The beginning of infinity.

[SciFly]

I want to explore what people here think about infinity, and if the concept of infinity has any place in science.

 

If I were to write down a binary number 0.1 and then a moment later append a 1 to it giving 0.11 and again 0.111 and so on for an infinite number of times.

 

Then would the number I create ever have an infinite number of digits, as an irrational number like pi, or would it always have a finite number of digits?

 

Also, do you think that this is a question about definitions or about necessary truth?

 

 

 

 

[wtf]

In binary, .0101010101... = 1/3. On the other hand some other bitstring might be irrational. This is just basic binary notation of real numbers. It's no different than decimal, in which .14159... is the decimal representation of pi - 3, which is irrational; and .3333333... is the decimal representation of 1/3, a rational.

 

Every such binary or decimal expression must denote a real number, because the set of finite truncations is a nonempty set bounded above by 1, hence has a least upper bound by the completeness of the real numbers. In other words the set {.1, .14, .141, .1415, ...} is bounded above by 1, hence has a least upper bound.

 

This is the basic theory of the real numbers. Has nothing to do with the mathematical study of infinity.

 

What is true is that in order to get the real numbers off the ground, we have to allow a "completed" infinity of the natural numbers {1, 2, 3, 4, 5, ...}. Of course nobody is making any claim that such a thing exists in the real world. No current physical theory includes infinite collections, but we don't know what physicists of the future will think.

 

However when it comes to physical science, everyone uses standard math to model physical theories. And the standard math of the real numbers does include the assumption that there is an infinite set. So there's a bit of a philosophical puzzler. If math is based on an assumption (the existence of an infinite set) that's manifestly false about the real world, why does math work so well to describe the real world?

Edited October 2, 2016 by wtf

[studiot]

 

 

wtf post#2

No current physical theory includes infinite collections,

 

 

Maybe but what is the 'proper length' of a lightwave according to Einstein's relativity?

[wtf]

Maybe but what is the 'proper length' of a lightwave according to Einstein's relativity?

I have no idea. Perhaps you can explain your remark.

 

I'm aware that quantum physics takes place in Hilbert space, an abstract infinite dimensional function space. I don't know anything about how philosophers of physics reconcile the apparent contradiction of using vast infinite spaces to model the real world, when there is no evidence for infinite collections. For example there are 10^80 hydrogen atoms in the known universe. If you know something about how this apparent contradiction is reconciled, I'd be happy to learn.

Edited October 2, 2016 by wtf

[studiot]

The proper length of a lightpath is infinite since it is accomplished in zero proper time.

[wtf]

The proper length of a lightpath is infinite since it is accomplished in zero proper time.

I don't know much physics and I don't know what you're talking about.

 

I am sincere in wanting to understand your point. As far as I know, no theory of physics posits an actual infinity in the real world. 10^80 atoms and all that. How can a "light path," whatever that is, have an infinite length? How does any infinite length fit into a finite universe?

 

Perhaps you think I know more physics than I actually do. I have no idea what you're talking about but I am interested in understanding. I don't know what you mean by proper path or proper length. I don't know what you mean by lightpath. The only thing I know about infinities in physics is that they need to be renormalized so that they go away. And the bit I mentioned about Hilbert space. I know Hilbert space only as an abstract mathematical structure. I don't have any idea how quantum physics is reconciled with a finite universe. Love to find out though.

Edited October 2, 2016 by wtf

[SciFly]

we have to allow a "completed" infinity

 

 

Then you would say that our 0.111... has a completed infinity of digits.

 

We can then expand our thought experiment a bit. Instead of writing just one line of digits, we will write a matrix of digits like this-

 

1st we write

 

0.0

0.1

 

Then we will expand our matrix like this

 

0.00

0.10

0.01

0.11

 

Next

 

0.000

0.100

0.010

0.110

0.001

0.101

0.011

0.111

 

And so on forever.

 

The question I want to ask is "will every row of this matrix have an infinite number of digits?" If they do, then will a number like pi/4 be in one of those rows?

[wtf]

The question I want to ask is "will every row of this matrix have an infinite number of digits?"

Yes, every real number has a decimal representation consisting of infinitely many digits (to the right of the decimal point). There's one digit for each of the real numbers 1, 2, 3 ...

