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Mathematicians Measure Infinities, Find They’re Equal

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Re: Mathematicians Measure Infinities, Find They’re Equal

#11

Post by ColinWright »

zelah wrote: post #9
ColinWright wrote: post #7

Earlier quoted context omitted.

So I asked: is it possible to have m:N -> F such that for every f in F, there is an n in N such that m(n)=f? Your reply says yes, but then your construction does not do it. In particular you said: > The infinite string of zeros and ones is then prepended with a 1 and interpreted as a transfinite natural number in binary notation. But the set N does not have transfinite natural numbers, so q does not map F to N, it ma…

>But the set N does not have transfinite natural numbers, so q does not map F to N, it maps F to something else. How many natural numbers are there? How many bits does it take to represent the average natural number? If you believe the natural numbers do not include transfinite numbers then how do you pick a successor when counting? There are infinite picks to be made so some of the picks must be transfinite. What I…

> How many natural numbers are there?

Infinitely many, more than any finite number.

> If you believe the natural numbers do not include transfinite numbers then how do you pick a successor when counting?

Just add one.

> There are infinite picks to be made so some of the picks must be transfinite.

No, adding one to a finite number does not result in a transfinite number.

> What I am calling a transfinite natural number must exist in N because N is an infinite set.

The definition is that N is the smallest set that contains 0, and every successor of something already in N. Every finite non-negative integer is there, and nothing else - they are all finite.

> Assume that N has only finite numbers in it but is itself an infinite set. Would you care to tell me which number (or numbers) are listed twice?

No number has to be listed twice - just because each thing in the the set is finite, that does not mean that the set has to be finite. There is no contradiction in N being infinite, but all its elements being finite.

You appear to be using words in a non-standard manner. As such, quite simply, you need to be amazingly careful, or you will not be understood.

Certainly I don't understand you.

Re: Mathematicians Measure Infinities, Find They’re Equal

#12

Post by lisper »

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…

> Without the decimal point these real numbers just become natural numbers.

No, they don't, because the vast majority of them have an infinite number of digits to the right of the decimal point. That's the key: there are more numbers with an infinite number of non-zero digits (the reals) than there are numbers with a finite number of non-zero digits (the naturals).

> The problem with Cantor's argument comes down to the fact that the procedure he uses to find a number not in the set is essentially the same as the procedure he uses for creating the infinite set in the first place.

Again, no. Cantor doesn't create the set, you do. The proof is like a game. It says: give me any procedure for (putatively) making a list of all of the real numbers, and I will give you back a number that is not in the list.

Re: Mathematicians Measure Infinities, Find They’re Equal

#13

Post by Cyph0n »

zelah wrote: post #8
gus_massa wrote: post #6

Earlier quoted context omitted.

[I'll try a non technical argument to convince you. It's also not a complete argument, so you must think about this for a while.] > If you believe n is a natural number then you must also believe that n x 10 is a natural number. One more digit! If you interpret the natural number in this way, the important property is that they have only a finite amount of "interesting" digits. Almost all their digits are zero. You c…

>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..." This trick makes much easier to prove that the diagonal ´+1 in each…

Yes they could, but note that these real numbers are all between 0 and 1. So the "infiniteness" of R is "higher" than that of N.

Re: Mathematicians Measure Infinities, Find They’re Equal

#14

Post by relate »

zelah wrote: post #8
gus_massa wrote: post #6

Earlier quoted context omitted.

[I'll try a non technical argument to convince you. It's also not a complete argument, so you must think about this for a while.] > If you believe n is a natural number then you must also believe that n x 10 is a natural number. One more digit! If you interpret the natural number in this way, the important property is that they have only a finite amount of "interesting" digits. Almost all their digits are zero. You c…

>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..." This trick makes much easier to prove that the diagonal ´+1 in each…

Suppose I have the set F = { all integers x where x can be written with a finite number of digits }

Are you saying the set F is finite?

Or are you saying that the set F contains transfinite numbers?

Re: Mathematicians Measure Infinities, Find They’re Equal

#15

Post by empath75 »

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…

> 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.

There are no natural numbers with infinite digits, so this is not correct.

Re: Mathematicians Measure Infinities, Find They’re Equal

#16

Post by pitaj »

zelah wrote: post #8
gus_massa wrote: post #6

Earlier quoted context omitted.

[I'll try a non technical argument to convince you. It's also not a complete argument, so you must think about this for a while.] > If you believe n is a natural number then you must also believe that n x 10 is a natural number. One more digit! If you interpret the natural number in this way, the important property is that they have only a finite amount of "interesting" digits. Almost all their digits are zero. You c…

>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..." This trick makes much easier to prove that the diagonal ´+1 in each…

The difference is that `0.7656536...` is smaller than 0.7656537, whereas a similar number with every digit moved left of the decimal point has no bound.

Re: Mathematicians Measure Infinities, Find They’re Equal

#17

Post by bmh_ca »

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…

> There is only one infinity.

Fractions are countable. Real numbers are not.

In other words, fractions, integers, positive integers all belong to the set of countable infinities, meaning there is an isomorphic function that bidirectionally maps each positive integer (the count) to every item in the target, countable infinity.

There is no isomorphism between real numbers and any countable set. If you create one before you are 40, you will get a Field Medal.

The isomorphism and the distinction between sets that have them and sets that do not has proven useful.

You could think of infinity as one concept, but it is usefully divided into countable and uncountable versions.

Re: Mathematicians Measure Infinities, Find They’re Equal

#19

Post by throwawayjava »

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…

It's OK not to be convinced by an argument, even a mathematical proof, and especially an informal proof.

But Cantor's argument is correct.

If you don't find a mathematical proof convincing even though all the trained mathematicians seem to believe the proof is correct, here's my advice. You should FIRST convert the proof into a sequence of valid deductions in a fixed logic. (If you cannot do this, you don't understand the proof/theorem; perhaps (re-)take a few mathematics courses.) If you can find a mistake in the completely formal proof, then you can convert that mistake into an informal explanation. If your disagreement boils down to a disagreement with a axiom in the formal system even though most mathematicians accept it as an adequate foundations, realize you're getting close to philosophy.

This advice is meant for people at the first phase in Terry Tao's hierarchy of mathematical maturity. So not great advice if you're a genius or a trained mathematician (read: people call you doctor).

> There is only one infinity. It means "repeat". It is simply the interplay of finite state with process. You can think of it as an "infinite loop" in programming. To say that one infinity is smaller than another is to deny that the smaller is infinite. Infinite means without bound.

But "modeling non-terminating loops in a computer program" is NOT the motivation for real numbers, so this foundational criticism of completing the rationals makes absolutely no sense.

Re: Mathematicians Measure Infinities, Find They’re Equal

#20

Post by cjslep »

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…

There is one English definition of infinity, but when applied to ordinals ("degrees of freedoms"), you get different mathematical concepts of infinity.

Vsauce's explanation is approachable: https://youtu.be/SrU9YDoXE88

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