I would have expected "lehetséges" and "lehetségetlen" to be the opposites. Wouldn't "lehetetlen" mean (literally) "unmaybe"? I guess in that case, the "ség" got dropped for some reason, although I don't get the impression that Hungarian speakers worry about words getting too long. :)
I'll explain what I can!
-hat / -het is a suffix that makes what you could call a "verb of possibility." Írhat = "can write", hívhat "can call", énekelhet "can sing" and so on. This generally has to do with physical, logical, or logistical possibily; it's not about knowing how and it's not about permission or being allowed to do something.
Some common verbs have slightly irregular -hat / -het forms that come from their slightly irregular infinitives. Among those are ehet "can eat", ihat "can drink"; mehet "can go" from the infinitives enni, inni, and menni respectively.
And lenni (to be) gives us lehet. So lehet is a verb meaning "can be" or "is possible."
The endings -hatatlan / -hetetlen added to verbs form adjectives that describe the impossibility of doing something. They often translate into "in-...-able" or "un-...-able" words in English.
ehetetlen - inedible (unable to eaten)
olvashatatlan - unreadable
láthatatlan - "unseeable" / invisible
hihetetlen - unbelievable (from hinni to believe)
...so lenni gives us lehetetlen - "un - be - able" or "impossible", more or less directly from the verb.
The verb also gives rise to the abstract noun lehetőség which means "possibility." (Érdekes lehetőséget látok itt - I see an interesting possibility here.) There's an intermediate step involving the adjective lehető which I think is not so common except in a certain construction, but -ság / ség need to go on an adjective.
Anyway a noun with the +s ending that I mentioned a few times in the birthday song lyrics the other day gives rise to an adjective. So lehetőség + -s (basically) gives us lehetséges the adjective "possibility-ish" or in normal English, possible. You can continue chasing your tail with these things for a nearly indefinite time, adding suffixes to make new forms but at some point all you're doing is making overly complicated versions of things you had built more simply at an earlier stage.
lenni - to be (v.)
lehet - can be / is possible (v.)
lehetetlen - "unable to be" / impossible (adj.)
lehetőség - possibility (n.)
lehetséges - "possiblity-ish" / possible (adj.)
Now it is also true that you can add "+tlan/tlen" to a noun to get a new sort of negative adjective. This usually means "without" or "-less."
Remény (hope) gives reménytelen (hopeless)
só (salt) gives sótlan (unsalted / saltless) and so on. You can have a lot of fun with this.
Anyway if you apply this to the abstract noun lehetőség you could get lehetségtelen "possibility-less" which is feasible, but the simpler derivation lehetetlen is the form that's used most. There might be a shade of difference between them, but you could probably go a long time without ever encountering lehetségtelen. And there does appear to be a difference between that and lehetőségtelen which is more like "devoid of possibilites."
This is why we love this language.
Yes, lehet has personal forms - just think of it as "can be". Szombaton sajnos nem lehetek veletek - "On Saturday, unfortunately, I can't be with you." Or Te is ott lehetsz a színpadon! - You can be there on the stage too!
Other verbs with the -hat/-het endings also conjugate totally normally and express "can do something."
Mondhatsz, amit akarsz. "You can say what you want."
Beléphetek? "Can I step in?" Megcsinálhatjuk - We can do it.
It's a neat construction but there are a variety of ways of expressing "can", possibility, and permission in Hungarian so -hat/-het verbs aren't always the right solution.
(By the way, better wait for the stamp of approval from a native speaker before you take everything I've said up above as gospel. I wouldn't be surprised if there are some subtleties or inaccuracies in all of that.)
Here is my stamp of approval, with just a tiny correction. :)
You mentioned this in your first post: "it's not about permission or being allowed to do something." But here, in your second post, you yourself also provide examples that show permission or being allowed to do something: "Can I step in?". Or, rather, may I? Children ask for permission like this: "Aaaanyuuuu, lemehetek játszani?" May I go down to play?
The other little thing is "lehetségtelen". I would say there is no such word. :) Rather, "lehetőség nélküli". "Nélkül" means "without", and it is another postposition. The "-i" ending makes it an adjective, just like with many other words. (For example, when talking about nationalities.)
And the "-ság"/"-ség" suffix creates a noun from other words. I can compare it to the "-ness" and "-ship" suffixes in English:
friend-friendship - barát-barátság
happy-happiness - boldog-boldogság
Now, this one is a weird exception:
mother-mothership - anya-anyahajó - and not anyaság
(that was a joke)
On cutting some suffixes when negating. Yes, Hungarian does that, more so than English. English likes sometimes to just stick an "un-" or something like that to the front and be done with it:
inhabitable - uninhabitable
faithful - unfaithful
(Sometimes the members of these pairs even mean the same thing:
valuable - invaluable
flammable - inflammable)
Hungarian modifies the tail end, cutting out some stuff:
lakható - lakhatatlan
hű(séges) - hűtlen
lehető/lehetséges - lehetetlen
There are variations of the "-tlan" ending, it can be atlan/etlen/tlan/tlen/talan/telen. I'm sure there is some rule, most importantly the last sound of the root word, but some of them may need to be memorized on a case by case basis.
