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Hello,

I've been getting into the concept of hyperreal numbers lately, and I've got tons of questions. What I understand about the hyperreals is that they are numbers larger than any real number or smaller than any real number. I'm sure you can imagine how counterintuitive this sounds to someone like me who's new to the concept. It's like talking about numbers greater than infinity. I always thought that was impossible. So it shouldn't be surprising that someone like me would have a ton of questions. I'll start with a couple.

1) Assume that R is a hyperreal number greater than any real number. What does 2 x R equal? It's clear what 2 x n means where n is a real number because there is a 0 value for reference--i.e. 2 x n is a number twice the distance from 0 as n is from 0. But do the hyperreals have their own 0 point? How could they if they are greater than any real number (I realize some hyperreals are smaller than any real number, but for this question I'm only focused on the infinitely large hyperreals)? If 2 x R means twice the distance from 0 the real number as R is from 0 the real number, the you get a number another infinite distance away--sort of like a hyper-hyperreal number. <-- Does that make sense? Do the infinitely large hyperreals have their own infinity beyond which are numbers that are hyperreal even to the hyperreals?

2) I remember watching a vsauce episode on youtube where Michael Stevens explained the difference between cardinals and ordinals, which as I understand it is the difference between numbers that represent quantities and numbers that represent orders. He explained that while there is no cardinal number greater than infinity, you could talk about ordinal numbers greater than infinity. He didn't explicitly link ordinals to hyperreals but it seemed like the same idea. He stressed that since ordinals don't stand for quantities, you cannot use ordinals to speak of "how much" something is, but simply whether they come "before" or "after" another number. Is this true of hyperreals as well? If so, this would seem to imply that there is no 0 point on the hyperreal number line as that would mean you could quantify any hyperreal number R (the ones greater than infinity). It's quantity would just be how many whole hyperreal numbers it is away from "hyper-zero" (just as we say the number 5 represents the quantity of whole numbers it is away from 0). But if there is no such "hyper-zero" number, then there isn't a reference point relative to which we can say "how much" a hyperreal number (greater than infinity) represents (except that it's greater than any real number). We could still quantify the difference between any two (greater than infinity) hyperreal numbers. So we could say R+3 is 3 greater than R, but without knowing how much R really is, we don't really know how much R+3 is either. So I guess the question is: should hyperreal numbers greater than infinity be thought of as ordinals only--they represent orders of number, not quantities--or is there a way of talking about their quantities as well?

I'll stop there for now. Thanks for any forthcoming responses.

Edited by gib65

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I would guess that a hyperreal infinity would end up just being infinity if you took the limit of it.  Then you could treat it as a normal infinity after finding the limit of it.

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Before I answer some of these questions, I'd like to note that it may be easier if you learn how the hyperreals are defined formally. However, the formal definition is somewhat complex and based in advanced set theory (namely ultrafilters and the existence of free ultrafilters on the natural numbers).

8 hours ago, gib65 said:

1) Assume that R is a hyperreal number greater than any real number. What does 2 x R equal? It's clear what 2 x n means where n is a real number because there is a 0 value for reference--i.e. 2 x n is a number twice the distance from 0 as n is from 0. But do the hyperreals have their own 0 point?

Yes. In fact, every real number is also a hyperreal number (if you have an exacting philosophy of math, the statement might be more accurately stated as "Every real number has a hyperreal counterpart", but I'm going to ignore philosophical issues for now). Slightly more formally, we really consider the hyperreals as an extension of the real numbers - the set of hyperreals includes the real numbers, and then some.

8 hours ago, gib65 said:

Does that make sense? Do the infinitely large hyperreals have their own infinity beyond which are numbers that are hyperreal even to the hyperreals?

You can create "hyper-hyperreals" through a similar formal process to the way the hyperreals are constructed. However, 2*R isn't special - it is "merely" hyperreal, in the same way that 0 is "merely" real. You can do any formal algebraic operation on the hyperreals that you could do on the reals - add them, subtract them, multiply them, etc. Formally, they are an ordered field just like the real numbers.

8 hours ago, gib65 said:

He explained that while there is no cardinal number greater than infinity, you could talk about ordinal numbers greater than infinity.

