I know almost nothing about infinite sets but if we define X(t) as the sum of the first t twin primes then is it possible that X(t) might be close to infinity or at least a lot bigger than your ((2)) for t equal to, say, one billion?
Edit Edit: It seems I double misinterpreted the parent. I think we’ve agreed that (2) is finite, so if I ramble about it being infinite below, ignore me.
Being “close to infinity” doesn’t really make sense in standard real analysis, if you’re talking about any given finite number, because you can always produce another number arbitrarily larger than whatever you wanted to show was “really close to infinity”.
Edit: So if you’re asking “what’s bigger, the sum of the first billion twin primes, or (2)”, this question doesn’t make sense because (2) isn’t finite, but that sum is.
What’s even more interesting, though, is that you can meaningfully compare different sized infinities. Look into Cardinality.
EDIT: I misunderstood the op, as can be seen from this post and the child.
I don’t understand why no one else is objecting to treating (2) as a number.
If F(x, s) = {s-sided function of x}, e.g. F(2, 3) = /2\, F(2,5) = [2>, then clearly F(2,x) > 2^x for x > 3.
(2) is the limit of F(2, x) as x approaches infinity; just as 2^x is infinite in the limit, so is (2). I’m not even sure whether ((2)) is well-defined, because we haven’t been told how it approaches the limits, and it’s not clear to me that all methods yield the same function.
Oh actually you’re right. I didn’t interpret the op correctly. I thought it was just some weird extension of Knuth’s up arrow notation but now I see what’s going on.
In that sense, (2) isn’t a real number, as infinity isn’t a real number, it’s an extended real number.
And I think you’re right again, ((2)) isn’t well defined I don’t think.
Hmm, now I think you might be right, and that I misunderstood the poster’s original intention. The paragraph currently reads
…
I’ll spare the next [X] operators, and go right to (X) (“circle-X”). (X) follows the process that took us from triangle to square to pentagon, iterated an additional [X] times.
Is that an edit? I do not remember the phrase following the last comma. The notation is at least confusing, in that triangle->square->pentagon->...->circle ought to represent a limiting process, rather than a finite one.
Thank you. I would ask the op to use a less confusing notation, but I will go ahead and edit my other objections.
(The point being that your use of the term “standard real analysis” here is a bit off, specifically in the form of being insufficiently meta. “Real analysis” is a subject in which one considers such ideas as “metrics” (and many other things) in general; the term doesn’t just refer narrowly to the properties of the real line equipped with all the most “standard” structures.)
They may not know the term “metric space” (in which case you just explain that it’s a setting where we can measure distance), but if they think larger numbers are “closer to infinity” than smaller numbers, that means they are intuitively thinking in terms of the extended real line (metrized in one of the usual ways).
As a ward against any confusion—because I expect at least one person will make this mistake after reading your comment—it should be noted that while as a topological space it’s metrizable, the resulting metric on the reals would necessarily have pretty different properties (it would be the same topologically, but not uniformly, e.g.).
I know almost nothing about infinite sets but if we define X(t) as the sum of the first t twin primes then is it possible that X(t) might be close to infinity or at least a lot bigger than your ((2)) for t equal to, say, one billion?
Edit Edit: It seems I double misinterpreted the parent. I think we’ve agreed that (2) is finite, so if I ramble about it being infinite below, ignore me.
Being “close to infinity” doesn’t really make sense in standard real analysis, if you’re talking about any given finite number, because you can always produce another number arbitrarily larger than whatever you wanted to show was “really close to infinity”.
Edit: So if you’re asking “what’s bigger, the sum of the first billion twin primes, or (2)”, this question doesn’t make sense because (2) isn’t finite, but that sum is.
What’s even more interesting, though, is that you can meaningfully compare different sized infinities. Look into Cardinality.
EDIT: I misunderstood the op, as can be seen from this post and the child.
I don’t understand why no one else is objecting to treating (2) as a number.
If F(x, s) = {s-sided function of x}, e.g. F(2, 3) = /2\, F(2,5) = [2>, then clearly F(2,x) > 2^x for x > 3.
(2) is the limit of F(2, x) as x approaches infinity; just as 2^x is infinite in the limit, so is (2). I’m not even sure whether ((2)) is well-defined, because we haven’t been told how it approaches the limits, and it’s not clear to me that all methods yield the same function.
Oh actually you’re right. I didn’t interpret the op correctly. I thought it was just some weird extension of Knuth’s up arrow notation but now I see what’s going on.
In that sense, (2) isn’t a real number, as infinity isn’t a real number, it’s an extended real number.
And I think you’re right again, ((2)) isn’t well defined I don’t think.
Count me wrong. You understood correctly the first time. See Vladimir’s comment; the notation is confusing, but it is a finite process.
It’s not. As I understand from the post, in your notation, (2)=F(2,[2])=F(2,F(2,4)).
Hmm, now I think you might be right, and that I misunderstood the poster’s original intention. The paragraph currently reads
Is that an edit? I do not remember the phrase following the last comma. The notation is at least confusing, in that triangle->square->pentagon->...->circle ought to represent a limiting process, rather than a finite one.
Thank you. I would ask the op to use a less confusing notation, but I will go ahead and edit my other objections.
Sure it does—the extended real line is easily made into a metric space.
(The point being that your use of the term “standard real analysis” here is a bit off, specifically in the form of being insufficiently meta. “Real analysis” is a subject in which one considers such ideas as “metrics” (and many other things) in general; the term doesn’t just refer narrowly to the properties of the real line equipped with all the most “standard” structures.)
Fair enough, I was simply trying to appeal to what is probably his most familiar intuition regarding the real number line.
Most people that are going to have confusion about a big number being close to infinity probably aren’t going to know what a metric space is.
They may not know the term “metric space” (in which case you just explain that it’s a setting where we can measure distance), but if they think larger numbers are “closer to infinity” than smaller numbers, that means they are intuitively thinking in terms of the extended real line (metrized in one of the usual ways).
As a ward against any confusion—because I expect at least one person will make this mistake after reading your comment—it should be noted that while as a topological space it’s metrizable, the resulting metric on the reals would necessarily have pretty different properties (it would be the same topologically, but not uniformly, e.g.).