The meaning of infinity

[Phil Scott]

This was originally about whether or not the reals form a continuum. I was taught that they do in a very rigorous calculus course.

Weere you given a proof? Rigorously showing that the reals form a continuum would require a rigorous definition of "continuum." What we have instead is rigorous definitions for things like "real number" and "hyperreal number". Whether these collections get at the nature of the continuum is more into the metaphysicals (as Speakpigeon observed).

So I need some reference that your claim is true.

Which claim in particular? I'm afraid I won't be giving references for what is undergraduate level mathematics. I can link you to my PhD thesis in formalised mathematics (where the rigor is dialled as high as it goes), and if that's not enough, I'll just leave you folks to it.

As for the equals symbol. Like I asked Juma, is it only an equals symbol for reals? If so, then I have no issue with 1 = 0.99.... If not, then the previous claims about 1 not being the same as 0.99... using hyperreals naturally concerns me when I see 1 = 0.99....

The equals symbol is, as you say, just the equals symbol. It's used to say that the two expressions on either side denote the same object.

The hyperreals are a proper superset of the reals, and so you will still use your decimals to refer to the subset of the hyperreals which are real. To refer to finite non-real hyperreals, you need to state what the infinitesimal part is.

 

[Speakpigeon]

Here you seem to be saying the opposite:

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.

I wouldn't know how to interpret this if not as you saying that 0.999999... and 1.0 are two different numbers.

A recurrent problem with your pronouncements.
EB

Read the posts again. Everything is there. You just have to actuslly read what is there, not what you expect to read.
A hint: ”decimal numbers are a textual representation” is central.

Sorry, I don't get it.

Hint: We use textual representations to talk about things represented, not about the representations themselves. Else, they would stop being representations at all.

2nd hint: "0.999999..." and "1.0" are two different decimals textual representations.

3rd hint: 0.999999... and 1.0 are equal, meaning they are the same decimal, just as 2/3 is the same 4/6 because 2/3 stands for the result of 2 divided by 3, and same for 4/6.

One more hint:

Show

How is this the same thing as my other issue? 2/1 = 2, so... ?

I couldn't say. To be honest, I'm not entirely sure what your issue is. There appears to be a lot of confusion in this thread generally, which is the usual for threads like this, which are often led by people who's experience with real numbers doesn't extend behind playing about with a calculator in high school. No offence meant there: these foundational issues over the real numbers didn't get settled until the mid-19th century, which is incredibly late in the history of maths. It takes a bit of sophistication to finally settle them.

Maybe this helps, maybe not: one of the best bits of ranting I ever read of Quine was in his Mathematical Logic, where he spent a good deal of time talking about the distinction between use and mention. The idea is that one uses signs to mention what's signified, and that we should never confuse the two. To avoid this confusion with the pedantry required of a mathematical logician, we have to be exact in our use of quotation, quotes being devices which we can use to mention signs.

For example, a dog is a four legged carnivore. "Dog", however, is a word which I can use to mention dogs. I can even iterate this, and say that "'dog'" (I just used double quotation) is a quotation I use to mention the word "dog", which I use in turn to mention dogs.

Now we can use this to get all pedantic and correct something Juma said earlier:

Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

What one should say is that "0.999..." and "1.0" are two different decimals. "0.999..." is not equal to "1.0". They are different decimals. But the reals which these decimals mention are 0.999... and 1.0, which are, in fact, the same real, a matter we can express with the equation "0.999... = 1". Indeed, 0.999... = 1.

There's no "=~" involved in this discussion (does not mean "approximately equal to"?). 0.999... and 1.0 are identical reals, just as 1/2 and 2/4 are identical reals, and just as 1 + 1 and 10 - 8 are identical reals.

See?
EB

 

[ryan]

Weere you given a proof? Rigorously showing that the reals form a continuum would require a rigorous definition of "continuum." What we have instead is rigorous definitions for things like "real number" and "hyperreal number". Whether these collections get at the nature of the continuum is more into the metaphysicals (as Speakpigeon observed).

I was told that the reals are "complete" and that they form a continuum (by my professor that got his PhD from Princeton ). Is there a proof that the hyperreals has a larger cardinality, or is it still just a hypothesis?

As for the equals symbol. Like I asked Juma, is it only an equals symbol for reals? If so, then I have no issue with 1 = 0.99.... If not, then the previous claims about 1 not being the same as 0.99... using hyperreals naturally concerns me when I see 1 = 0.99....

The equals symbol is, as you say, just the equals symbol. It's used to say that the two expressions on either side denote the same object.

... except when it comes to the hyperreal "set".

The hyperreals are a proper superset of the reals, and so you will still use your decimals to refer to the subset of the hyperreals which are real. To refer to finite non-real hyperreals, you need to state what the infinitesimal part is.

Interesting that you are using "subset". Is that for the reals of the finite hyperreals? If so, then that would seem to contradict what I was taught.

