Is this a bad definition of the rationals? Considers (-2/-1)

>>7942229
>Am I the only one that thinks this is a paradox?
>paradox
Yes, you are.

>>7942234
Oh interesting, how would you define a paradox and why would this not be one?

i think it's just referring to the fact that negative quotients -(n/z) are usually thought of as having a negative numerator (-n/z) even though this is purely a formality

You seem to have a wrong understanding of the concept.

Just because you can represent 2/1 in a form that doesn't fit the definition, doesn't mean that 2/1 isn't an element of the set.

Think of even numbers- you can define an even number as any number of the form 2k (when k is an integer). Then 12=3*4, which is not of the form 2k, but that doesn't mean it's not an integer- you can also write it as 2*6.

Also,
>Is this a bad definition of the rationals?
Kind of.
It only makes sense if you're defining it inside of a larger number system, like the real numbers. In that case, saying "rational numbers are all real numbers such that ____" is perfectly valid.

But think- If we only defined the integers, what does an expression such as 5/2 actually mean? We haven't defined it yet.

You can find the formal construction online, here's one page describing it: http://www.math.wustl.edu/~freiwald/310rationals.pdf

Basically, you define rational numbers as ordered pairs of integers, under the intuition that (a,b) represents the number a/b. Then you define the arithmetical operations on that: For an example, (a,b)+(c,d)=(ad+bc,bd) (try to write it out in fractional notation and see why it makes sense).
There's a slight catch, which is, we need to make sure we don't count the same fraction as different elements. For an example, 7/8 and 21/24 both represent the same number. So we impose an "equivalence relation" on all pairs, and then each rational number is actually a class of equivalent pairs.

>>7942229
>>7942247
you're mixing your own personal definition of what ratioals are with his. in his definition youre nt allowed to have a negative in the denominator. this is probably just him being lazy, but theres no paradox

his definition is not common, the most common in to define a rational as an equivalence class of pairs of integers

>>7942247
Topologically or computationally?

rememeber

{a,a,c,b,a,b}

equals the set

{a,b,c}

2=2/1=(-2)/(-1) is in the set because 2/1 is in it.

However, defining Q as a set alone is silly, you must equip it with the structure and you'd do that using more logical language or a type theory.

>>7942273
The latter?

>>7942276
A paradox is the assumption that time can conflict with its natural direction. Contradiction is asserting that time/computation doesn't exist.

Thanks everyone, I think I get this a little better now :

One question though, grounded on the points posters like >>7942262 brought up.

Say I want to define the set of fractions where both the numerator and denominator are positive, if I define it as p/q where p belongs to positive integers and so does q, would it not be possible to exclude fractions such as (-2/-1) from this set since, if what >>7942262 rather convincingly says is true :

>Just because you can represent 2/1 in a form that doesn't fit the definition, doesn't mean that 2/1 isn't an element of the set.

>Think of even numbers- you can define an even number as any number of the form 2k (when k is an integer). Then 12=3*4, which is not of the form 2k, but that doesn't mean it's not an integer- you can also write it as 2*6.

>>7942287
>>7942262
>2262▶>>7942287
>You seem to have a wrong understanding of the concept.
>Just because you can represent 2/1 in a form that doesn't fit the definition, doesn't mean that 2/1 isn't an element of the set.
>Think of even numbers- you can define an even number as any number of the form 2k (when k is an integer). Then 12=3*4, which is not of the form 2k, but that doesn't mean it's not an integer- you can also write it as 2*6.

Oh I'm also assuming what you mean by :

>Just because you can represent 2/1 in a form that doesn't fit the definition, doesn't mean that 2/1 isn't an element of the set.

Was also that, 2/1 in this different form (-2/-1) would also be a member of this set.

"Paradox" is not rigorous. It is used in unscientific contexts to mean something like "contradictory", and is used in physics to describe things that are apparently contradictory, but not actually.

If you want to be rigorous about it, use the term "contradiction." Demonstrate that this definition of rational leads to some other conflict in mathematics.

Here's why you won't, though: the author defines what *is* in the set Q, not what *is not* in Q. The author populates Q based on the provided definitions. Your proposition of including negative denominators provides no additional generality.

>>7942293

>invoking rigor

This is /sci/

>>7942290
Syntax-centric definitions for sets tend to be more or less antithetical to /sci/ and mathematics.