According to Rickless, what the text of the Parmenides actually gives, as the conclusion of the TMA, is that each form is unlimited (or infinite) in number. The usual interpretations, he claims, go astray and incorrectly take the conclusion to be, rather, that the number of forms is unlimited (or infinite).
The Greek is: kai\ ou)ke/ti dh\ e(\n e(/kasto/n soi tw=n ei)dw=n e)/stai, a)lla\ a)/peira to\ plh=qoj.
But I think the usual interpretations are correct, and the Greek should be understood in the way they presume, not as Rickless would have it. Here’s why.
1. The expression ‘each of the Forms’ has a definite and established meaning in Parmenides: it signifies the Forms that Socrates wishes to postulate, for each of the kinds of things that we naturally distinguish. ‘Each of the Forms’, then, indicates distributively Large, Many, One, Good, Beauty, and so on. Here is a passage that clearly shows that this is so:
Socrates, that is because you are trying to mark off something beautiful, and just, and good, and each one of the Forms (e(n e(/kaston tw=n ei)dw=n), too soon…before you have been properly trained. (135c)
Here are other passages which show the same thing.
And yet, Socrates…the Forms inevitably involved these objections and a host of others besides—if there are those characters for things, and a person is to mark off each form as ‘something itself’ (o(riei=tai/ tij au)to/ ti e(/kaston ei)=doj). (135a)
Yet on the other hand, Socrates,…if someone, having an eye on all the difficulties we have just brought up and others of the same sort, won’t allow that there are Forms for things and won’t mark off a Form for each one (mhde/ ti o(riei=tai ei)=doj e(no\j e(ka/stou), he won’t have anywhere to turn his thoughts…(135)
Because I think that you, Socrates, and anyone else who posits that there is for each thing some being, itself by itself (o(/stij au)th/n tina kaq' au(th\n e(ka/stou ou)si/an ti/qetai ei)=nai), would agree, to being with, that none of these beings is in us. (133c)
I assure you,…that you do not yet, if I may put it so, have an inkling of how great the difficulty is if you are going to posit one form in each case every time you make a distinction among things (ei) e(\n ei)=doj e(/kaston tw=n o)/ntwn a)ei/ ti a)forizo/menoj qh/seij). (133b)
2. Relying on this expression, Parmenides sets up the TMA in this way. Socrates wants to say that each of his Forms is one, because he thinks that only one Form should be postulated (i.e only one entity which exists separately from particulars and is ‘itself by itself’) for each kind of thing that we naturally distinguish (oi)=mai/ se e)k tou= toiou=de e(\n e(/kaston ei)=doj oi)/esqai ei)=nai: o(/tan po/ll' a)/tta mega/la soi do/ch| ei)=nai, mi/a tij i)/swj dokei= i)de/a h( au)th\ ei)=nai e)pi\ pa/nta i)do/nti, o(/qen e(\n to\ me/ga h(gh=| ei)=nai). But the TMA shows that there are an infinite number of such things. If it was true to say that “Large is one”, on the grounds that we should postulate just one Form for the large things that we naturally distinguish, then it is true to say that “Large is infinite”, if, as the TMA shows, we should postulate an infinite number of Forms for the large things we naturally distinguish. So ‘each of your Forms’, that is, not simply Large, but also Many, One, Good, Beauty, etc. turns out to be infinite in number, not one.
3. Parmenides expresses this by saying that each of the the Forms will ‘no longer’ be one, but ‘will be’ infinite in number, because he is thinking of the infinite regress of the TMA as the result of a process of postulation, involving an intellectual viewing, as is clear from his language (e)a\n w(sau/twj th=| yuxh=| e)pi\ pa/nta i)/dh|j).
4. It is important, I think, that Parmenides says ‘each of your Forms' (e(/kasto/n soi tw=n ei)dw=n) rather than simply ‘each of the Forms’. This suggests precisely the opposite of what Rickless thinks. Rickless thinks that ‘your’ functions to indicate each of the Forms which Socrates has agreed to exist, in agreeing to the steps of the TMA. But at 131c7 Parmenides uses, rather, ‘our Form’, when he wants to say something about the Forms that is the result of his ad hominem argument against Socrates (to\ e(\n ei)=doj h(mi=n th=| a)lhqei/a| meri/zesqai). If, in the TMA, Parmenides wanted to say that the each member of the infinite hierarchy was infinite in number, presumably he would again have said that ‘each of our Forms is infinitely multiple’. He says, in contrast, ‘each of your Forms’, precisely because he wants to refer to the theory that has been marked out and identified outside the context of the TMA.