 

[Phil Scott]

I was told that the reals are "complete" and that they form a continuum (by my professor that got his PhD from Princeton ).

The reals are Cauchy-complete. They are complete in that they contain all the limits of their Cauchy sequences. They are also Dedekind-complete, in that all sets bounded above have a least upper bound.

The complex numbers are also "complete", in that they are algebraically complete: they contain all solutions to algebraic equations, such as x^2 = -1.

Is there a proof that the hyperreals has a larger cardinality, or is it still just a hypothesis?

The standard construction of the hyperreals carves them out of sequences of real numbers. The space of all sequences of real numbers has the same cardinality as the real numbers. The hyperreals are a superset of the reals. Therefore, they have the same cardinality.

... except when it comes to the hyperreal "set".

What?

Interesting that you are using "subset". Is that for the reals of the finite hyperreals? If so, then that would seem to contradict what I was taught.

What I wrote was very clear. If you don't quite understand the term "subset", we need to start simpler.

 

[ryan]

The reals are Cauchy-complete. They are complete in that they contain all the limits of their Cauchy sequences. They are also Dedekind-complete, in that all sets bounded above have a least upper bound.

The complex numbers are also "complete", in that they are algebraically complete: they contain all solutions to algebraic equations, such as x^2 = -1.

The standard construction of the hyperreals carves them out of sequences of real numbers. The space of all sequences of real numbers has the same cardinality as the real numbers. The hyperreals are a superset of the reals. Therefore, they have the same cardinality.

That makes sense.

... except when it comes to the hyperreal "set".

What?

Why is this such a problem for you???

1 = 0.999... except, as you seemed to imply, for the hyperreals. So obviously it begs my question of what = strictly implies.

Interesting that you are using "subset". Is that for the reals of the finite hyperreals? If so, then that would seem to contradict what I was taught.

What I wrote was very clear. If you don't quite understand the term "subset", we need to start simpler.

Wasn't it you who said that 0.999... is a hyperreal and not a real?

 

[Phil Scott]

Why is this such a problem for you???

Because honestly, I don't understand where this confusion is coming from. I don't mind such confusion, but when it is paired with the sort of attitude I've read in this thread, I tend to get blunt.

1 = 0.999... except, as you seemed to imply, for the hyperreals. So obviously it begs my question of what = strictly implies

1 = 0.999... in the hyperreals too. "1" and "0.999..." are just ways to refer to 1. Basic decimal notation doesn't change for the hyperreals. And fundamental concepts like equality certainly don't change.

Wasn't it you who said that 0.999... is a hyperreal and not a real?

Nope. 0.999... is 1.

1 is a natural number, an integer, a rational, a real, a complex number and a hyperreal.

 

[ryan]

Because honestly, I don't understand where this confusion is coming from. I don't mind such confusion, but when it is paired with the sort of attitude I've read in this thread, I tend to get blunt.

1 = 0.999... in the hyperreals too. "1" and "0.999..." are just ways to refer to 1. Basic decimal notation doesn't change for the hyperreals. And fundamental concepts like equality certainly don't change.

Wasn't it you who said that 0.999... is a hyperreal and not a real?

Nope. 0.999... is 1.

1 is a natural number, an integer, a rational, a real, a complex number and a hyperreal.

Okay then where are the hyperreals needed for there to be a continuum? This is what I originally had an issue with.

 

[Phil Scott]

Okay then where are the hyperreals needed for there to be a continuum? This is what I originally had an issue with.

I never said they did. Here's what I wrote:

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.

I am saying that the reals are an adequate model of our informal notion of the continuum. I'm saying that the hyperreals are an adequate model of our informal notion of the continuum. I put it down to taste which you prefer. Historically, the ancient Greeks worked with Archimedean conceptions (which are the reals). Newton and Leibniz worked with non-Archimedean conceptions (which are the hyperreals). The first set-theoretic formalisations of the continuum were the reals. The hyperreals were figured out in the 20th century, by which time, everyone was already pretty happy with the reals.

 

[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.

This is what I am talking about. Where are the hyperreals needed?

 

[Phil Scott]

This has now become a tedious joke.

Have fun folks! This mathematics subforum is a colossal waste of time, IMO.

 

[ryan]

This has now become a tedious joke.

Have fun folks! This mathematics subforum is a colossal waste of time, IMO.

Oh and you also said this which made me jump in, " I'd say that both the reals and the finite hyperreals form adequate models of our informal notion of the continuum".

 

[untermensche]

Mathematics is just a tool.

Examining it closely is as exciting as examining a hammer.

Very exciting to some.

Tedious to most.

 

[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.

I will be honest; looking back, the rest of your post supports what you said all along. I guess I started down this road with you saying that you "would dispute that". And then with that in mind it looked like you would have meant that both reals and hyperreals are neccessary for one model, but I see it was plural.

So then what exactly are you disputing about with what juma said?

 

[steve_bank]

Mathematics is just a tool.

Examining it closely is as exciting as examining a hammer.

