If you define omnipotent as, "The power to do anything." then you aren't proving that God is impossible, you are proving that your word, "omnipotent" is impossible.

We can define words however we like. But they only have meaning in reality if they can apply to reality. One thing we can know for certain is that a contradiction is impossible. If you define omnipotent as, "The most power a being in this universe can have", then you're good.

Some may state, "This limits God!" No, it makes God a real possibility. Compared a a human, an "omnipotent being" might have limits, but it would still be greater than us.

Also, I think its quite obvious God has limits. God tells human beings to act a particular way. Why not just design the human beings to act that way? The simple solution is, that's the limit of designing a human being. A human has the capability to act in a way God would like, but God can't force the human being to act the way God would like. That's a pretty obvious limit that few would argue.

that is logical, but quite not what major religious sources would agree with

The idea of omnipotence is a joke.

doesn't this also entail a limit on god? — Augustusea

And even more, if omnipotent, then not good, because to be good is to operate under constraint, the constraint to be good. But all roads lead back to the words themselves, which means that the whole thing needs to be returned to its cage and a blanket thrown over it. Because out in the open it's a meaningless waste of time.

As to "most religions" this both directly and indirectly figures in the history of Christianity.

I don't think the definition of "everything" can include the illogical. That would be like using the lack of a definition to create a new definition. In order to understand an idea, that idea must have some sort of logic to it. Illogical things are nonexistent to the human mind. Logic is the only language our brains can comprehend.

↪Philosophim that is logical, but quite not what major religious sources would agree with — Augustusea

I suppose this can be the case. I think this is an indicator of those who are willing to bring logic to their religion, and those who are not.

Say you define omnipotent as, "Having unlimited power" and omniscient as, "All knowing." You believe God is omnipotent and omniscient. Limitations would cease to exist as the definition of omnipotent implies, and logical vs. illogical in God's eyes would also cease to exist as he is also omniscient. What may seem impossible to us, anything illogical, is most definitely logical to God. To put this into an argument form,

1. If a being is omnipotent and omniscient using the definitions above, then that being knows "everything" and the "illogical" is logical to them.
2. God is omnipotent and omniscient.
3. Therefore, "everything", including the illogical, is logical to God.

It really matters, in this case, whose perspective we're looking at both "everything" and "the illogical" through. God's "everything" includes what we may view as illogical.

As for where you say,

Illogical things are nonexistent to the human mind.

the example of the square that is also a circle is something that is commonly thought to be illogical, but still exists in the human mind.

The word "illogical" does not refer to things which are difficult to understand. It refers to situations where reasoning is missing. For example, the following statement is illogical:

2 + 3 = 10

Why would two and three have a sum of ten? The statement has a logical disconnect between the beginning and the end result. It fails to coherently illustrate the use of arithmetic.

One could try using language and storytelling to distract the reader's attention away from the logical disconnect:

When God thinks about two plus three, he sees how it can become ten.

The preceding sentence is linguistically correct. But the correct use of language does nothing to improve the underlying logic. The sentence also succeeds at telling a story coherently. The main character in the story is God. What does he do? He reflects on how three can be added to two. Then what happens next? He realizes that two and three is ten. But stretching out the original (2 + 3 = 10) statement into a story still does nothing to explain how 2 + 3 becomes 10. The logic is still missing.

You are right that limitations about knowledge would cease to exist when one is omnipotent and omniscient. However, having a perfectly complete set of knowledge does not push illogical statements into the realm of the logical. On the contrary, an all-knowing being would possess the knowledge that 2 + 3 = 10 is an illogical statement. If they were to think that two plus three really does equal ten, then you would probably start to question how much knowledge they possess about basic mathematical operations. Isn't it correct to say that 2 + 3 = 10 is incorrect math? How wouldn't God know that?

Why can't two plus three also equal ten? Because it already equals five. Once the statement 2 + 3 = 10 is allowed to become true, then you have abandoned the rules of mathematical addition. You haven't really made the illogical statement logical, you've only redefined "logical" to mean something else entirely.

To clarify, the argument I’m making is through the perspective of God, an omnipotent and omniscient being, alone, and is excluding the human perspective as humans are neither omnipotent or omniscient.

Say I use your definition of illogical, “situations where reasoning is missing.” This definition doesn’t describe what kind of reasoning has to accommodate an illogical situation, it only suggests that there is no explanation. Now, if something being logical refers to situations where reasoning is present, a few things come to mind.

Think of the story of The Feeding of the 5000, linked here if needed. Jesus, through God’s power, turns five loaves of bread and two fish, which could only feed a few, into food plentiful enough for 5000. In this story, if we bring it to the basics, it possesses a statement such as:

5 + 2 + God = 5000

God has no numerical value to our understanding and can be simplified to 5 + 2 = 5000

As you suggest,

For example, the following statement is illogical:

2 + 3 = 10 — Bird-Up

We can follow the parallels between the two statements and determine that the statement 5 + 2 = 5000 is also illogical, but is supported by reasoning as God made it happen. This is quite contradictory to the definition of illogical as something cannot be explained by reasoning and be illogical at the same time.

If we go back to the statement I started with, "To clarify, ..." I suggest that God is omnipotent and omniscient while humans are neither. Because humans are neither, we can never truly admit that what we know, for example: 2 + 2 = 4, is the end all. There is still more to be known.

If you want to get technical, yes, the apparently-illogical may exist as "logic" outside of our known logic. As humans we cannot (for certain) distinguish between unknown situations and impossible situations. Regarding religion, however, it does raise the question of an all-knowing being expecting humans to know things outside of their limitations of understanding.

If human logic does not provide an avenue for 5 to become 5,000, then should the creator expect his humans to believe this outcome? I do not think we can even suspect possibilities for which we see no logic. If we cannot see ultraviolet light with our human eyes, then should we suspect that ultraviolet light resembles a certain color? Which color would that be? Or are we helpless to know that for which we are blind to?