and don't actually know the answer to the question: Grab any analysis book and read the first two chapters. It will explain what a Dedekind cut is and most likely build the real field with them.

Rational numbers are all real numbers that can be expressed as the ratio of two whole numbers.
If a real number cannot be expressed as a ratio of two whole numbers, it is irrational. The two are mutually exclusive by definition.

> this statement is false
It is not a valid statement.

The sqrt is a function over the reals. 2 is a number. Therefore, sqrt is a number in the reals. Otherwise sqrt wouldn't be a function! So sqrt 2 is definitely a number.

My point was that it among others irrational is convenient when working with them.

"This statement is false" is a paradox

Sqrt 2 is rational can be proven false. That is different from being unable to prove it, or prove it true and false.

My point about atoms is that wild burgers insistence on using physical limitations of our universe to impose limits on mathematics is retarded.

You answered your own question. 2^0.5 is not rational therefore by definition its irrational.
Whats the point of this thread ?

The question is whether or not we should define irrationals as numbers that are not rational. Obviously the answer is yes. Op just wanted a semantic circlejerk.

I drew a square on a piece of paper, and then drew a diagonal in the square. I looked at it very closely to make sure it was there. It was, so it's definitely real. Then I measured it with a ruler. It seemed to have a definite length that didn't change, so I guess it represents a number. Since it's real and a number, I guess that makes it a real number, right?

>should we define *thing* by *definition*
Well if we used a different definition it wouldnt be the same thing unless the definitions are basically the same
Unless hes talking about reserving the actual words 'rational' and 'irrational' for something else .
OPs post is unclear af

Thats not the definition of real numbers.
Also squares are geometrical abstractions and dont exist outside geometry\mathematics so drawing one is impossible

Either a number can be expressed as a fraction of 2 finite integers or it can't. There is no in-between or neither nor here.

It's so easy to district a Cauchy sequence converging to sqrt(2). This pointless pedantry is so contrary to the point of math. Nobody gives a shit about the constructions of number systems.