I strongly suspect that you misunderstood something here; the ordinals, cardinals, and hyperreals are all in some way or another generalizations of the idea of infinity. All of them have some idea of "greater than infinity".

8 hours ago, gib65 said:

So I guess the question is: should hyperreal numbers greater than infinity be thought of as ordinals only--they represent orders of number, not quantities--or is there a way of talking about their quantities as well?

If I had to give an answer, I'd say that they are like quantities in that they can be added, subtracted, multiplied, divided, etc. However, I'd more say that ordinals, cardinals, and hyperreals are simply attempts to extend different collections of properties of finite things - in the case of ordinals, ordering; in the case of cardinals, counting; in the case of hyperreals, the arithmetic.

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5 hours ago, uncool said:

Before I answer some of these questions, I'd like to note that it may be easier if you learn how the hyperreals are defined formally. However, the formal definition is somewhat complex and based in advanced set theory (namely ultrafilters and the existence of free ultrafilters on the natural numbers).

Yes, the definitions and explanations on the web are very complex. They just confuse me. I'm a better learner if I just ask someone and they explain it in plane English.

5 hours ago, uncool said:

Yes. In fact, every real number is also a hyperreal number (if you have an exacting philosophy of math, the statement might be more accurately stated as "Every real number has a hyperreal counterpart", but I'm going to ignore philosophical issues for now). Slightly more formally, we really consider the hyperreals as an extension of the real numbers - the set of hyperreals includes the real numbers, and then some.

Ah, but you're essentially saying the 0 of the hyperreals is just the 0 of the reals, correct?

5 hours ago, uncool said:

You can create "hyper-hyperreals" through a similar formal process to the way the hyperreals are constructed. However, 2*R isn't special - it is "merely" hyperreal, in the same way that 0 is "merely" real. You can do any formal algebraic operation on the hyperreals that you could do on the reals - add them, subtract them, multiply them, etc. Formally, they are an ordered field just like the real numbers.

But is it correct to think of 2 x R as twice the distance from 0 as R is from 0? If so, and if 2 x R is just another hyperreal number, then is it fair to say that there can be infinite distances between hyperreal numbers (assuming both hyperreal numbers are greater than any real number).

5 hours ago, uncool said:

I strongly suspect that you misunderstood something here; the ordinals, cardinals, and hyperreals are all in some way or another generalizations of the idea of infinity. All of them have some idea of "greater than infinity".

...

If I had to give an answer, I'd say that they are like quantities in that they can be added, subtracted, multiplied, divided, etc. However, I'd more say that ordinals, cardinals, and hyperreals are simply attempts to extend different collections of properties of finite things - in the case of ordinals, ordering; in the case of cardinals, counting; in the case of hyperreals, the arithmetic.

Do you mean to say even the cardinals can be extended passed infinity? As in, there are cardinal numbers (representing quantities) greater than infinity?

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20 minutes ago, gib65 said:

Yes, the definitions and explanations on the web are very complex.

I prefer to use the word complicated rather than complex partly because complex has its own special meaning in both English and Science and partly because it avoids confusion with complex numbers (especially in this case).

Complex means made of of more than one part, which cannot be removed without destroying the whole.

However I agree the web sometimes (often ?) overcooks the goose.

So here are some simpler definitions.

Cardinal does not mean it represents quantity.

A cardinal number is a measure of the size of a set that does not take into account the order of its members.

An ordinal number is a measure of the size of a set that does take into account the order of its members.

Note I have said ordinal and cardinal number. This is more specific and avoids confusion with the adjective cardinality and ordinality.

A hyper-real number is a symbol appended to a set of  numbers as an 'extension'.

Usually only  a handful of extra symbols are appended (such as ∞)  so that special rules for manipulating them can be set up and applied.

This is in some ways similar to the one special rule we have in ordinary arithmetic forbidding division by zero.

Is this English plane plain enough?

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2 hours ago, gib65 said:

Yes, the definitions and explanations on the web are very complex. They just confuse me. I'm a better learner if I just ask someone and they explain it in plane English.

Unfortunately, some of the concepts involved in the definition require that complexity.

2 hours ago, gib65 said:

Ah, but you're essentially saying the 0 of the hyperreals is just the 0 of the reals, correct?

