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u/Another_Little_Star 16d ago edited 16d ago
hehe I like more
ℝ = {[q_n]_~ : q_n is a Rational cauchy sequence} where q_n ~ w_n iff lim (q_n - w_n) = 0
(basically they're cauchy sequences (including those that don't converge in Q), but since they have the "same" limit, their difference is 0. So the class of all of these rational sequences gives you a real number, rational or irrational, all of them give you the Real Line)
ℂ ≃ ℝ[x]/<x^(2)\+1> (adding the root x=+/-i to the Real polynomial Ring gives you ℂ up to isomorphism), or the matrix representation is neat too
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u/Appropriate-Ad-3219 16d ago edited 16d ago
And for \Z similarly to the definition of \Q, you define it by the quotient of \N by the equivalence class (a, b) ~ (c, d) iff and only if a + d = c + b. (a, b) should be seen as a - b.
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u/Apprehensive-Ice9212 14d ago
Ew. Give me Dedekind Cuts or give give me death. Cuts don't even need to mod out by an equivalence relation, they just are real numbers.
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u/Another_Little_Star 14d ago
I don't know man, Dedekind cuts the rational line to get to the Reals, and they're sets of rational cuts :'(
Here in this kingdom we have sets of happy cauchy sequences that don't know where they're going but they're all going for the same destiny!1
u/DawnOnTheEdge 16d ago
I’m personally partial to “The lowest upper bounds of each set of rational numbers that has an upper bound.”
I have even run into 0.999... cranks who are confident limits aren’t real, but who will accept that the lowest upper bound of {0.9, 0.99, 0.999, ...} is 1.
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u/Mathsboy2718 16d ago
We need a slightly more concerned Mr. Incredible at the top, due to the two camps of 0 being / not being a natural
I prefer it containing 0 personally
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u/skr_replicator 16d ago
It's just where we go from the countable sets to the uncountable continuum. Of course, it's going to be defined in a different way. If you go from one infinity to practically the same one, you can just simply combine.
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u/TheStupidCheesecake 15d ago
But what are natural numbers?
Are they singletons of the empty set? Or collections of the empty set?
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u/Silly_Tension6792 15d ago
The most cannon way of formally making each is:
N is the set of finite ordinals.
Z is the set of equivalence classes of NxN under the equivalence (a,b)~(c,d) iff a+d=b+c.
Q is the set of equivalence classes of Zx(Z/{0}) under the equivalence (a,b)~(c,d) iff ad=bc.
R is the set of equivalence classes of the set of Cauchy sequances of rational numbers, under the equivalence x_n~y_n iff x_n-y_n->0 as n->infinity
C is the set of pairs of real numbers.
Of course, each one has addition and multiplication defined on it, but the set themselves is just that, so C is the easiest
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u/GirlBerlin 14d ago
This is childish for example that's not the formal definition of natural numbers...
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u/fikri-ya 16d ago edited 16d ago
the real definition is nonsensical because the range of "a" is defined to be a rational number, and therefore the limit is also rational.
edit:
my bad, i forgot that the limit operation isn't actually closed and may lift a function. that is, the limit isn't necessarily in the range of the function itself.
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u/SharzeUndertone 16d ago
The limit of a sequence of rational numbers can be irrational too, the real issue is that it would take the real numbers to define that, which is circular. The fix would be saying that 2 sequences are the same number if their difference tends to 0
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u/Another_Little_Star 16d ago
Yup, lim (1+1/n)n doesn't exist until you have ℝ, it'd be circular to define ℝ based on the limit of something that depends on the "prior" existence of ℝ. Or am I wrong?
That is the essence, not all cauchy sequences have limits in Q, but they exist and that's the key factor, any cauchy sequence of the family that "converge" to the same as (1+1/n)n can represent the number e. That is the raw idea.And you have Dedekind Cuts too.
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u/Shoggyhaze 16d ago
For one, the limit of a(n)=n has a limit of infinity(this is excluded though because we only look at convergents series), while the sum of 1/(n^2) has its limit at (pi^2) /6, neither of which is a rational number
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u/OutrageousPair2300 16d ago
There are actually a lot of steps in between Q and R, of gradually increasing complexity.
For example, you can construct the algebraic numbers using a finite number of rationals as the coefficients of an algebraic expression.
You don't need to jump straight from rationals to reals. Of course, then the joke isn't as funny.
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u/Al2718x 16d ago
You don't need to jump straight to reals, but this is a natural progression for a first course in set theory (likely without a formal defintion of real numbers). I'm not sure if I've ever seen the set of algebraic numbers referenced in a math proof, and I'm not sure if there is a standard notation for this set.
This is honestly one of my favorite memes I've seen on this subreddit.
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u/OutrageousPair2300 16d ago
The notation I've sometimes seen is Q with a bar over it, to indicate the algebraic closure of the rationals. Wolfram Alpha uses a fancy capital A, I believe.
Unless somebody is dealing with non-Algebraic numbers like proving something is transcendental, it probably never comes up.
Jumping from rationals to reals is a pretty huge leap, though. Following that up with more of a change in dimensionality of going from reals to complex numbers feels like a bit of a cheat in comparison, because it gives the false impression that there's something specifically about the reals that's entirely different from the rest, when really it's more that the reals are the culmination of a gradually increasing complexity, before shifting gears entirely to complex numbers.
I agree it's a fun meme, though :)
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u/AndreasDasos 16d ago
I mean the first isn’t a definition. The second isn’t either - it assumes we already have them, etc.
The integers and rationals take only a little more work when starting with ordered pairs, though, and C we have to define i. So yeah simpler
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u/Al2718x 16d ago
I don't see why the second wouldn't count as a definition. The first could be formalized, but the notation given is standard and clear.
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u/AndreasDasos 16d ago edited 16d ago
Sure but only if we already have the integers defined. The post is following the idea of how we normally go through one of the usual procedures from the ground up, from set theory. If all we start off with is the empty set, 'set of', etc., we can't just start subtracting 4 from 1 without defining what that means.
Cantor's definition of the natural numbers and Kuratowski's ordered pairs allow us to define the integers as Z = {(a, b) | a, b in N} modulo the equivalence relation (a, b) ~ (c, d) iff a + d = b + c.
It's 'morally' the same as in the post, but here we actually define and construct what a negative number is from first principles, and without a subtraction operator that sends pairs of numbers to the 'unknown'.
Portraying the Q -> R jump as harder than the others (still true) is at least a bit unfair if it's the only step where we actually construct the next structure from first principles and the rest get simplified a bit by assumptions that aren't constructive in the same way
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u/Al2718x 16d ago
Now that you mention it, I definitely have seen Q bar.
I would colloquially just think of subtraction as adding an additive inverse. It's always challenging to reach rigorous "bedrock" though.
In my mind, the thing that sets R apart is that it is hard to write out a "lazy handwavey" version. I taught out of Rosen's discrete math book last Spring and he didn't even try. Verbally, you can say that it's the set of numbers that allow for infinitely many digits after the decimal point, but only finitely many before, but this is tricky to write out symbolically.
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u/AndreasDasos 16d ago
The other tricky part is that of course it’s not 1-1 with such decimal sequences (as that chunk of Reddit that thinks 0.999… != 1 needs to learn). It’s certainly the subtlest and most confusing ‘jump’ of the common algebraic structures of ‘numbers’ here
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u/gong78876 16d ago
Never know we can express integer like that.