The meaning of infinity

[Speakpigeon]

You can't blame me if I'm confused as to whether mathematicians think of the Real numbers as integers plus a possibly a non-zero decimal part, or if they think they're somehow all the quantities in a continuum. I would have thought you'd need to choose one or the other. And this being mathematicians, I would suggest they keep with the decimal numbers. Seems we know what we're talking about there. Less metaphysical, so to speak.

How could we even prove that things like the decimal numbers map any continuum?
EB

I'd respond, but I honestly think it would be a waste of time for both of us.

You have no idea.
EB

 

[Juma]

prove the decimal numbers map any continuum.

Since Real numbers are the continuum by definition we just have to prove that any real number can be written as a decimal number (not necesarily with finite number of decimals).

To prove that intuitively :

start with a line marked with integers.
Pick any point on this line. (Between, or on the integers)

Now pick the nearest smaller integer a0 and the nearest larger integer b0
Divide the interval between a0 and b0 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a1 and the nearest to the right b1
Divide the interval between a1 and b1 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a2 and the nearest to the right b2
Divide the interval between a2 and b2 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Etc

Working like this you get a serie of numbers a0 a1 a2 a4 etc that will be the decimal number.

 

[Speakpigeon]

prove the decimal numbers map any continuum.

Since Real numbers are the continuum by definition we just have to prove that any real number can be written as a decimal number (not necesarily with finite number of decimals).

To prove that intuitively :

start with a line marked with integers.
Pick any point on this line. (Between, or on the integers)

Now pick the nearest smaller integer a0 and the nearest larger integer b0
Divide the interval between a0 and b0 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a1 and the nearest to the right b1
Divide the interval between a1 and b1 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a2 and the nearest to the right b2
Divide the interval between a2 and b2 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Etc

Working like this you get a serie of numbers a0 a1 a2 a4 etc that will be the decimal number.

Thanks.

I have no difficulty accepting this could be done, ad libitum. Decimals are obviously at the very least a very good approximation of the Real line. Yet, the point would be to prove the two sets are either identical, or somehow equivalent. How do you even prove that all transcendentals are equivalent to decimals? Intuitively, not much of a problem. Any proof, though?
EB

 

[Phil Scott]

Since Real numbers are the continuum by definition

I'd dispute that, since it leaves me asking "what are the hyperreals suppposed to model?" I'd say that both the reals and the finite hyperreals form adequate models of our informal notion of the continuum, and suggest that the only reason the former dominates is because of an accident of history.

These philosophical arguments don't move me a whole lot though. I want a structure that allows me to do all the algebraic geometry that I want, and I want to be able to do calculus. The reals are adequate for that purpose, as are the hyperreals, with the choice between them being, ultimately, arbitrary.

 

[Juma]

prove the decimal numbers map any continuum.

Since Real numbers are the continuum by definition we just have to prove that any real number can be written as a decimal number (not necesarily with finite number of decimals).

To prove that intuitively :

start with a line marked with integers.
Pick any point on this line. (Between, or on the integers)

Now pick the nearest smaller integer a0 and the nearest larger integer b0
Divide the interval between a0 and b0 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a1 and the nearest to the right b1
Divide the interval between a1 and b1 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a2 and the nearest to the right b2
Divide the interval between a2 and b2 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Etc

Working like this you get a serie of numbers a0 a1 a2 a4 etc that will be the decimal number.

Thanks.

I have no difficulty accepting this could be done, ad libitum. Decimals are obviously at the very least a very good approximation of the Real line. Yet, the point would be to prove the two sets are either identical, or somehow equivalent. How do you even prove that all transcendentals are equivalent to decimals? Intuitively, not much of a problem. Any proof, though?
EB

The sets are not identical. There are more decimal numbers then there are reals.
Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

But I have already shown that each real has a decimal representation.
That each decimal corresponds to a really the same; think if the decimals as ever smaller scale markings.
For finite number of decimals it is obvious that a decimal always maps to a real.
For infinite numbers of decimals its a bit trickier but it is not so hard to see that it must converge (it is always within the interval of the decimal your are on).

 

[ryan]

Since Real numbers are the continuum by definition

I'd dispute that, since it leaves me asking "what are the hyperreals suppposed to model?" I'd say that both the reals and the finite hyperreals form adequate models of our informal notion of the continuum, and suggest that the only reason the former dominates is because of an accident of history.

These philosophical arguments don't move me a whole lot though. I want a structure that allows me to do all the algebraic geometry that I want, and I want to be able to do calculus. The reals are adequate for that purpose, as are the hyperreals, with the choice between them being, ultimately, arbitrary.

It may just be a continuum from 1 to 0, no hyperreals needed.

 

[ryan]

Thanks.

I have no difficulty accepting this could be done, ad libitum. Decimals are obviously at the very least a very good approximation of the Real line. Yet, the point would be to prove the two sets are either identical, or somehow equivalent. How do you even prove that all transcendentals are equivalent to decimals? Intuitively, not much of a problem. Any proof, though?
EB

The sets are not identical. There are more decimal numbers then there are reals.
Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

But 1 = 0.99999...

 

[Phil Scott]

It may just be a continuum from 1 to 0, no hyperreals needed.

And it may not be a continuum without the hyperreals. Hyperreals needed.

 

[Juma]

Thanks.

I have no difficulty accepting this could be done, ad libitum. Decimals are obviously at the very least a very good approximation of the Real line. Yet, the point would be to prove the two sets are either identical, or somehow equivalent. How do you even prove that all transcendentals are equivalent to decimals? Intuitively, not much of a problem. Any proof, though?
EB

The sets are not identical. There are more decimal numbers then there are reals.
Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

But 1 = 0.99999...

