it shouldn't be too hard to see that (accepting all that) you could just as easily add another one (one the right) to get ω + 2, a second infinity.
You might see where this is going.
So you can keep on doing that ω + n, for any natural number n.
let's just say you do that
an infinite number of numbers bigger than ω
for convenience we'll call that ω2 (for ω times 2 because we kind of have two copies of ω, one next to the first one)
so, is that enough numbers?
and again and again and again
You might see where this is going
add an infinity more and you get:
Look man it's Friday, we're not stopping now.
OK got the pattern: ω3 + n
1 min ago, by
Mitch You might see where this is going
That's infinity squared number of infinities
See, if I were smoking dope this would be more difficult.
You really weren't doing anything important before dinner tonight anyway.
So there's no more infinities, right?
Hey you, there in the back, reading comics, what did you say?
Oh I kinda skipped a few ordinals. They're just the same thing but with a few more details, like (ω^17)+35 + (w^9)*2 + 4, essentially polynomials over ω
oops I made a mistake above.
instead of ω^ω it should have been ω^2
and that was sort of a hint as to how to continue
you can get ω^3, ω^4, etc... up to ω^n, do it a bunch more and you get ω^ω
you can get ω^(ω+1), ω^(ω2), ω^(ω^ω), ω^(ω^(ω^(ω^(ω...))))
Long story short... sure you can keep going, you keep labeling the 'limit' with a new name or notation.
41 mins ago, by
Mitch In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation
ε
=
ω
ε
,
\varepsilon =\omega...
It keeps going on after that, but the notation gets to be a little messier.
By the way, all these ordinals are still equinumerate with the cardinal \aleph _{0}.
There are much bigger ordinals that are also the size of the continuum (therefore much bigger than $\epsilon_0$)