Allow me to pre-face my reply with this quote, from myself, from the post you replied to:

“If you would argue that near-infinite = infinite, you’ve missed the definition.
If you would argue that no number is near-infinite, then you’ve missed the definition as well, since near-infinite does not apply to any number you can name.”

“Can you ???”

Yes. Would you like some examples?

“I am an electronics engineer, in my line of work, and throught my career I have used quite a bit of math, (a HUGE amount).
Therefore, my skills are in practicle or APPLIED math, so yes I am aware of thigs such as set theory, I understand infinite I think more clearly than you claim you do.”

The more I read of your posts, the more I doubt your qualifications. The proof, they say, is in the pudding, and I’m quite willing to show you my pudding.

“I know all about real and complex and imaginary number, vectors, ‘rate of change’, integration, and differentiation.”

Congratulations, you can pass a 1st year math course.

“Here is a question for you:

How many real numbers exist between the numbers 1 and 2 ??????”

The answer is Aleph 1; otherwise known as “unaccountably many”.

“Would the set of real numbers that exist between 1 and 2 be greater than or smaller than or the same size as the set of real numbers between 3 and 4 ???”

Exactly the same size; Proof: F(x):f->g = x + 2

“how many sets of real numbers do you think exist ?”
I’m not sure what you intend to ask; your question is poorly defined in several ways.
To answer the most useful question, the powerset of the set of real numbers has cardinality aleph 2.

“would those sets exceed the number of real integers ?”
See problem with previous question.
Would the size of the powerset of real numbers exceed the size of the set of real numbers? Yes. Alpeh 2 > Aleph 1

“are the integers 1, 2, 3 or 4 ‘near infinite’ OR
do they form a part of an infinite set ? or infinite sets???”
No nameable number can be near-infinite, since a near-infinite number is one which can only be described as a range, with infinity as the upper bound.
Would you like a strict definition?

“is the result of 10 divided 3 and infinite number ?”
no.

“is ZERO (0) near infinite, or is it infinite, is it even a number ???”
Zero is clearly a number, and is not near-infinite or infinite.

“can you conceive that anything that is “near infinite” is itself infinite?”
Can I imagine something which contradicts it’s own definition?

“is there a special math symbol for ‘near infinite’ ?”
Is there a special symbol for ‘arbitrary’? How about ‘random’? What if I want to differentiate between the different types of random?

“Have you heard your professors use that term or seen them derive a ‘near infinite’ proof ?”
In computer science, we do look at problems which require near-infinite time, how to recognize them, and how create algorithms which minimize the expected time & memory. They also take advantage of this subject to introduce us to how to manage memory & CPU usage properly. IE: to make a maximum that the program will not exceed.

“is it a countable or uncountable infinite set of numbers, either real, integer or imaginary?”
Is what? Near-infinity?
Besides all the other stupidity around the question, this implies that you think that infinity is a set of numbers?
As it is, I cannot discern what this question is intended to mean, or even make a best guess.

“Would not a series of sets created by an infinite group (monkeys) over infinite time also be infinite”
∞ * ∞ = ∞

“Would an infinite number of combinations of letters created over infinite time result in an infinite number of combinations, therefore all combinations would be created instantly.”
Would somethign created over an INFINITE amount of time be created in an INFINITESIMAL amount of time?
errrr . . . no.

But you probably meant to ask, that if we had infinite monkeys creating combinations, would all combinations be created infinitesimally? The answer is still no! All combinations would be created in the time it takes a single monkey to make a single combination.

“do you use ‘estimations’ like “near” and “far” or “big” or “small”, or “close too” when you are working on a pure math problem ?”

Ever hear of ‘neighbourhoods’? Yeah, we do.
And there’s this whole area of pure & applied math, called Statistics, which hinges completely on ‘good enough’.
Estimations, as it happens, has nothing to do with it.

“How often have you had a “near infinite” result ?”

When I’m trying to find out how long a problem will need before it can be solved by an algorithm? Often enough to need to know how to recognize and deal with it.

“My proof that the Monkeys are in fact, a near-infinite time, is this:
Give me a single number of iterations, after which we can guarantee that your group of monkeys have written Shakespeare.

What, that is your proof !!!

Define “a single number of iterations”??”

An iteration = the amount of time it takes a monkey to punch a single key.

The single number then, is the number of keys each monkey will have pressed when they have written shakespeare.
So, give me a number of keys, such that if the monkeys have pressed that number of keys each, then one of them will have written shakespeare.

“So if I give you a single number, like 10 trillion or 10^trillion gogols, is that number “near infinity”.”

No, they would not be. After 100^^^googleplex,(knuth arrow notation), there will still be groups of monkeys who have not finished, so those smaller numbers also do not fulfill the question.

“I do not think you have grasped the concept of infinity at all !!!.”

I can form well-defined questions about it at the very least, which puts me head and shoulders above yourself.

“A VERY big number, no matter how big, is not and will NEVER BE ‘near infinity’.”

I agree, that conflicts with the definition of near-infinity.

“And you call yourself a ‘would be’ math grad !!!!

Now let me take a wild guess, you are the product of the US education system right ??”

I never called myself that; I said I had one course left to graduate. For all you know, I’ve tried that course several times, and failed each time. As it is, I’ve never tried it, since I decided that I wasn’t ready for the masters level, and that I’d rather be studying a field related to complexity theory. Thus computer science.

Also, I’m German-Canadian.

“1 is as “near infinite” as 100,000,000,000,000, billion, trillion, trillion and 1

no matter what the number is, you have to go an infinite way to be infinite, therefore no finite number is ‘near infinite’ every number (including infinity itself) is an infinite distance away from infinity… (and every other number !!).”

See the quote at the top of the reply, would you?

“Maybe that is the course you have not done yet !”

The course I haven’t done yet is General Topology.

“also set theory “infinity” and ‘normal infinity’ are (as you should know two different entities, which is why you are so tied up with ordinal’s and cardinal’s, and “transfinite numbers”.”

Depends on what you mean by ‘normal’ infinity. Definition, please?

“but when you look into it, as you know (having claims you have studied this) I do not see the term “near infinite” in ANY texts.”

*pulls “Data Structures & Algorithms: Edition 5” from the shelf*
Page, 488, on bogosort:
” . . . this algorithm will run for near-infinity, but has an expected value that scales with N!. If the infinite universes theorem of Quantum Physics is true, then the algorithm can take advantage of the infinite universes by running once, and then destroying any universe in which the list is not sorted. This is known as the quantum bogosort, and the result will be a sorting algorithm that scales linearly, even after using infinite computational power, so that is clearly the lower bound for sorting efficiency of an unknown list of known size.”

Bogo-sort:
http://catb.org/jargon/html/B/bogo-sort.html

“My proof that the Monkeys are in fact, a near-infinite time, is this:
Give me a single number of iterations, after which we can guarantee that your group of monkeys have written Shakespeare.

How do you propose to ‘prove’ that ‘single number’ no matter how large is “near infinity” ??

or, DISPROVE, that any single number is not an infinite distance away from infinity.”

The point, which I’ll say again for your benefit, is that no single number will fulfill the conditions set forward in my question.

Now, let’s see if you posted something new while I was typing this up.