Exploring the Properties of Hyperreal Numbers

[Phrak]

The hyperreals consists of elements having the property that

h > 0
h < 1/2
h < 1/3
h < 1/4
...
and their negatives.

I'm curious to know if this continues within the definition of the hyperreals. That is, for all h,

g > 0
g < h/2
g < h/3
g < h/4
...

 

[g_edgar]

The original hyperreals will still work for that. For example, if $h$ is infinitesimal (that is, $h<1/n$ for all positive integers $n$ but $h > 0$), then $g = h^2$ satisfies $g < h/n$ for all positive integers $n$ and $g > 0$.

 

[Phrak]

I think you're telling me that numbers that satisfy g and h are all hyperreal numbers, right.

You know, I thought I had a ranking system worked out for this--magnitude classes of hyperreals: g⁰ would be a real number, then g¹, g²... Now, I'm not so sure I haven't used a circular argument. I think I need something better than wikipedia...

 

[g_edgar]

There is also $\sqrt{g}$ and $g |\log g|$ and many more...

 

[Hurkyl]

I think I need something better than wikipedia...

Have you read Keisler's calculus text?

 

[Phrak]

Thanks for the reference! I've just now completed the relevant sections in Ch1 (1.4-1.6) concerning the algebra. The calculus is scattered throughout the text, so more difficult to pick-up.

The scheme I've been developing appears to be equivalent to hyperreal numbers--oh well, that's OK. I simply introduced an additional structure so that indeterminate forms such as

$$e \cdot H$$

are determined by the structure, where H is an infinite, and e is an infinitessimal.
(btw, recall that indeterminate means that the number could be a) finite and real, b) finite and infinitessimal, c) infinite.

 

[g_edgar]

he scheme I've been developing appears to be equivalent to hyperreal numbers.

Maybe you would be interested in some of the other systems ... ordered fields containing infinite and infinitesimal elements ...

https://www.amazon.com/dp/0201038129/?tag=pfamazon01-20

http://www.math.ohio-state.edu/~edgar/preprints/trans_begin/"

 

[Phrak]

Very nice, g_edgar. I like your writing style, as well. I've bookmarked it for reading.

I didn't understand this, "There is also $\sqrt{g}$ and $g |\log g|$ and many more..."

Rather than talk around things, what I have done is set up a hiarchy of infinitessimals, and infinities by defining a unit infinitessimal. I believe I am dealing with hyperreal numbers with the addition of subscripts, to indicate magnitude scales:

Real numbers have no subscripts (or a subscript of zero). There is an infinitessimal element that acts similarily to 1 that is represented as 1_-1.

1/1_-1 = 1₊₁ defines infinite unity.

1_-1² = 1_-2

In general 1_-1ⁿ = 1_-n

Multiplication: for c = a b, c_p = a_m b_n , where p = n + m

Additive identity: 0_n = 0_m

Addition: a_n + b_n = c_n and (... + a_-2 + a_-1 + a₀ + a₁ + a₂ + ...) + (... + b_-2 + b_-1 + b₀ + b₁ + b₂ + ...) = (... + c_-2 + c_-1 + c₀ + c₁ + c₂ + ...)

a_n > b_m for all a and b, where n > m amd a ≠ 0

I could add more, but I'm sure you get the idea.​

Although, √1_-1 could defeat this scheme or require irregular forms with funny subscipts. a_n = √(1_-1), n = ?

Were you suggesting there is more than one way to define a scaling hierarchy?

 

[g_edgar]

Were you suggesting there is more than one way to define a scaling hierarchy?

No, actually I was mistakenly assuming that by "hyperreals" you meant the already-existing system with that name (also called nonstandard analysis) ... http://mathforum.org/dr.math/faq/analysis_hyperreals.html", for example.

 

[Hurkyl]

Rather than talk around things, what I have done is set up a hiarchy of infinitessimals,

Isn't that just the real function field (i.e. the fraction field of the ring of real polynomials) along with the ordering that makes the indeterminate variable infinite?

i.e. the thing you call 1_-1 is just the rational function x^-1 (where x is the indeterminate)

Other interesting fields are the field of real formal Laurent series, and the field of real Puiseaux series.



But if you want to define something that has the same analytical behavior as the reals, then a nonstandard model of real analysis is your only option.

 

[Phrak]

According to Keisler's calculus text, the square root of an infinitesimal is still an infinitesimal. I was afraid it would lie half-way in-between reals and infinitesimals.

Isn't that just the real function field (i.e. the fraction field of the ring of real polynomials) along with the ordering that makes the indeterminate variable infinite?

i.e. the thing you call 1_-1 is just the rational function x^-1 (where x is the indeterminate)

With 1_-1 replaced with the symbol 'u', we can dispense with the cumbersome subscripts for infinitesimals infinities.

I'm afraid I don't know enough mathematics to follow you quit, but you bring up something I hadn't noticed. With this scaling scheme, a hyperreal, a* can be expressed as

a* = ... + a_-2 u^-2 + a_-1 u^-1 + a₀ u + a₁ u¹ + a₁ u² + ... ,

where the subscripts resume their usual role of distinguishing variables, and the a's are real valued. Other than the a's being real valued rather than quotients, is this what you were talking about?

But if you want to define something that has the same analytical behavior as the reals, then a nonstandard model of real analysis is your only option.

that seems to be the case, though comming at it from a different direction, I'm trying to add to it. After scanning Keisler I didn't find any referral to infinitesimals with numerical values. That is, where are the equivalent counting numbers 1, 2, 3, ... in the hyperreal infinitesimals?

More, I wonder if u^1/u can stand in for the identity element of addition without inconsistency.