Originally posted by DoctorScribbles All of a sudden, I do.
I just realized that according to me, the set of integers which are multiples of 3 (or 4, 5, 6, 7...) constitutes half of the set of integers, complemented by the set of non-multiples. Applying my analysis of the proposed wager, I'd have to accept an even odds wager on drawing a multiple of 7 at random. But I'd obviously e at. Does the notion of asymptotic density suffice to remedy this?
God damn infinite sets.
Fun with numbers.
Typically two halves make a whole, yet all the even numbers are
infinite, as all the odd ones, as all of them when they are together
too. It is what it is, this is more fun then talking about a piece of
chock, you break it in two and you get two pieces of chock. 🙂
Kelly
Originally posted by DoctorScribbles All of a sudden, I do.
I just realized that according to me, the set of integers which are multiples of 3 (or 4, 5, 6, 7...) constitutes half of the set of integers, complemented by the set of non-multiples. Applying my analysis of the proposed wager, I'd have to accept an even odds wager on drawing a multiple of 7 at random. But I'd obviously e ...[text shortened]... at. Does the notion of asymptotic density suffice to remedy this?
God damn infinite sets.
Yes, the notion of asymptotic density does rectify this.
For any positive real number x, the number of integers between 0 and x which are divisible by three is close enough to x/3 to not need to bother formalising what I mean by close, given I'm talking to you. The proportion of integers less than x which are divisible by three thus tends to 1/3 as x --> infinity.
Doing the same for the even integers gives 1/2.
In general, the proportion of integers less than x which are in some set is a function which grows more slowly than x does, and it's usually best to look at this function rather than at its limit to see how dense the set is.
Originally posted by KellyJay I'm addressing the point of it being relative, and really only that
point! The ink dot no matter how big it gets relative to the sheet of
paper cannot be put into any percentage over the infinite, because
that is true, it cannot be looked at that way, and have it make sense.
The important part of this is it does not mean that the dot of ink isn’t
real, ...[text shortened]... percentage of the
infinite that does not mean that it wasn’t there before the Big Bang.
Kelly
Is still don't quite get how infinity has anything to do with whether or not time existed before the big bang.
Originally posted by DoctorScribbles God damn infinite sets.
And that was my point all along.. I didn't want to souun arrogant. Using everyday words to not everyday concepts always brings us to this kind of mess.
Originally posted by adam warlock And that was my point all along.. I didn't want to souun arrogant. Using everyday words to not everyday concepts always brings us to this kind of mess.
What was your point? That I neglected to include in my definition the condition that if the original set has infinite cardinality, each of the complementary sets must have an asymptotic density of 1/2 therein? I must have missed that in your rebuttals.
I hereby revise my definition accordingly, and I presume we are now in agreement.
Originally posted by DoctorScribbles What was your point? That I neglected to include in my definition the condition that if the original set has infinite cardinality, each of the complementary sets must have an asymptotic density of 1/2 therein? I must have missed that in your rebuttals.
I hereby revise my definition accordingly, and I presume we are now in agreement.
you missed my point because i didn't put it explicitly in my rebutals. i don't think many people wil quite understand what that means so i prefered not to. sometimes confusing is worst than misleading.
Originally posted by adam warlock i didn't put it explicitly in my rebutals. i don't think many people wil quite understand what that means so i prefered not to.
Originally posted by DoctorScribbles LOL. Hilarious!
hilarious why?
do you think that everbody that will read this in the forum will understand it?
maybe you don't belive i understand it too. but if you look carefuly to what i posted you'll see some hints to that respect.
i see you don't take me seriously so i guess i'm finished with this.
Originally posted by adam warlock hilarious why?
do you think that everbody that will read this in the forum will understand it?
maybe you don't belive i understand it too. but if you look carefuly to what i posted you'll see some hints to that respect.
i see you don't take me seriously so i guess i'm finished with this.
have fun
Ok, why don't you humor me and formalize that which ChronicLeaky declined to formalize a few posts back regarding asymptotic density.
Originally posted by DoctorScribbles Ok, why don't you humor me and formalize that which ChronicLeaky declined to formalize a few posts back regarding asymptotic density.
i'm not a mathematician. i have a "licenciatura" in physics. in portuguese that's what you do before the phd or masters degree. i think that in the USA its called undergraduation.
so i don't know how to formalize it right away but if i catch a book with it i can understand it. or at least most of it. but i'll see his previous posts and try to formalize it. just give me some time.
Originally posted by ChronicLeaky Yes, the notion of asymptotic density does rectify this.
For any positive real number x, the number of integers between 0 and x which are divisible by three is close enough to x/3 to not need to bother formalising what I mean by close, given I'm talking to you. The proportion of integers less than x which are divisible by three thus tends to 1/3 as ...[text shortened]... 's usually best to look at this function rather than at its limit to see how dense the set is.
To Doctor S: Is this the post you were talking about?
Originally posted by DoctorScribbles Yes. Don't bother -- I'll believe you can do it.
"For any positive real number x, the number of integers between 0 and x which are divisible by three is close enough to x/3 to not need to bother formalising what I mean by close, given I'm talking to you."
Be n(x) the number of numbers (sorry for the expression) between 0 and x which are divisble by 3. Then as x goes to infinity n(x)-x/3->0.
i think this is a quick way to formalize it.
have fun
Edit: sorry for the unusual amount of words in a mathematical exposition but that's the best i could do given that some symbols are hard to do in a keyboard.
\lim_{x->\infty}n(x)-\frac{x}{3}=0 in latex code. with some possible minor mistakes
Originally posted by adam warlock "For any positive real number x, the number of integers between 0 and x which are divisible by three is close enough to x/3 to not need to bother formalising what I mean by close, given I'm talking to you."
Be n(x) the number of numbers (sorry for the expression) between 0 and x which are divisble by 3. Then as x goes to infinity n(x)-x/3->0.
i thin ...[text shortened]... d.
\lim_{x->\infty}n(x)-\frac{x}{3}=0 in latex code. with some possible minor mistakes
The limit you describe does not exist, because the function n(x)-(x/3) oscillates; it does not grow arbitrarily close to 0 as x increases.
Originally posted by DoctorScribbles The limit you describe does not exist, because the function n(x)-(x/3) oscillates; it does not grow arbitrarily close to 0 as x increases.
You asked him to describe what Leaky said, not to correct what he said.
A relative failure
22 Mar '07 06:531 edit
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