Ignoring the philosophical issues, yes.

2 hours ago, gib65 said:

But is it correct to think of 2 x R as twice the distance from 0 as R is from 0? If so, and if 2 x R is just another hyperreal number, then is it fair to say that there can be infinite distances between hyperreal numbers (assuming both hyperreal numbers are greater than any real number).

Correct.

2 hours ago, gib65 said:

Do you mean to say even the cardinals can be extended passed infinity? As in, there are cardinal numbers (representing quantities) greater than infinity?

I mean that "infinity" is not a single concept by itself, so it doesn't really make sense to say "past infinity" in the same way as, e.g., "past 2". The closest thing I can say is that for any cardinal, there is a larger cardinal.

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4 hours ago, studiot said:

A hyper-real number is a symbol appended to a set of  numbers as an 'extension'.

Usually only  a handful of extra symbols are appended (such as ∞)  so that special rules for manipulating them can be set up and applied.

Bullpucky. You're confusing the hyperreals with finite extensions of the reals, or perhaps the extended reals. Why not take the trouble to find out what the thread's about before posting utterly false information?

Edited by wtf
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10 hours ago, uncool said:

Yes. In fact, every real number is also a hyperreal number (if you have an exacting philosophy of math, the statement might be more accurately stated as "Every real number has a hyperreal counterpart", but I'm going to ignore philosophical issues for now). Slightly more formally, we really consider the hyperreals as an extension of the real numbers - the set of hyperreals includes the real numbers, and then some.

From the wiki it sounded like hyperreals would only be the inverse of irrational numbers, and irrational numbers are a subset of the real numbers.

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.[1]

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...irrational numbers are not infinitesimals. They are real numbers.

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10 hours ago, studiot said:

A cardinal number is a measure of the size of a set that does not take into account the order of its members.

An ordinal number is a measure of the size of a set that does take into account the order of its members.

Here's the youtube video where Michael Stevens talks about cardinals and ordinals:

Note that at 7:55, he explains how only cardinals refer to amounts. Note the statement: "Omega plus one isn't bigger than omega, it just comes after omega."

Another question I have is: infinitesimals--are they divisible?

In other words, is an infinitesimal defined as the "smallest possible number" (in terms of magnitude, not how far below 0 it is)? Or is it more of a set of numbers that are infinitely smaller than any real number?

I would think its a set. Just as for any infinitely large hyperreal number R, you can have R + 1, R + 2, etc., and R - 1, R - 2, etc., I would think for any infinitely small hyperreal number e, you can have e/2, e/3, etc. or 2e, 3e, etc. That is, e doesn't represent a limit to how small numbers can get, it just represent an infinite amount of division you would have to do on a real number to get to it. That means that no matter how many times you multiply e, you will still only have an infinitely small hyperreal number.

What happens if you multiply the infinitely small hyperreal number e by the infinitely large hyperreal number R? Do you get a real number?

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1 hour ago, gib65 said:

Another question I have is: infinitesimals--are they divisible?

In other words, is an infinitesimal defined as the "smallest possible number" (in terms of magnitude, not how far below 0 it is)? Or is it more of a set of numbers that are infinitely smaller than any real number?

I would think its a set. Just as for any infinitely large hyperreal number R, you can have R + 1, R + 2, etc., and R - 1, R - 2, etc., I would think for any infinitely small hyperreal number e, you can have e/2, e/3, etc. or 2e, 3e, etc. That is, e doesn't represent a limit to how small numbers can get, it just represent an infinite amount of division you would have to do on a real number to get to it. That means that no matter how many times you multiply e, you will still only have an infinitely small hyperreal number.

What happens if you multiply the infinitely small hyperreal number e by the infinitely large hyperreal number R? Do you get a real number?

Yes, you can divide any two infinitesimals in the hyperreals as along as you don't divide by zero. You have that right. And therefore, there's no smallest positive infinitesimal in the hyperreals.

The hyperreals are a field, just as the real numbers are. A field is a mathematical structure in which you can add, subtract, multiply, and divided (except by zero). The rational numbers are a field. The reals are a field. The complex numbers are a field. And the hyperreals are a field.

