are there transfinite equivalents to non-integer real numbers? Cantor first envisioned the transfinite ordinals as a kind of 'extension' to the finite integers, where  followed on where ℕ left off and continued the sequence. In this way, we can see them as 'equivalent' to the finite integers in this way.
I was wondering whether there are transfinite numbers (which are useful, in that they are used/have been used in set theory) which are in this sense equivalent to the rest of the reals? Does it make sense to talk about fractional transfinite numbers such as ( + 1/2), or to talk about uncountably many irrational transifinte numbers between  and  +1? If so, are they useful?
 A: As upvoted in the comments, one candidate for what you are looking for would be the "surreal numbers" (a.k.a. "the surreals").
The surreals are a system of "numbers" (which admit ordering, addition, division by nonzero numbers, etc.) that includes a canonical copy of the ordinals, while also including a canonical copy of the reals. Exactly as the question proposes, the surreals contain numbers reasonably named by "$\omega + \frac12$" and satisfy $\omega<\omega + r<\omega +1$ for all reals $r\in(0,1)$. There are actually more surreals between $\omega$ and $\omega+1$ than could fit in any set (they form a "proper class", just like the ordinals do).
A dry but detailed introduction to the surreals can be found in Claus Tøndering's paper "Surreal Numbers – An Introduction". A colorful but less-detailed introduction can be found in Don Knuth's book "Surreal Numbers (How two ex-students turned on to pure mathematics and found total happiness)".
A discussion/justification of the analogy the question raises of the form "naturals:ordinals::reals:surreals" is given in my answer to the MathSE question "Can we embed the ordinal and cardinal number systems into larger, more convenient systems of arithmetic?".
