Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Mathematicians Measure Infinities, Find They’re Equal

quantamagazine.org

Post reply on HN
Print view View original

131–140 of 170 posts

Re: Mathematicians Measure Infinities, Find They’re Equal

#131

Post by dragonius »

kmill wrote: post #83
logfromblammo wrote: post #76

Earlier quoted context omitted.

I don't see the problem with having a first digit and a last digit and an infinite number of digits in between. Edit: Infinitesimal divided by two is infinitesimal, in the same way that infinity multiplied by two is infinity. So 0.000...0001 / 2 = 0.000...0001 . Infinitesimal multiplied by any finite number is infinitesimal. Infinitesimal multiplied by infinity is every number in the interval from infinitesimal to in…

There's not a problem if you can answer this: What is 0.000...1 + 0.000...9? (The 1 and 9 are digits after infinitely many zeros.) You are not allowed to say "undefined" if these are real numbers, because you are supposed to be able to add any two real numbers. You are also not allowed to say 0.000...10 since that changes the place values. Well, you are allowed to say 0.000...10 if you are imagining a real number is…

He's already answered you:

If you are going to allow this kind of notation, then

0.000...1 = 0 is equivalent to 0

0.000...9 = 9 x 0 = 0

0.000...1 + 0.000...9 = 0 + 0 = 10 x 0 = 0.000...10

(0.000...10) / 2 = 0/2 = 0

You're getting hung up on notation and missing the concept.

Re: Mathematicians Measure Infinities, Find They’re Equal

#132

Post by kmill »

dragonius wrote: post #131
kmill wrote: post #83

Earlier quoted context omitted.

There's not a problem if you can answer this: What is 0.000...1 + 0.000...9? (The 1 and 9 are digits after infinitely many zeros.) You are not allowed to say "undefined" if these are real numbers, because you are supposed to be able to add any two real numbers. You are also not allowed to say 0.000...10 since that changes the place values. Well, you are allowed to say 0.000...10 if you are imagining a real number is…

He's already answered you: If you are going to allow this kind of notation, then 0.000...1 = 0 is equivalent to 0 0.000...9 = 9 x 0 = 0 0.000...1 + 0.000...9 = 0 + 0 = 10 x 0 = 0.000...10 (0.000...10) / 2 = 0/2 = 0 You're getting hung up on notation and missing the concept.

Neither has logfromblammo answered me nor am I hung up on notation. The notation is only incidental.

They are claiming that it is a well-defined number system with numbers "having a first digit and a last digit and an infinite number of digits in between." I say show that it works.

You are saying the way this works is to disregard the digits after the infinitely many digits. Sure, that would make a consistent system.

They seem to be saying something distinct from your interpretation. It's possible they mean to take the real numbers and adjoin a new "infinitesimal" element, for instance.

Re: Mathematicians Measure Infinities, Find They’re Equal

#134

Post by calebm »

samoright wrote: post #79

> In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size I thought there are only two types of infinity and Cantor already proved that they are different. * Uncountable infinity which is the cardinality of the set of real numbers * Countable infinity which is the cardinality of the set of integers Cantor has alr…

My question exactly. I still don't get it.

Re: Mathematicians Measure Infinities, Find They’re Equal

#135

Post by kmill »

logfromblammo wrote: post #104
kmill wrote: post #101

Earlier quoted context omitted.

> Don't confuse the limitations on mathematical notation with a limitation on imagination Good luck proving or calculating anything. You can define "infinitesimal/2 == infinitesimal", but nothing good will come out of it. A definition is no good unless it lets you do something. Letting e=infinitesimal, you have e/2==e, so e==2e so 0==2e-e so 0==e. This definition is inconsistent with being able divide by non-zero int…

That's not the definition, that's just what it does. The definition of infinitesimal is "the smallest-magnitude number that is greater than zero". If you divide a finite number by infinity, infinitesimal is what you get, but don't go thinking that if you multiply it by infinity again that you will get the same number back, because you won't. The floating point standard does not include a representation for infinitesi…

> The definition of infinitesimal is "the smallest-magnitude number that is greater than zero"

That is hardly a definition. The real numbers are defined either as Dedekind cuts or as equivalence classes of Cauchy sequences of rationals. If you say "e is defined to be a real number such that e>0 and for all c>0, c>e", you would get a contradiction purely from the definition of the reals since "for all c>0, c>e" implies "e The only consistent scenario I can think is that you are actually extending the real numbers with a new element called "infinitesimal." Go ahead, but don't pretend that it is an element of the set of real numbers. Also, don't get the idea that there is some "true" set of numbers that we are trying to approximate with better accuracy. Modern mathematics has blown this idea wide open by introducing a wide array of mutually-inconsistent number systems.