It's a common misunderstanding that there is a "smallest infinitesimal." In the hyperreals there is not. If you have some positive infinitesimal in the hyperreals. you can just divide it by 2 to get a smaller one ... just as you can in the reals.

The way to think of the hyperreals is to imagine the real number line; and around each real number there is a "cloud" of uncountably many infinitesimal elements. In fact it's called the [url=https://en.wikipedia.org/wiki/Monad_(non-standard_analysis)]halo[/url].

Even in the hyperreals, there is no smallest positive number. Any infinitesimal can be divided by 2 as many times as you like, just as in the real numbers.

In fact subject to technical considerations, everything true about the reals is true about the hyperreals; and vice versa.Of course, "contains an infinitesimal" is true about the hyperreals and not the reals; so there are rules for when it's legal to transfer back and forth. This is the famous [url=https://en.wikipedia.org/wiki/Transfer_principle]transfer principle[/i]. This is the trick that makes nonstandard calculus work. You can take a problem in standard calculus, solve it in the hyperreals, and transfer the result back to the standard world.

If there's a tl;dr, it's that the hyperreals are a lot like the reals except for a halo of infinitesimals around each real number. That's a good way to think of them.

Edited by wtf
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12 hours ago, wtf said:

Bullpucky. You're confusing the hyperreals with finite extensions of the reals, or perhaps the extended reals. Why not take the trouble to find out what the thread's about before posting utterly false information?

!

Moderator Note

The rules require a level of politeness that this post does not reach. Just the second sentence would have been enough.

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8 hours ago, gib65 said:

What happens if you multiply the infinitely small hyperreal number e by the infinitely large hyperreal number R? Do you get a real number?

I am not an expert, so if I say something incorrect, wtf may want to correct it. But i am pretty sure that it only works in one possible way. Namely among the infinitesimals you have to pick a standard one, say call it $$\varepsilon,$$ and likewise among the infinite hyperreals you pick an $$\omega,$$ and then you arrange that multiplication works so that $$\varepsilon \omega = 1,$$ that is to say, they are reciprocals. And in general, if $$a$$ is real and nonzero, then $$a\varepsilon$$ and $$\omega /a$$ would be other possible, and just as good, choices. And you do always get a real number by multiplying an infinitesimal $$a\varepsilon$$ by an infinite $$b\omega$$ when $$a$$ and $$b$$ are real.

What I think is perhaps a little interesting is that once you have made your choices of $$\varepsilon$$ and $$\omega,$$ then because you can multiply all hyperreals, there is an even much smaller infinitesimal (it is still called that, right?) $$\varepsilon^2$$ and en even larger infinite $$\omega^2$$ which you could have picked in place of $$\varepsilon$$ and $$\omega,$$ without anything working any different. Which indicates that there are even more brutal ways of making smaller infinitesimals than just by dividing by $$2$$ or any other large real. The exact converse situation would occur if we were to introduce new hyperreals $$\sqrt{\varepsilon}$$ and $$\sqrt{\omega}$$ which are hugely larger, respectively smaller, than $$\varepsilon$$, and $$\omega$$, themselves.  It would mean creating an extension of our original field, vaguely similar to how we create $$\mathbb{C}$$ as an extension of $$\mathbb{R}$$ by adding $$\sqrt{-1}.$$ Except in the case of hyperreals, it looks to me like the new extension field is pretty much the same as the one we had already.

Edited by taeto
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Quote

Wiki

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

1 + 1 + ⋯ + 1 {\displaystyle 1+1+\cdots +1} (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.[1]

Slightly more formal than what I said, but still defined as an 'extension'.

21 hours ago, studiot said:

A hyper-real number is a symbol appended to a set of  numbers as an 'extension'.

There is a slight issue with the Wiki definition since it defines a hyper real number in terms of a number and also uses the term "greater than".

Perhaps not so formal after all ?

I think taeto did a good concise job of explaining why you don't need a whole universe of new numbers (symbols). +1

As to the distinction between cardinal and ordinal.

10 hours ago, gib65 said:

Note that at 7:55, he explains how only cardinals refer to amounts. Note the statement: "Omega plus one isn't bigger than omega, it just comes after omega."

Thank you for the video. I will watch it when I have time.