So, back to the starting post: it would be "lehetséges" and "lehetségTELEN", but this latter word does not really exist, we have "lehetetlen" instead.
And there is a similar word: "tehetséges" - talented. This is interesting:
Tenni - to do something (that is one meaning, another is to put)
tehet - can/may/is able to do
tehetség - talent, the ability to do great things
And now comes the interesting stuff:
tehetséges - talented
tehetségtelen - not talented (untalented? :) )
tehetetlen - helpless/powerless/incapable
So, both negated versions (with and without "-ség") exist and they mean different things. Not so with "lehetséges" where "lehetségtelen" is simply not a real word.
I thought you'd never get here (RyagonIV)! :)
Time to throw away your grammar book. :) Or just add some notes on exceptions. If these are even exceptions:
"Páratlan" - without a pair, odd (as a number)
"Feneketlen" - bottomless
"Fejetlen" - headless
"Időtlen" - timeless
and probably many more...
None of these are verbs.
So, while the first part, about the verbs, seems to stand, the case of the nouns is no way that simple.
Update - just a few more of these interesting word pairs:
"Jellemes" - "jellemtelen"
(With/without a good moral character)
"Kellemes" - "kellemetlen"
(Pleasant - unpleasant)
And my favorite:
"Város" - "váratlan" (city - unexpected).
This is, of course, a fake pair. "Vár" is both a verb ("wait") and a noun ("castle"). The noun "város" was probably derived from the noun "vár", meaning "with a castle").
One more observation: there is a difference on the positive side of these pairs between nouns and verbs. While nouns go with "-(V)s" (a vowel plus "s"), verbs get the "-t"/"-tt" ending (with possible buffer vowel), similar to the past tense suffix. This kind of derivative of a verb is called "befejezett melléknévi igenév" (for grammar book writers), and it is probably the "past participle" in English (when it is used as an adjective).
"A várt eredmény" - "the expected result".
But this is not that simple, as I just realize. It looks like there are two cases. With the root verb, it is what I just said: the "-t"/"-tt" ending:
"olvas" - read:
"olvas-o-tt" - "olvas-atlan"
With the "-hat"/"-het" ending added, the positive side will get an "-ó"/"-ő" ending:
"olvas-hat-ó" - "olvas-hat-atlan".
This type of word ("olvasható"), is a "folyamatos melléknévi igenév" - most probably the "present participle" in English (when it is used as an adjective).
One more note: while, in theory, we can create these forms from any verb, many of them do not actually exist, that is, we never use them. And sometimes, some of these forms take up a special meaning, and come to mean something else. They become idiomatic. It is possible because there are more than one ways of creating the negative form. As in "lehetetlen", "nem lehetséges", etc. I guess the same thing can be said for English, as well. For example:
"látható" - visible
"láthatatlan" - invisible
"nem látható" - not visible
"Invisible" and "not visible" are not necessarily the same thing.
"olvasható" - readable
"olvashatatlan" - unreadable
"nem olvasható" - not readable, can not be read
Sometimes, one form is preferred over the other. For example:
"várható" - "nem várható"
"can be expected" - "can not be expected"
We can create the form:
"várhatatlan", but it is not, or just in certain situations, used.
And then there are "traps", where you have to really think. For example, I have this verb:
"kímél" - it means something like "preserve", "keep in good condition", etc. For example, you can say this of your car: "megkímélt" - meaning slightly used, like new, well preserved.
So, let's see:
"olvasott" - "olvasatlan"
Following the pattern:
"(meg)kímélt" - "kíméletlen".
"Kíméletlen" means "merciless". And it does not come directly from "kímél". No. It comes from a noun that comes from the verb "kímél":
"Kímél" - preserve (a verb)
"Kímélet" - mercy, care (a noun)
"Kíméletes" - merciful, careful
"Kíméletlen" - merciless
And if we want to negate the word "megkímélt", we have to use another form: "nem megkímélt". Or use another word: "használt" (used), "kopott" (worn, faded), "lestrapált" (really worn down), etc.
And so on. There are most probably many many examples like this one.
So, as a suggestion, while it is definitely a good idea to learn the technique of creating these derivative words, and learn their meanings, watch out for which ones are actually used by natives, and in what sense. Verify the actual meaning if something seems to not make sense.