How To Use The Word ‘Aspire’ Correctly？?

1. She aspires to greatness. "To" is a preposition. She strives toward greatness. (noun)

3. She aspires to achieving greatness. She strives toward achieving greatness. (gerund)

Both of these are the same. A gerund functions as a noun. Both nouns are object of the preposition, "to." I'd say the prepositional phrase is the direct object of the verb.

2. She aspires to achieve greatness. In this case, I'd say "to achieve" is the infinitive, and (again) acts as direct object of the verb. You could probably argue that this is "to" as preposition, plus the bare infinitive. All three structures act as direct object of the verb "to aspire." What you call "verbs" in your analysis are really "verbals," or non-finite verbs.

The meaning is the same in all three.

As you choose to aspire to different things, you may find that one form works better than another.

May you tell me how to correct

1, People aspire to speak for many poor.(why not use speaking)

2, I can only aspire to speaking Italian as well as you speak English. (why not use speak)

Bye

1, People aspire to speak for many poor.(why not use speaking)

2, I can only aspire to speaking Italian as well as you speak English. (why not use speak)

There's no reason why they can't be interchangeable.

As an aside, "aspire" may be misused in the first sentence. It usually involves a looking forward to achieving something.
The sentence seems to suggest that a certain group of people claim to be speaking for many poor at present.
(Certainly not "people in general.")

It's not a clear sentence either way.

Just out of curiosity I Googled "aspire to", and I did not find any examples of aspire to + v-ing within the first 10 pages of hits. They were all to + noun or to-infinitives.

I conclude that to + v-ing is much less used.

It seems to me, therefore, that the more commonly used form would be

I can only aspire to speak Italian as well as you speak English.

CJ

Hmmm, I'll have to look into that. - A. (curiosity)

It seems like we aspire to a condition as well as to an action.

AvangiIt seems like we aspire to a condition as well as to an action.

We do! We do! Well, I do, anyway.

CJ

Right!
Pardon the dumb comment. Of course the "conditions" would be nouns.

Edit. Good'ole Google!

I ran another ten and gave up! Then I Googled "aspire to *-ing" and got 24.5 mil.
("Aspire to" yields only 16.7 mil. Go figure!)

What am I doing wrong? O I C ! Not all "-ing" words are gerunds! (Eg, "Aspire to something!")
Still, the math doesn't work. (Does it ever?)

Examples:
"But even the most prestigious university can no longer aspire to having departments in every human endeavor ."
. . . . but it seems sometimes to aspire to becoming a diagnostic handbook . . . .
Another related activity is to aspire to filling high-level strategic positions,

AvangiStill, the math doesn't work. (Does it ever?)

No. I'm convinced you need an advanced degree to understand "Google math".

CJ