Meanwhile here is an older definition of cardinal number which avoids the circular definition of defining a number by using the term number.
(My first definition came from my mathematical dictionary)

Quote

Littlewood

We define the cardinal number of a class A to be the class of all classes similar to A.

You would need to understand by similar he meant bijective.

The order is irrelevant to this definition and the set may not be ordered.

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15 hours ago, uncool said:

...irrational numbers are not infinitesimals. They are real numbers.

Infinitesimals have to continue on forever past the decimal point.  Then that makes them irrational numbers, because they cannot be expressed as a fraction.  That is the definition of an irrational number.

A hyperreal could just be a number that continues on forever without a decimal point or an ordinary irrational number.  If you put that number on the bottom of a fraction it would make it an infinitesimal, because you would be dividing 1 by an infinite number of numbers.  The process would never stop, so you would end up with an infinite number of numbers past the decimal.  Therefore, it would become an infinitesimal or irrational number.

Edited by Conjurer
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10 minutes ago, Conjurer said:

A hyperreal could just be a number that continues on forever without a decimal point or an ordinary irrational number.  If you put that number on the bottom of a fraction it would make it an infinitesimal, because you would be dividing 1 by an infinite number of numbers.

That's not how infinitesimals work.

1/pi is not infinitesimal.

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2 minutes ago, Conjurer said:

Infinitesimals have to continue on forever past the decimal point.  Then that makes them irrational numbers, because they cannot be expressed as a fraction.  That is the definition of an irrational number.

A hyperreal would just be a number that continues on forever without a decimal point.  If you put that number on the bottom of a fraction it would make it an infinitesimal, because you would be dividing 1 by an infinite number of numbers.  The process would never stop, so you would end up with an infinite number of numbers past the decimal.  Therefore, it would become an infinitesimal or irrational number.

To "continue forever past the decimal point" is not quite the same as being irrational. The decimal expansion of the rational number $$1/3$$ is $$.333\ldots$$.

You seem a little too hung up on the representation of numbers. The numbers themselves do not "have decimal points" nor do they "continue forever". These kinds of descriptions only apply to whatever representation we choose, which starting from elementary school happens to be mostly the decimal point representation. But the actual properties of numbers do not depend on the way in which we happen to choose a representation of them. In the decimal notation we represent $$1/3$$ as $$.333\ldots$$ whereas in ternary notation we would represent the exact same number as $$.1$$.

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11 minutes ago, taeto said:

To "continue forever past the decimal point" is not quite the same as being irrational. The decimal expansion of the rational number 1/3 is .333

You seem a little too hung up on the representation of numbers. The numbers themselves do not "have decimal points" nor do they "continue forever".

The way he was saying it made it sound like 31 could be a hyperreal.  When it is not.  I guess 333,333,... could be a hyperreal that is rational.

16 minutes ago, uncool said:

That's not how infinitesimals work.

1/pi is not infinitesimal.

The answer I got from putting it into my calculator is 0.3183098862...

It appears to continue on forever repeating randomly.  It becomes infinitesimally close to a rational number, since each digit after the decimal point is a smaller and smaller description that approaches a rational number.

3 hours ago, studiot said:

Slightly more formal than what I said, but still defined as an 'extension'.

It appears someone has taken the liberty of changing the wiki to be more informal, since you made that response.

Now the wiki says,

The transfer principle, however, doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that

but there is no such number in R. (In other words, *R is not Archimedean.) This is possible because the nonexistence of ω cannot be expressed as a first order statement.

Edited by Conjurer
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7 minutes ago, Conjurer said:

It appears to continue on forever repeating randomly.  It becomes infinitesimally close to a rational number, since each digit after the decimal point is a smaller and smaller description that approaches a rational number.

Actually it does not continue "randomly", since you can write a computer program which will tell you the $$n$$'th digit after the decimal point if you input $$n.$$

And the only real number that $$1/\pi$$ "becomes infinitesimally close to" is the number $$1/\pi$$ itself. All other real numbers have a positive real number distance. For the simple reason that the decimal expansion that you are talking about is precisely the decimal expansion of $$1/\pi$$ and nothing else. Also, it is only the real numbers that have usual decimal expansions, other hyperreals do not.

