6 ms·

> In their 1872 papers, though, Cantor and Dedekind had found a way to construct a number line that was complete. No matter how much you zoomed in on any given

by dkarl 7mo ago

> In their 1872 papers, though, Cantor and Dedekind had found a way to construct a number line that was complete. No matter how much you zoomed in on any given stretch of it, it remained an unbroken expanse of infinitely many real numbers, continuously linked. > Suddenly, the monstrosity of infinity, long feared by mathematicians, could no longer be relegated to some unreachable part of the number line. It hid within its every crevice. I'm vaguely familiar with some of the mathematics, but I have no idea what this is trying to say. The infinity of the rational numbers had been known a thousand years prior by the Greeks, including by Zeno whom the article already mentioned. The Greeks also knew that some quantities could not be expressed as rational numbers. I would assume the density of irrational numbers was already known as well? Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2. I don't get what "suddenly" became apparent.

terminalbraid 7mo ago
The density does not dictate cardinality which is what this article is about.
pfortuny 7mo ago
The continuum. Connectedness.
zeroonetwothree 7mo ago
Complete just means the limit of every sequence is part of the set. So there’s no way to “escape” merely by going to infinity. Rational numbers do not have this property. How to construct the real numbers as a set with that property (and the other usual properties) formally and rigorously took quite a long time to figure out.
- lupire 7mo ago
  Critically, "Complete" also means that the supposed limit necessarily exists.
  - freehorse 7mo ago
    ... for "Cauchy sequences", which are basically sequences whose terms become "closer and closer together". You can still have sequences with no limits (a_n:=n, going to infinity, where all successive terns differ by 1 and which does not have a limit in the usual metric), as well as sequences with multiple limit points (in which case, subsequences can be considered). Btw this is "Cauchy completeness", so it is a bit different (but equivalent) way to approach the construction of the real numbers from Dedekind's, but it is also one that can apply to more general metric spaces.
markisus 7mo ago
> Before their papers, mathematicians had assumed that even though the number line might look like a continuous object, if you zoomed in far enough, you’d eventually find gaps. I'll try to interpret this sentence. We all have some mental imagery that comes to mind when we think about the number line. Before Cantor and Dedekind, this image was usually a series of infinitely many dots, arranged along a horizontal line. Each dot corresponds to some quantity like sqrt(2), pi, that arises from mathematical manipulation of equations or geometric figures. If we ever find a gap between two dots, we can think of a new dot to place between them (an easy way is to take their average). However, we will also be adding two new gaps. So this mental image also has infinitely many gaps. Dedekind and Cantor figured out a way to fill all the gaps simultaneously instead of dot by dot. This method created a new sort of infinity that mathematicians were unfamiliar with, and it was vastly larger than the gappy sort of infinity they were used to picturing.
- dkarl 7mo ago
  We've known since Zeno that all of our ways of visualizing infinity in finite terms are incomplete and provably incorrect, despite being unavoidable in human thinking. In other words, we knew the "gaps" reflected incomplete reasoning, not real emptiness between "consecutive" numbers. If Dedekind and Cantor only changed how we visualize infinity, I don't understand why it would cause a stir. > This method created a new sort of infinity that mathematicians were unfamiliar with, and it was vastly larger I understand that the construction of the reals paved the way for the later revolutionary (and possibly disturbing, for people with strongly held philosophical beliefs about infinity) discovery that one infinity could be larger than another. But in the narrative laid out by the article, that comes later, and to me it's clear (unless I misread it) that the part I quoted is about the construction of the reals, before they worked out ways to compare the cardinality of the reals to the cardinality of the integers and the rationals.
  - AndrewKemendo 7mo ago
    > If Dedekind and Cantor only changed how we visualize infinity, I don't understand why it would cause a stir. Because scientific progress is explicitly the process of changing the general mental model of how people approach a problem with a more broadly capable and repeatable set of operations This is philosophy of science 101
wrsh07 7mo ago
You can construct sequences of rational numbers where the limit is not rational (eg it's sqrt 2) Trivially, the sequence of numbers who are the truncated decimal expansion of root 2 (eg 1.4, 1.41. 1.414, ...) although I find this somewhat unsatisfying. With the real numbers there are no gaps. There are no sequences of reals where the limit of that sequence is not a real number
JadeNB 7mo ago
> Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2. That's only obviously irrational if x and y are rational. (But maybe you meant that, given an arbitrary interval a < b, you first shrink it to a rational interval a < x < y < b?)
Chinjut 7mo ago
I don't like the way it's written, but what they are talking about is completeness in the sense of "Dedekind completeness"; i.e., that given any two sets A and B with everyone in A below everyone in B, there is some number which is simultaneously an upper bound for A and a lower bound for B. Note that this fails for the rationals: e.g., if we let A be the rationals below sqrt(2) and B be the rationals above sqrt(2).
- buttermeup 7mo ago
  In school, we talked about “Dedekind cuts” but we never formalized the definition. Kind of disappointed now because your explanation is very simple and elegant.
deleted 7mo ago
[deleted]
antasvara 7mo ago
Take something like the integers (1,2,3,etc.). They are infinite; given an integer, you can always add 1 and get a new integer. However, there are "gaps" in that number line. Between 1 and 2, there are values that aren't integers. So the integers make a number line that is infinite, but that has gaps. Then we have something like the rational numbers. That's any number that can be expressed as a ratio of 2 integers (so 1/2, 123/620, etc.). Those ar3 different, because if you take any two rational numbers (say 1/2 and 1/3), we can always find a number in between them (in this case 5/12). So that's an improvement over the integers. However, this still has "gaps." There is no fraction that can express the square root of 2; that number is not included in the set of rational numbers. So the rational numbers by definition have some gaps. The problem for mathematicians was that for every infinite set of numbers they were defining, they could always find "gaps." So mathematicians, even though they had plenty of examples of infinite sets, kind of assumed that every set had these sorts of gaps. They couldn't define a set without them. Cantor (and it seems Dedekind) were the first to be able to formally prove that there are sets without gaps.
- lupire 7mo ago
  Right, but that's the opposite of what the Quanta article says. The article says that Cantor and Dedekind discovered infinity in bounded intervals. What they discovered (really, what they concocted) was uncountable infinity.
  - littlestymaar 7mo ago
    Quanta messing things up isn't a particularly rare occurrence, unfortunately.
- dkarl 7mo ago
  I just don't understand why this was disturbing. Prior to the construction of the reals, the existence of irrational and transcendental numbers was disturbing, because they showed that previous constructions (rational numbers and algebraic numbers) were incomplete. If those gaps were disturbing, a construction without gaps should have been satisfying, reassuring, a resolution of tension. Was there some philosophical or theological theory that required the existence of gaps, that claimed that a complete construction of the number line was mathematically impossible, because of some attribute of God or the cosmos?
sandslides 7mo ago
could I just leave my favourite thing ever here? thanks :) https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...
- effed3 7mo ago
  and do not forget this: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox way more technical, but wonderfull
hearsathought 7mo ago
> > Suddenly, the monstrosity of infinity, long feared by mathematicians, could no longer be relegated to some unreachable part of the number line. It hid within its every crevice. Think of the number line stretching from negative infinity to positive infinity and let C represent the cardinality/size/count of numbers on that number line. Now just take portion of the number line from 0 to 1. Let C1 represent the cardinality/size/count numbers from the truncated line from 0 to 1. You would assume that C > C1. But in fact they are equal. There are just as many infinite real numbers from 0 to 1 as there are on the entire number line. Even worse, this hold true for any portion of the number line, how small or big you make the line. Rather than infinity being in a far distance place at the edge of the line in either direction, there is infinity everywhere along the number line. > I don't get what "suddenly" became apparent. It appeared suddenly because prior to cantor/dedekind, mathematics only understood the countably infinite ( natural numbers, integers, rationals, etc ) . By constructing a complete number line, cantor/dedekind showed there is a cardinality greater than infinity ( countable ). The continuum. Cantor also showed that there is an infinite number of cardinalities.