Yes? I said so.

 

[ryan]

It may just be a continuum from 1 to 0, no hyperreals needed.

And it may not be a continuum without the hyperreals. Hyperreals needed.

I don't think that is true.

- - - Updated - - -

But 1 = 0.99999...

Yes? I said so.

Then does the = sign only work for reals? Or do we have to put =~ ?

 

[Phil Scott]

I don't think that is true.

Okay, sure. But suppose I think it is. Where are we now?

 

[Phil Scott]

Then does the = sign only work for reals? Or do we have to put =~ ?

How do you feel about 1l2 = 2/4.

 

[steve_bank]

Tje results of numerical calculations in a general sense are all approximations. In practical work what matters is having enough decimal places that results do not become inaccurate from rounding. You can see interesting results with a hand calculator from rounding and finite number size.


0.9999 is a real number. 0.9999... is not, it is an infinite sequence.

 

[skepticalbip]

Then does the = sign only work for reals? Or do we have to put =~ ?

How do you feel about 1l2 = 2/4.

Or maybe better would be:

1/3 ~ 0.33333333333333333333

but:

1/3 = 2/6

 

[Speakpigeon]

Thanks.

I have no difficulty accepting this could be done, ad libitum. Decimals are obviously at the very least a very good approximation of the Real line. Yet, the point would be to prove the two sets are either identical, or somehow equivalent. How do you even prove that all transcendentals are equivalent to decimals? Intuitively, not much of a problem. Any proof, though?
EB

The sets are not identical. There are more decimal numbers then there are reals.
Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

OK, that's a good point.

Personally, I take 0.999... to be equal to 1, and I take equality between numbers to be the same as them being exactly the same thing.

This is because I tend to take "0.999..." and "1" as two different formal expressions of the same number. Same things for "1/3" and "0.333...".

So, as I see it, the set of fractions is potentially the same as the set of all rational decimals. This needs to be proved because it's absolutely evident but the result is just that.

Still, I understand what you're saying. I could use the word "quantity" instead and reserve the word "number" for the formal expression of a quantity. I guess that's what you're doing and this may be more in line with how most mathematicians think of it.

So, if there are five children in the courtyard, there is no number in the courtyard, but there is a quantity, which is a quantity of children...

But I have already shown that each real has a decimal representation.

No, you haven't. You've merely expressed how you intuitively think of the problem. It's no proof.

That each decimal corresponds to a really the same; think if the decimals as ever smaller scale markings.

I understand your point. I just don't take it as a proof.

For finite number of decimals it is obvious that a decimal always maps to a real.

I think this isn't the issue. The question is whether decimals map all the Reals, i.e. whether there aren't any Reals somehow left out.

For infinite numbers of decimals its a bit trickier but it is not so hard to see that it must converge (it is always within the interval of the decimal your are on).

Sorry, you can't assume any "convergence" without having specified the series you're talking about.

If you're talking about a series of intervals of decimals only defined by the intervals being included in each other, the this series doesn't converge. It's obvious there is an infinity of possible ways it can go.
EB

 

[Speakpigeon]

Tje results of numerical calculations in a general sense are all approximations. In practical work what matters is having enough decimal places that results do not become inaccurate from rounding. You can see interesting results with a hand calculator from rounding and finite number size.


0.9999 is a real number. 0.9999... is not, it is an infinite sequence.

What's a "real number"?
EB

 

[Speakpigeon]

Then does the = sign only work for reals? Or do we have to put =~ ?

How do you feel about 1l2 = 2/4.

Or maybe better would be:

1/3 ~ 0.33333333333333333333

but:

1/3 = 2/6

Well, if so, then why not 1/3 = 0.333... and 1 = 0.999...
EB

 

[Phil Scott]

The question is whether decimals map all the Reals, i.e. whether there aren't any Reals somehow left out.

Any mathematical answer to this requires a formal definition of "real numbers". There are such definitions, all equivalent. An appropriate abstract definition is that the reals are, up to isomorphism, any ordered collection together with two operations "+" and "x" satisfying the axioms of an ordered field, and with the crucial axiom that any subset of this collection that is bounded above has a smallest such bound.

The use of "the reals" is justified because all collections which satisfy these axioms are isomorphic. The two traditional examples of such collections are the set of all Cauchy sequences and the set of Dedekind cuts, both of which can serve as definitions of reals themselves.

The set of infinite decimals, with recurring sequences x.999... being identified with x.000..., can also be shown to satisfy the abstract definition of reals. Alternatively, it can be shown that every Cauchy sequence or Dedekind cut maps to one of these decimals and back again (with the caveat that x.999... and x.000... map to the same sequence or cut).

All the proofs are somewhat abstract and a little tedious, but are generally reviewed in undergraduate real analysis.

Questions about the continuum are not strictly mathematical questions, since "continuum" lacks the formal definition that "real numbers" have. This question belongs more to philosophy of maths.

 

[Juma]

I don't think that is true.

- - - Updated - - -

But 1 = 0.99999...

Yes? I said so.

Then does the = sign only work for reals? Or do we have to put =~ ?

Eh? The decimal number is a textual representation. 1 and 0.9999... are different textual representations.
Both representations represents the same real number.

 

[Juma]

Tje results of numerical calculations in a general sense are all approximations. In practical work what matters is having enough decimal places that results do not become inaccurate from rounding. You can see interesting results with a hand calculator from rounding and finite number size.


0.9999 is a real number. 0.9999... is not, it is an infinite sequence.

Sigh...
0.9999 is a decimal representation of a Real number.
0.9999... is another decimal representation of (another) Real number.