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Mathematicians Measure Infinities, Find They’re Equal

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Re: Mathematicians Measure Infinities, Find They’re Equal

#21

Post by traitormonkey »

ColinWright wrote: post #4
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…

So if I understand you correctly, you find it unconvincing, and therefore generations of mathematicians who study these things must all be wrong. Perhaps you simply don't understand the argument in detail, and are relying on your intuition. And perhaps your intuition is faulty. Which seems more likely? So let me try to provide a better insight for you. Consider the collection of natural numbers, including 0. Call it…

>Hundred of thousands of mathematicians...

Surely there is no need for argumentum ad verecundiam?

Re: Mathematicians Measure Infinities, Find They’re Equal

#22

Post by jakef »

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…

The natural numbers are each finite. In set theory, the standard way they are defined is that they can be constructed by assuming the existence of the empty set (or 0) and assuming that if you "insert a set into itself" the result will be a set. (So they can be thought of as {} = 0, {{}} = 1, {{}, {{}}} = 2, etc.).

The natural numbers are simply the (smallest) set that contains the empty set and is closed under this "insert a set into itself" operation (successor). It only contains finite sets since the successor operation will never turn an finite set into an infinite set.

The existence of this set of ALL natural numbers (an infinite object) relies on an axiom, the axiom of infinity.

There is no transfinite natural number.

Re: Mathematicians Measure Infinities, Find They’re Equal

#24

Post by deadmetheny »

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…

The hubris in this thread is spectacular. Multiple people are demonstrating that your grasp of the math here is wrong, yet you keep arguing that no, the mathematicians must be wrong because you haven't grasped the idea. You could argue that only one infinity exists/actually matters in devland, but in mathematics it's absolutely not the case.

Re: Mathematicians Measure Infinities, Find They’re Equal

#25

Post by traitormonkey »

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…

Cantor draws a diagonal and says 'See this number is not in the list'. However he hasn't sorted the list properly.

Re: Mathematicians Measure Infinities, Find They’re Equal

#26

Post by v64 »

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…

That's not how infinity works, which is why many results involving infinity are counterintuitive.

For instance, consider the natural numbers and just the even natural numbers. Intuition says these two sets must differ in size, because I took one set and removed half of its elements. But the mapping f(x) = 2x is a trivial bijection from the natural numbers to the even natural numbers, showing the two sets are of equal size.

Do you agree with the statement that two infinite sets are of different cardinality if it can be shown that no bijection exists between them? Let's consider an uncountable set that's simpler to imagine than the real numbers: the power set of the natural numbers, that is, the set of all subsets of N. It can be shown that no bijection exists between any set and its power set (Cantor's theorem [1]). Do you agree with that theorem? It can also be shown that a bijection does exist between the power set of N and R [2], implying they are of the same cardinality, and are both of a larger cardinality than N.

> There is only one infinity. It means "repeat".

Then what does it mean to you that infinite sets can be constructed that can be shown to have no bijection between them?

[1] https://en.wikipedia.org/wiki/Cantor%27s_theorem

[2] https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstei...

Re: Mathematicians Measure Infinities, Find They’re Equal

#27

Post by dogecoinbase »

zelah wrote: post #5
ColinWright wrote: post #4

Earlier quoted context omitted.

So if I understand you correctly, you find it unconvincing, and therefore generations of mathematicians who study these things must all be wrong. Perhaps you simply don't understand the argument in detail, and are relying on your intuition. And perhaps your intuition is faulty. Which seems more likely? So let me try to provide a better insight for you. Consider the collection of natural numbers, including 0. Call it…

It is possible. The inverse of m is called q. The function q takes an infinite sequence of coin flips and one by one changes every heads to 1 and every tails to 0. The infinite string of zeros and ones is then prepended with a 1 and interpreted as a transfinite natural number in binary notation. This will not take forever because each change will only take half as long as the previous one. A transfinite natural numbe…

Hundreds of thousands of mathematicians are wrong.

No. No natural number has an infinite number of digits, and you are wrong.

If you weren't, of course, it would be easy to prove me wrong -- simply name the natural number to which the successor function is applied that results in a "transfinite" number, whatever that is.

Re: Mathematicians Measure Infinities, Find They’re Equal

#28

Post by iamlucaswolf »

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…

Phew, you have some courage, questioning the foundations of modern Mathematics in a place like this. But I can relate to your concerns about Cantor's argument. When I first heard it, it also felt artificial and unconvincing to me.

What helped me (as with many proofs and concepts in Math) was an image, a visual metaphor if you like.

Imagine a very, very large paper on which you place infinitely many dots in a grid. That's the infinity you were referring to, the infinity of a for-loop, discrete infinity. Here's the trick: you can always add more dots, say by making the distance between grid points half as small, which would quadruple the number of dots in your grid. But no matter how many dots you place on the paper, ho matter how fine your grid, there will always be holes (imagine "zooming in" on a square of four grid points). In fact, most of the paper will be empty!

The other kind of infinity, continuous infinity, does not have any holes. Every spot is covered. You could not add any grid point, because the whole paper itself is painted.

I'm not a "full-time Mathematician", so this view may be entirely wrong. But it helped me understand and appreciate Cantor. Perhaps it did the same for you.

Cheers!

Re: Mathematicians Measure Infinities, Find They’re Equal

#29

Post by v64 »

thriftwy wrote: post #23

So they basically found a way to represent any real number as an unique sequence of bits? Doesn't sound too hard when spelled like that. I could probably do that, win some prize (not anymore). Do they have anything else like that?

If you've got some free time, can you look into whether or not the real part of every non-trivial zero of the Riemann zeta function is 1/2?

Re: Mathematicians Measure Infinities, Find They’re Equal

#30

Post by weichi »

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…

You need to write down the 1-1 correspondence you are proposing. Don't describe it in words, actually start writing the natural numbers in one column and the corresponding reals in the other. When you do this, you will find either that the 1-1 correspondence that you are envisioning doesn't actually work (i.e., it isn't 1-1), or that it does work ... but in that case, Cantor's diagonalization argument can be applied to it.
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