Hyperreal

What exactly is the difference between a monad and a galaxy of a hyperreal number? How do I determine the monad and galaxy of a given hyperreal number? I know, for example, what the monad and galaxy of 0 (the only real hyperreal) is, but I don't know how it is reached. Help please!

Yes.

Yes to what?

While we're at it, what does the following mean?:

Corollary

1.4. Any two galaxies are either equal or disjoint.

Proof.

For each x 2 R¤, the galaxy of x is the coset of x

modulo galaxy(0),

galaxy(x) = fx + a: a 2 galaxy(0)

g:

Theorem

1.5. The set monad(0)

of infnitesimal elements is a subring of

R

¤ and an ideal in galaxy(0)

[copied and pasted from Foundations, pg. 13]

Continuing in my desperate attempt to get an answer, I am now making a post that only serves to put this forum towards the top in hopes of getting more views.

My response to your inquiry, (the other much more intelligent, knowledgeable and relevant responses notwithstanding), is, as I said:

'Yes.'

Not at all familiar with this stuff, but the terminology makes it look interesting! I do know that the hyperreals are a subfield of the surreal numbers, the largest ordered field.

Thank you very much to CoachTom. I knew you were cool, but I never knew you knew this stuff! The "goasts of departed quantities" sounds vaguely familiar. Did you find this from a book?

More important technically, what here is "ln" and "Li". The only context I know ln is just as the natural log, and for Li Gauss and the primes. What are these here?

I think I read that quote in Where Mathematics Comes From. I'm reading Foundations right now. Do you know about the ln/Li thing?

Thanks for the clarification!