According to the texts there are, when it comes to successive cessations, four radiances (jhānā) and five bases (āyatanā). A popular take is to see four or five of these bases as "formless jhānā". But has this got any merit?
So, what is the difference between a radiance and a base?
The texts don't seem to explain, this might be the case when things are self explanatory. So let's look at it that way. And let's aim for their difference as the texts do not talk about the bases as (arūpa) jhānā.
The dictionary says:
Jhāna (nt.) [from jhāyati] conflagration, fire.
To almsmen was the advise to do jhāna. And there is the 1st, 2nd, 3rd and 4th one. So fire or conflagrate do grammatically not quite work but let's keep the connection with fire in mind. Outside the texts is it is said that it is about the burning is of 'defilements' and let's look into this first.
Greed, hatred and delusion are the three unwholesome roots. A feeling of happiness has passion dormant, a feeling of suffering has friction dormant and a feeling of neither happiness nor suffering has not knowing dormant, thus levels of greed, hatred and delusion are dormant. But the thing is, not all feelings must have them. And which ones have not? Those in the radiances (MN 44). Thus before entering a jhāna have these obstructions already been dealt with. Gone is absent it doesn't mean uprooted, but they can here become barren, which goes for the arahant (AN 9.36). It is then not that radiances in themselves burn the roots away, and it is not self explanatory as the bases carry this same feature yet are not called jhāna. Thus was the burning of the 'defilements' here not helpful. So let's look for a difference.
Āyatanā means base, and a base is a base for touch. Just as with the hexad base. These bases here can be gone through sequentially. These radiances here can be gone through sequentially. Both through engagement with perception. Both have a certain perception falling away. But unlike these bases these radiances also have something else falling away. For example when engaged in the fourth radiance inhaling and exhaling have stopped. So here is a difference, let's see if it helps to point them out as being different things.
Before the perception of boundless air there is a perception of form and from that point on the perceptions are without form. But is there form or isn't there? Suppose I want to read a certain book which can only be read in the library, then I must enter the library. After some reading I close the book and leave the story behind, which says something of what I am perceiving. I did not physically enter and exit the book and I am still at the library too, but the story is then not on my mind. But I could say that by reading the book I left the library, which just means I got into the story. If I take another book but it is not of my interest at all, I have not left the library and not entered the book. So from a first person point of view, when fully emerged in the story, is there then a book? and is there then a library? Thus talking about leaving one perception for another doesn't justify them to be similar things (library, book, story).
With the engagement in the forth radiance the perception of form falls away; thus are radiances with form rūpa. Arūpa means without form. Air (ākāsa) can be seen as the space above the ground so to speak, like sky, which then have no form here as it is a matter of perception. And based on what is this perception? Based on a view from earth; thus from form. So with leaving form boundless air becomes the base. And thus have we here something similar going on as with library, book and story. And therefore can a base not be a radiance, as arūpa jhāna (without form yet with form) is a contradiction.
And as for the connection with fire?
It seems that in ancient times fire was seen as having to to with clean, purification, and symbolizing life. So perhaps could jhāna back then be more easily understood. Maybe we can say that without lust there is just pure form as media to reflect from; to shine or radiate from.