Very exciting to some.

Tedious to most.

The intelectual isolated from reality. Math as a tool is manifest in everything around you manmade, yet you are ignorant of the simple fundamentals. I always found exploring both math and science far more entertaining than idle conversation. I heard it said the best entertainment is learning something new.

Repeating one line statements over and over seems rather dull. I'd think you'd pickup something just by immersion and osmosis.

 

[Juma]

Oh, this one? Sorry...


Sorry, I'm confused now. It seems to me you could do the same thing using the Rationals instead of the Decimals and therefore prove similarly that the Rationals is the same as the Reals and yet the Rationals are missing the part of the Reals that's called the Irrationals.

Can you explain where I'm wrong here?
EB

Sorry if was unclear...
The sequence of narrowing down intervals that I descibed was the build up of an infinite long decimal number as
In 6.7375674785365377...
as a contrast a rational number, as for example 567/397 is the completed number. The corresponding example with rational numbers would be a series 1/2+ 1/4+7/230+...

Sorry, but that doesn't address my point.
EB

I think it does.
Simplified an non-pedantic: The difference between reals and rationals is that reals doeasnt allow you to write ”...”
That is why decimals can represent more reals than rationals.

 

[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.

I will be honest; looking back, the rest of your post supports what you said all along. I guess I started down this road with you saying that you "would dispute that". And then with that in mind it looked like you would have meant that both reals and hyperreals are neccessary for one model, but I see it was plural.

So then what exactly are you disputing about with what juma said?

Thanks for going back over the thread.

I dispute the idea, at least in a thread such as this, that the reals are the continuum by definition. That was Juma's claim.

I'd say that the identification of the reals and the continuum was excusable when we had the likes of Bertrand Russell declaring infinitesimals to be incoherent. But then some decades after that overconfident pronouncement, it was discovered that the existence of hyperreals are equiconsistent with reals, and that they were the basis for a differential calculus with out need for limits, and a model to legitimise the logic you find in Newton and Leibniz' genuine infinitesimal calculus.

My dispute is over the idea that, when it comes to understanding our intuitive idea of continuous space, the continuum, the reals are the only game in town. I would say they are just one model.

And in a thread about infinity, the reals are the less interesting model, because only the hyperreals include a rich structure of both infinitesimals and infinite numbers, one which satisfies all the laws of high school algebra, unlike the arithmetically impoverished transfinite cardinals, that only extend the naturals, not the reals.

They are worth talking about here.

However, the hyperreals offer nothing new on the interminable 0.999… = 1 internet debate. 0.999… = 1 always.

 

[untermensche]

Mathematics is just a tool.

Examining it closely is as exciting as examining a hammer.

Very exciting to some.

Tedious to most.

The intelectual isolated from reality. Math as a tool is manifest in everything around you manmade, yet you are ignorant of the simple fundamentals. I always found exploring both math and science far more entertaining than idle conversation. I heard it said the best entertainment is learning something new.

Repeating one line statements over and over seems rather dull. I'd think you'd pickup something just by immersion and osmosis.

Show me where I repeated myself.

Yes, a basic knowledge of geometry and trigonometry and algebra can be very helpful in the world.

But beyond that it is esoteric.

I had to take calculus to get into pharmacy school.

Nothing but a weeder class.

It was not used one time in pharmacy school and I have not used it one time since I took it.

 

[Phil Scott]

Mathematics is just a tool.

Examining it closely is as exciting as examining a hammer.

Very exciting to some.

Tedious to most.

The intelectual isolated from reality. Math as a tool is manifest in everything around you manmade, yet you are ignorant of the simple fundamentals. I always found exploring both math and science far more entertaining than idle conversation. I heard it said the best entertainment is learning something new.

Repeating one line statements over and over seems rather dull. I'd think you'd pickup something just by immersion and osmosis.

Show me where I repeated myself.

Yes, a basic knowledge of geometry and trigonometry and algebra can be very helpful in the world.

But beyond that it is esoteric.

I had to take calculus to get into pharmacy school.

Nothing but a weeder class.

It was not used one time in pharmacy school and I have not used it one time since I took it.

I confess I haven't read many of your posts, but that is probably the dumbest thing I've read on this forum.

 

[untermensche]

Show me where I repeated myself.

Yes, a basic knowledge of geometry and trigonometry and algebra can be very helpful in the world.

But beyond that it is esoteric.

I had to take calculus to get into pharmacy school.

Nothing but a weeder class.

It was not used one time in pharmacy school and I have not used it one time since I took it.

I confess I haven't read many of your posts, but that is probably the dumbest thing I've read on this forum.

I've read a few of your posts and you have never once backed up an opinion you made with actual argument in any of them.

You are a complete waste a time.

Your opinion about mathematics is no better than any other opinion.

You're a sniper that takes shots and runs away.

I don't expect anything less now.

 

[untermensche]

Esoteric: intended for or likely to be understood by only a small number of people with a specialized knowledge or interest.