 

If they do, then will a number like pi/4 be in one of those rows?

Not necessarily. You can't list the real numbers. That's Cantor's diagonal argument. Any list of real numbers must leave some out.

 

 

https://en.wikipedia.org/wiki/Cantor's_diagonal_argument

 

A more precise statement is that any function from the natural numbers to the reals can not be surjective; that is, it can not hit every real.

Edited October 3, 2016 by wtf

[Strange]

I'm aware that quantum physics takes place in Hilbert space, an abstract infinite dimensional function space. I don't know anything about how philosophers of physics reconcile the apparent contradiction of using vast infinite spaces to model the real world, when there is no evidence for infinite collections. For example there are 10^80 hydrogen atoms in the known universe. If you know something about how this apparent contradiction is reconciled, I'd be happy to learn.

 

 

It is possible the whole universe is infinite.

Not that that is relevant to your question. I'm not sure there is any contradiction.

 

Hilbert spaces have infinite dimensions, the natural numbers have infinite cardinality, and there are other infinities that occur naturally in mathematics. That doesn't stop it being applied to the real world. Just because we could count an unbounded number of apples doesn't mean we have to (or that we can't count three of them).

[studiot]

I want to explore what people here think about infinity, and if the concept of infinity has any place in science.

 

If I were to write down a binary number 0.1 and then a moment later append a 1 to it giving 0.11 and again 0.111 and so on for an infinite number of times.

 

Then would the number I create ever have an infinite number of digits, as an irrational number like pi, or would it always have a finite number of digits?

 

Also, do you think that this is a question about definitions or about necessary truth?

 

 

 

 

 

Regarding your second question (underlined in the quote), why do you fail to discuss this further?

 

Regarding your first qustion about decimal representation

 

Hypothesis : Each decimal representation of the real numbers between 0 and 1 can be arranged to correspond to a unique whole number.

 

(This is what is meant by countable or enumerable.)

 

For the decimal corresponding to any whole number, n, let the n_th digit be a_n.

If a_n = 0,1,2,3,4,5,6 or 7 then take b_n = (a_n + 1)

If a_n = 8 or 9 then take b_n = 0

 

Then the decimal 0.b₁b₂b₃........_.b_n....... differs from every decimal in the enumerated set in at least one place, contrary to hypothesis.

 

Hence the hypothesis fails and the set of real decimals cannot be placed in an enumerable set.

Edited October 3, 2016 by studiot

[Strange]

Then would the number I create ever have an infinite number of digits, as an irrational number like pi, or would it always have a finite number of digits?

 

It would always have a finite number because you can always add another digit (same reason there is no largest integer).

[SciFly]

 

It would always have a finite number because you can always add another digit...

 

Then would you agree that in binary 1.0 > 0.111...

 

or in decimal 1.0 > 0.999...

[studiot]

 

Then would you agree that in binary 1.0 > 0.111...

 

or in decimal 1.0 > 0.999...

 

 

No

[Strange]

 

Then would you agree that in binary 1.0 > 0.111...

 

or in decimal 1.0 > 0.999...

 

No.

 

Although, 1 is larger than any number you can create by appending 9s to 0.9.

[Country Boy]

I want to explore what people here think about infinity, and if the concept of infinity has any place in science.

 

If I were to write down a binary number 0.1 and then a moment later append a 1 to it giving 0.11 and again 0.111 and so on for an infinite number of times.

 

Then would the number I create ever have an infinite number of digits, as an irrational number like pi, or would it always have a finite number of digits?

 

Also, do you think that this is a question about definitions or about necessary truth?

 

 

 

 

 

First, you poll is invalid because it asks a meaningless question- there is no such number to begin with. As for you question, because you create the numeral by adding the digit 1 "for an infinite number of times" the number, as written, obviously has an infinite number of digits". But numbers can be written in different ways. If we let x= 0.1111.... (continuing infinitely) then, because this numeral is in binary, multiplying by 2, 2x= 1.1111... (still continuing infinitely). Subtracting, x= 1, just as, in base 10, "0.999...." is the same as 1. You need to understand that these are questions about numerals (the way numbers are represented), not about numbers themselves.

 

Finally, mathematics itself is about definitions and logic, not "necessary truth".