by dkarl 7mo ago

terminalbraid 7mo ago

The density does not dictate cardinality which is what this article is about.

pfortuny 7mo ago

The continuum. Connectedness.

zeroonetwothree 7mo ago

Complete just means the limit of every sequence is part of the set. So there’s no way to “escape” merely by going to infinity. Rational numbers do not have this property. How to construct the real numbers as a set with that property (and the other usual properties) formally and rigorously took quite a long time to figure out.

lupire 7mo ago
Critically, "Complete" also means that the supposed limit necessarily exists.
- freehorse 7mo ago
  ... for "Cauchy sequences", which are basically sequences whose terms become "closer and closer together". You can still have sequences with no limits (a_n:=n, going to infinity, where all successive terns differ by 1 and which does not have a limit in the usual metric), as well as sequences with multiple limit points (in which case, subsequences can be considered). Btw this is "Cauchy completeness", so it is a bit different (but equivalent) way to approach the construction of the real numbers from Dedekind's, but it is also one that can apply to more general metric spaces.

markisus 7mo ago

> Before their papers, mathematicians had assumed that even though the number line might look like a continuous object, if you zoomed in far enough, you’d eventually find gaps. I'll try to interpret this sentence. We all have some mental imagery that comes to mind when we think about the number line. Before Cantor and Dedekind, this image was usually a series of infinitely many dots, arranged along a horizontal line. Each dot corresponds to some quantity like sqrt(2), pi, that arises from mathematical manipulation of equations or geometric figures. If we ever find a gap between two dots, we can think of a new dot to place between them (an easy way is to take their average). However, we will also be adding two new gaps. So this mental image also has infinitely many gaps. Dedekind and Cantor figured out a way to fill all the gaps simultaneously instead of dot by dot. This method created a new sort of infinity that mathematicians were unfamiliar with, and it was vastly larger than the gappy sort of infinity they were used to picturing.

