Monad in non-standard analysis

[nomadreid]

Quick two questions:
(a) In the hyperreals, is 0 considered an infinitesimal?
(b) Does a monad include the real number?
I seem to get contradictory answers in the Internet.
Thanks.

 

[Mark44]

(a) In the hyperreals, is 0 considered an infinitesimal?

No. Reconsidering, I'll say yes.

I don't know the answer to your b part. Can you provide more information

 

[nomadreid]

In the Wiki page on monads
https://en.wikipedia.org/wiki/Monad_(nonstandard_analysis)
it defines the monad with respect to a real number x as anything is difference to x is infinitesimal. To me, that would exclude x (since, as you wrote, 0 is not an infinitesimal).
Yet is also says
"the unique real number in the monad of x is called the standard part of x"
Which would seem to imply that r was in the monad.
So what am I reading incorrectly here?
Thanks.

 

[Frabjous]

Check out pages 1 and 2 of https://people.math.wisc.edu/~hkeisler/foundations.pdf

 

[nomadreid]

Thank you, Frabjous. Accordingly, 0 is an infinitesimal, so the monad definition in Wiki works after all .
Mark44 -- you answered that 0 is not an infinitesimal; are there two different definitions to be found?

 

[Mark44]

Mark44 -- you answered that 0 is not an infinitesimal; are there two different definitions to be found?

No. After reading the first few pages of the link that @Frabjous provided, I retract my earlier response. From that source, "the only real infinitesimal is 0."

 

[WWGD]

No. After reading the first few pages of the link that @Frabjous provided, I retract my earlier response. From that source, "the only real infinitesimal is 0."

The only real or Real infinitesimal?

 

[Mark44]

The only real or Real infinitesimal?

The only infinitesimal that happens also to be real. Again, according to the paper/book by Keisler.

 

[WWGD]

The only infinitesimal that happens also to be real. Again, according to the paper/book by Keisler.

Not doubting you, just curious as to what you meant.