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19 minutes ago, Conjurer said:

The way he was saying it made it sound like 31 could be a hyperreal.  When it is not.

It is a hyperreal number. It's also a real number. Every real number is a hyperreal number (more formally: there is a natural map embedding the real numbers in the hyperreal numbers).

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3 minutes ago, taeto said:

Actually it does not continue "randomly", since you can write a computer program which will tell you the n 'th digit after the decimal point if you input n.

You can also write a program that determines the calculation of pi to however many places you want.  It is still said that pi continues randomly on forever.  That just means that the sequence of numbers do not repeat the same way forever.

1 minute ago, uncool said:

It is a hyperreal number. It's also a real number. Every real number is a hyperreal number (more formally: there is a natural map embedding the real numbers in the hyperreal numbers).

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

{\displaystyle 1+1+\cdots +1} (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.[1]

You just proved to me that you have no idea what you are talking about.  When you have 1+1+...+1  that means it is an infinite series, and it would have to have an infinite number of digits.  I bolded and highlighted the part where it says the numbers are infinite.  The number 31 doesn't continue on forever or have an infinite number of digits.

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2 minutes ago, Conjurer said:

You can also write a program that determines the calculation of pi to however many places you want.  It is still said that pi continues randomly on forever.  That just means that the sequence of numbers do not repeat the same way forever.

Your version of something being random is very different from everyone else's. If I could write a computer program which predicts with absolute 100% certainty the winning lottery numbers in next weeks lottery, and it would be correct every time, would you still persist in a belief that the lottery numbers are drawn every week completely at random? Most people would say that they are completely predictable.

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11 minutes ago, Conjurer said:

You can also write a program that determines the calculation of pi to however many places you want.  It is still said that pi continues randomly on forever.  That just means that the sequence of numbers do not repeat the same way forever.

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

{\displaystyle 1+1+\cdots +1} (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.[1]

You just proved to me that you have no idea what you are talking about.  When you have 1+1+...+1  that means it is an infinite series, and it would have to have an infinite number of digits.  I bolded and highlighted the part where it says the numbers are infinite.  The number 31 doesn't continue on forever or have an infinite number of digits.

The fact that the set of hyperreal numbers contains infinite numbers doesn't mean it can't contain finite numbers. Read that definition again: the hyperreals are an extension of the real numbers. The set of hyperreal numbers includes the set of real numbers. The statement "Such numbers are infinite" refers specifically to "numbers greater than anything of the form", not to all hyperreal numbers

Edited by uncool
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11 minutes ago, taeto said:

Your version of something being random is very different from everyone else's.

The sheer notion of this completely boggles my mind.  Pie can also be expressed as a fraction of the circumference of a circle divided by it's diameter, but that doesn't make it a rational number.  You couldn't know the exact value of both the circumference and the diameter with 100% accuracy, so you could only approach a slightly better estimate for it by making more accurate measurements of a perfect circle.

Then there exist computer programs that can calculate it to a further degree of places.  That doesn't change the fact of it being irrational.  Everyone I have ever heard teach about this aspect  of math uses that type of lingo.

14 minutes ago, uncool said:

The fact that the set of hyperreal numbers contains infinite numbers doesn't mean it can't contain finite numbers. Read that definition again: the hyperreals are an extension of the real numbers. The set of hyperreal numbers includes the set of real numbers. The statement "Such numbers are infinite" refers specifically to "numbers greater than anything of the form", not to all hyperreal numbers

that contain anything greater in the form

1+1+...+1

31 doesn't contain anything greater in the form of adding an infinite number of 1's.  Therefore, it fails to meet the requirement.  It is trying to deal with infinities that are greater than an infinity of adding an infinite number on single digits or counting to infinity by integers.

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5 minutes ago, Conjurer said:

that contain anything greater in the form

1+1+...+1

31 doesn't contain anything greater in the form of adding an infinite number of 1's.  Therefore, it fails to meet the requirement.

Once again: The fact that the set of hyperreal numbers contains infinite numbers doesn't mean it can't contain finite numbers.

Not all hyperreal numbers are infinite. Infinitude is not a "requirement" for all hyperreal numbers. There are finite hyperreal numbers. Some hyperreal numbers are finite.

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