dkarl 7mo ago
We've known since Zeno that all of our ways of visualizing infinity in finite terms are incomplete and provably incorrect, despite being unavoidable in human thinking. In other words, we knew the "gaps" reflected incomplete reasoning, not real emptiness between "consecutive" numbers. If Dedekind and Cantor only changed how we visualize infinity, I don't understand why it would cause a stir. > This method created a new sort of infinity that mathematicians were unfamiliar with, and it was vastly larger I understand that the construction of the reals paved the way for the later revolutionary (and possibly disturbing, for people with strongly held philosophical beliefs about infinity) discovery that one infinity could be larger than another. But in the narrative laid out by the article, that comes later, and to me it's clear (unless I misread it) that the part I quoted is about the construction of the reals, before they worked out ways to compare the cardinality of the reals to the cardinality of the integers and the rationals.
- AndrewKemendo 7mo ago
  > If Dedekind and Cantor only changed how we visualize infinity, I don't understand why it would cause a stir. Because scientific progress is explicitly the process of changing the general mental model of how people approach a problem with a more broadly capable and repeatable set of operations This is philosophy of science 101

wrsh07 7mo ago

You can construct sequences of rational numbers where the limit is not rational (eg it's sqrt 2) Trivially, the sequence of numbers who are the truncated decimal expansion of root 2 (eg 1.4, 1.41. 1.414, ...) although I find this somewhat unsatisfying. With the real numbers there are no gaps. There are no sequences of reals where the limit of that sequence is not a real number

JadeNB 7mo ago

> Give x < y, it's easy to construct x + (y-x)(sqrt(2))/2. That's only obviously irrational if x and y are rational. (But maybe you meant that, given an arbitrary interval a < b, you first shrink it to a rational interval a < x < y < b?)

Chinjut 7mo ago

I don't like the way it's written, but what they are talking about is completeness in the sense of "Dedekind completeness"; i.e., that given any two sets A and B with everyone in A below everyone in B, there is some number which is simultaneously an upper bound for A and a lower bound for B. Note that this fails for the rationals: e.g., if we let A be the rationals below sqrt(2) and B be the rationals above sqrt(2).

buttermeup 7mo ago
In school, we talked about “Dedekind cuts” but we never formalized the definition. Kind of disappointed now because your explanation is very simple and elegant.

deleted 7mo ago

[deleted]

antasvara 7mo ago

Take something like the integers (1,2,3,etc.). They are infinite; given an integer, you can always add 1 and get a new integer. However, there are "gaps" in that number line. Between 1 and 2, there are values that aren't integers. So the integers make a number line that is infinite, but that has gaps. Then we have something like the rational numbers. That's any number that can be expressed as a ratio of 2 integers (so 1/2, 123/620, etc.). Those ar3 different, because if you take any two rational numbers (say 1/2 and 1/3), we can always find a number in between them (in this case 5/12). So that's an improvement over the integers. However, this still has "gaps." There is no fraction that can express the square root of 2; that number is not included in the set of rational numbers. So the rational numbers by definition have some gaps. The problem for mathematicians was that for every infinite set of numbers they were defining, they could always find "gaps." So mathematicians, even though they had plenty of examples of infinite sets, kind of assumed that every set had these sorts of gaps. They couldn't define a set without them. Cantor (and it seems Dedekind) were the first to be able to formally prove that there are sets without gaps.

lupire 7mo ago
Right, but that's the opposite of what the Quanta article says. The article says that Cantor and Dedekind discovered infinity in bounded intervals. What they discovered (really, what they concocted) was uncountable infinity.
- littlestymaar 7mo ago
  Quanta messing things up isn't a particularly rare occurrence, unfortunately.
dkarl 7mo ago
I just don't understand why this was disturbing. Prior to the construction of the reals, the existence of irrational and transcendental numbers was disturbing, because they showed that previous constructions (rational numbers and algebraic numbers) were incomplete. If those gaps were disturbing, a construction without gaps should have been satisfying, reassuring, a resolution of tension. Was there some philosophical or theological theory that required the existence of gaps, that claimed that a complete construction of the number line was mathematically impossible, because of some attribute of God or the cosmos?

sandslides 7mo ago

could I just leave my favourite thing ever here? thanks :) https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...

effed3 7mo ago
and do not forget this: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox way more technical, but wonderfull

hearsathought 7mo ago

> > Suddenly, the monstrosity of infinity, long feared by mathematicians, could no longer be relegated to some unreachable part of the number line. It hid within its every crevice. Think of the number line stretching from negative infinity to positive infinity and let C represent the cardinality/size/count of numbers on that number line. Now just take portion of the number line from 0 to 1. Let C1 represent the cardinality/size/count numbers from the truncated line from 0 to 1. You would assume that C > C1. But in fact they are equal. There are just as many infinite real numbers from 0 to 1 as there are on the entire number line. Even worse, this hold true for any portion of the number line, how small or big you make the line. Rather than infinity being in a far distance place at the edge of the line in either direction, there is infinity everywhere along the number line. > I don't get what "suddenly" became apparent. It appeared suddenly because prior to cantor/dedekind, mathematics only understood the countably infinite ( natural numbers, integers, rationals, etc ) . By constructing a complete number line, cantor/dedekind showed there is a cardinality greater than infinity ( countable ). The continuum. Cantor also showed that there is an infinite number of cardinalities.