> If you divide a finite number by infinity, infinitesimal is what you get

So you say. This would need to be part of the definition, or at least provable from it. Quoting Timothy Gowers, a mathematical object is what it does. How was I supposed to know that twice infinitesimal is equal to infinitesimal?

Elaborating extension: there is a way to add ("adjoin") an infinitesimal element to the real numbers. Let R(e) be the set of rational functions in e, a formal constant. For instance, 2+3e or 5e^2. I think there is a way to give R(e) a total order by saying 0is the real numbers, however.

> It's probably easier to think of quantities like zero, one, infinity, and infinitesimal as the base vectors in mutually orthogonal dimensions

In what way? In a vector space, I can divide by two, and I am apparently not able to divide infinitesimal by two (in the sense that e/2=e implies e=0).

> in the same way that slightly changing the Euclidian parallel lines property can produce elliptic and hyperbolic geometries

At least right now, there is a rather large difference: many interesting theorems follow from hyperbolic geometry.

What interesting things follow from asserting that there is a smallest-magnitude real number greater than zero? (If you say "I never said the number was real," then there has been no point to this discussion, because it started when you claimed the real numbers were countable by multiplying "infinitesimal" by other numbers.)

Re: Mathematicians Measure Infinities, Find They’re Equal

#136

Post by fooker »

Others wrote: post #123
abnry wrote: post #94

Earlier quoted context omitted.

If you take the power set of an infinite set, you /always/ get a cardinality bigger than the original set. So there are an infinite number of infinities.

This is a good point. But to be super pedantic, there isn't a size of infinity large enough to describe how many sizes of infinity there are.

Why not? The aleph numbers have a bijection with naturals.

Re: Mathematicians Measure Infinities, Find They’re Equal

#137

Post by Others »

fooker wrote: post #136
Others wrote: post #123

Earlier quoted context omitted.

This is a good point. But to be super pedantic, there isn't a size of infinity large enough to describe how many sizes of infinity there are.

Why not? The aleph numbers have a bijection with naturals.

See this answer on the mathematics stack exchange (it's really a book excerpt, but I don't know where else to find it.) https://math.stackexchange.com/a/5390/220797

Re: Mathematicians Measure Infinities, Find They’re Equal

#138

Post by zelah »

moyix wrote: post #31
zelah wrote: post #3

I find Cantor's diagonal argument unconvincing. The claim is that there are more real numbers in the range from zero to one than there are natural numbers. To see that this is false simply realize that you don't actually have to write a decimal point to specify the real numbers in this range. Without the decimal point these real numbers just become natural numbers. Can a rational person believe that there are infinit…

Based on your other replies, I suspect it's not worth engaging, but for anyone else reading, here's a great article by a math journal editor writing about all of the different attempts to disprove Cantor's diagonal argument he received and tracing out some common mistakes he saw: http://www.logic.univie.ac.at/~ykhomski/ST2013/Hodges.pdf

Thank you - I have read this and do realize I am out of my depth.

Re: Mathematicians Measure Infinities, Find They’re Equal

#139

Post by haddy6235 »

zelah wrote: post #3

I find Cantor's diagonal argument unconvincing. The claim is that there are more real numbers in the range from zero to one than there are natural numbers. To see that this is false simply realize that you don't actually have to write a decimal point to specify the real numbers in this range. Without the decimal point these real numbers just become natural numbers. Can a rational person believe that there are infinit…

Why do mathematicians normally not distinguish between numbers for counting and numbers for measuring (length, volume, etc)? They are fundamentally different, and conflation makes many arguments hard to grasp.

Re: Mathematicians Measure Infinities, Find They’re Equal

#140

Post by lisper »

zelah wrote: post #124
lisper wrote: post #120

Earlier quoted context omitted.

> Cantor will never get a list because I will never really be done generating it. That would be true even if you weren't diagonalizing because the list is infinite. In fact, just a single item in the list is potentially infinite. So you can't generate the whole list regardless. What you have to do is to produce an algorithm that takes any two natural numbers i and j as input and produces as output in finite time the…

Why is it that Cantor can do an infinite procedure of diagonalization but I can't? If he can diagonalize then I can too. Is it possible that Cantor's "algorithm" is not really an algorithm? Knuth says an algorithm must be correct and must also terminate. Regardless, I have a lot to ponder.

Cantor is playing by the exact same rules that you are. His burden is: given an integer i, produce in finite time the i'th digit of a real not in your list. (He can't produce the whole thing in finite time because it's infinite, obviously.) He does this by using your algorithm to produce the i'th digit of the i'th row and adds one to it (mod 10).
Post reply on HN