The basic tenet of advaita is
ब्रह्म सत्यं जगन्मिथ्या जीवो ब्रहैव नापरः - ब्रह्म ज्ञानावलीमाला - २०
Brahma Satyam Jagan Mithya Jivo Brahmaiva Na Aparah" - Brahma Jnanavali mala - 20
The falsity of the world is argued by one sect of Buddhism, where nothing is real. Advaita does not agree with the falsity of the world
Satya (सत्य)= one that is present at all times - only Brahman.
A-satya (असत्य)= one that is not present at any time - son of a barren woman. This is false.
Mithya (मिथ्या)= the one which is not there, but it is experienced. Rope/snake, mirage in desert etc.
anitya means the one which is destroyed or transformed into something else. For example, cloth is burnt to ash and can not return to cloth. But mithya is different, it is a light (Jnana) is shown, the snake gets destroyed and only the rope remains. But here, no trace of snake is left because there was no snake in the first place.
Shankara states in his commentary on Bhagavad gItA 2.16:
sarvatra buddhidvayopalabdheH, sadbuddhiH asadbuddhiH iti |
yadvishhayA buddhiH na vyabhicharati tat.h sat.h, yadvishhayA
vyabhicharati tat.h asat.h|
There are two cognitions everywhere - cognition of the real and
cognition of the unreal. That cognition which does not change is real
(sat) and that which changes is unreal (mithyA).
What does it mean to negate something in the world, say a pot? The real underlying portion of the pot is clay and can NEVER be negated. Only the nama rupa of the pot is negated. Thus, the complete falsity of the world can not be accepted by advaita.
Madhusudana Sarasvati says in advaita siddhi
tAtvikatve .api na-advaitahAnikaratvam.h | na cha tAtvikAbhAva-
pratiyoginaH prapaJNchasya tAtvikApattiH, tAtvikAbhAvapratiyogini
shuktirajatAdau kalpite vyabhichArAt.h |
The negation of the world is non-different from Brahman which is the
substratum of the negation of the world...
For more discussion on advaita siddhi, you can refer to mithyAtve visheShAnumAnam
(14) मिथ्यात्वं, ब्रह्मतुच्छोभयातिरिक्तत्वव्यापकम्,
mithyAtva is present in every object except Brahman and the absolutely
non-existent (tucCha / asat) because a) all objects that are mithyA
have mithyAtva or b) where mithyAtva is absent, this is absent. The
example for such a concomitance is knowability.
Knowability is present in every location except Brahman and the
non-existent. It is also present in every mithyA object (first hetu),
and where mithyAtva is absent, knowability is absent. So both hetu-s
and sAdhya are invariably concomitant.
(17) उभयसिद्धमसद्विलक्षणं मिथ्यात्वासमानाधिकरणधर्मानधिकरणम्,
That which has been accepted by both (the
dvaitin and the advaitin) as different from the non-existent (e.g the
world), is not the locus of any attribute that is not colocated with
mithyAtva, because it is a locus, like the shell silver. Every
attribute in shell-silver is colocated with mithyAtva.
According to Nagarjuna, the character of the phenomenal world is declared to be neither real nor unreal, but logically indeterminable and all phenomena are empty (sunyata) of an inherent self. However, both Dvaita and Advaita does not agree with this.
Mithya is used to explain that it is neither sat (Brahman) or asat (hare's horn or son of barren woman). MithyAtva is present in every object except Brahman and the absolutely non-existent. Since the world is only dependently real, when it is negated altogether, having no reality whatsoever, it is known to be mithyA. The world has thus a dependent reality while Brahman’s is Independent Reality. Without the Reality of Brahman the world would be simply naught. Thus, one has to admit that it is Brahman that appears as the world.
Shankaracharya brings out the ‘asanga’ or unattached nature of Brahman. The rope, even though ‘supports’ the snake, does not inhere in the snake as it has no contact with the illusory snake.
So, the answer to your question remains that the world is mithya. And this mithya can not be classified as sat or asat. The ajnani perceives the world but perceives it wrongly as dual with name-form and does not look at the substratum. It is perceived as sat (Brahman) by a jnani. But the jnani also see the world as name-form but knows it is unreal. It is like a mirage. The mirage appears to both the knower and non-knower. The former knows that there is no water there while the latter thinks the water is there.
In advaita siddhi, Madhusudhana saravati gives five definitions of mithyātva.
False is something that appears and is later negated, the unreal is never an object of experience, the concept of unreal is not possible. Vacaspati of the Bhamati school states that whereas illusion conceals, mithyātva signifies 'concealment'. Padmapada of the Vivarna school adds to the sense of concealment the sense of inexpressibility and says that mithyātva is anirvacaniya as the nature of being different from sat and asat in essence.
Having established that the world is neither sat nor asat, the advaita says the world is mithya.
Now, you are asking whether Mithya itself is mithya. It is like asking whether an illusion is also an illusion. You can, however, ask whether this mithya is sat or asat. But that also can not be answered. Because if mithya is sat, then there are two sats (Mithya and brahman) and if mithya is asat, then the world should be sat.
The critic now asks, Is the illusoriness of the world itself illusory or not? If it is illusory, then the world must be real. If it not illusory, is it one with Brahman or different from it? The former is the stance of Vedanta Desika in vishita advaita, who hold that the illusory world is different from Brahman but inseparably related to it. The alternate is the illusoriness appears in the same locus as where it does not exist as in the case of the mirage where water does not exist at all.
The answer by the advaita is that the mithya is neither real or unreal. The universe is seen and thus it is real but it is sublated so it is unreal. Thus it is anirvacaniyatva. Dvaita says that there is no entity that is unreal and real. Either it is real or unreal. There is nothing that is neither nor both.
The advaita answers it by resorting to two levels of reality. What is pertinent to one level is not applicable to the other level. The illusoriness of the world is illusory. But this does not mean that the world is real. The world is illusory, the illusion is illusory. All this pertains to the jiva’s point of view. It is the ajnani who thinks that the illusion is real. When avidya is removed, the mithya is also removed and the cognized (snake) disappears. The universe of illusion does not exist in Brahman (there was never a snake in the rope) though it appears as superimposed on Brahman. In this sense, it is illusory. However, from the ultimate point of view of the jnani, there is only Brahman. There is no world, creation etc. from the paramarthika view and the creation is itself is ajata. Brahman, without undergoing any change, appears as the universe. The universe depends on Brahman for its being just a snake cannot appear if there is no rope.
For more details on this, please read The Seven Great Untenables: Sapta-vidhā Anupapatti