A concept in which an infinite force is also limited

Edit (complete rewrite):

OK so I'm completely rephrasing the question.

Let's begin with declaring the concept in hand - A is an infinite, willing, creation force. A "wants" to create all the natural numbers. I'd like here to use the concept of ordinal numbers suggested in the comments, just to deny its use here - A is one complete set, it cannot "move" to a different set (by saying this I admit that I am not sure about the validness of this sentence, and I have zero experience with the set theory, only general guidelines).

Just for clarification first: the infinity of A is it's "will" to create, not the fact that it "wants" to create exactly the idea of all numbers. But, it cannot "change" its will.

Now, let's assume A was able to create all natural numbers (this, might be the invalid assumption I thought of at first. But, as I said, I think this won't exactly be assuming A reached what it "cannot" [according to the definition of infinity] reach, as I've said the infinity in A is its creation force and not the exact creation of numbers). If A created all numbers it can, the very creation force that moves it, the very will that enlivens it, won't have what to "want" anymore. Hence, it's will will be limited.

• To be honest now that I think about it, I'm probably misusing the idea "infinity" by giving it a possibility to have an end. Just wanted to know other people's thoughts about it. – Yechiam Weiss Jan 3 '18 at 15:17
• There seems to be two problems with this. First, you seem to be confusing infinite quantities with absolute infinity (there are infinite sets of integers, for example, that nevertheless do not contain all integers; in fact, you could exclude an infinite number of them... for example, there's an infinite number of odd numbers). The second issue is that you're trying to propose that infinite sets are limited, but the thing you show actually is limited is not an infinite set (suppose you just take counting numbers up to some n; no matter what your n is, you have a finite set). – H Walters Jan 3 '18 at 15:31
• With "A being the array..." what do you mean ? A sequence of number ? The collection of all numbers ? In the first case, it makes sense to say that "it goes to infinity"; in the second case no: it is an infinite "complete" collection (if we agree with Cantor's point of view) taht does not "go" nowhere. – Mauro ALLEGRANZA Jan 3 '18 at 16:50
• What is this "striving?" Does the sequence 1, 2, 3, 4, 5, ... "strive" to be 6? Or perhaps it strives to be 7 but fails. What do you mean by strive? – user4894 Jan 3 '18 at 21:33
• "A is an infinite, willing, creation force" -- WTF does that mean? – user4894 Jan 6 '18 at 0:48

I think what you may be after is modeled by the class of all ordinals, which does exist, in traditional mathematical logic, but cannot be a set.

One definition of the ordinals is the model called L.

L starts by defining the integers in terms of sets:

0 = {}

1 = {0}

2 = {0, 1}

3 = {0, 1, 2}

...

omega is the union of 0, 1, 2, 3, etc.

Then every whole number is in omega, and omega is infinite. But omega is only 'infinite' it is not 'infinity'. We can easily create the object which is "omega union {omega}", and that object is larger than omega itself, the first 'transfinite successor'.

We can continue building from there, and get the class Omega (capital) of all 'ordinals'. Each of these is an infinity that is also limited. So there is no inherent conflict to that concept.

You can prove that every potential model of inclusion is represented somewhere in that class. And any real network of sets is equivalent to a combination of ordinals. So 'L' is a model of all of set theory.

The problem is that if Omega is a set, we could cram it into another set. That would allow us to construct "{Omega} union Omega". Unfortunately that would be another model of inclusion, not isomorphic to any of the ordinals. The way modern set theory avoids this is by declaring Omega not to be a set, only a 'proper' class of sets.

But that is not losing anything from most naive notions of infinity, because one of the standard properties of a realized infinity is already that it would not fit inside anything else.

You can do a very similar and only sightly more confusing construction with pairs of sets, if you want a broader sense of 'all the numbers', which includes models of everything most folks can consider a number. The result is the class of J. H. Conway's 'Hyperreals' as elaborated in Donald Knuth's "Surreal Numbers".

• I assume this is set theory? Please review my edit, as I am aware of these concepts (in other terms), but it isn't what I meant to ask. – Yechiam Weiss Jan 4 '18 at 6:29
• That we have this math is still an indication that many humans see no conflict between the idea of infinity and being limited. Which is at least part of what you asked. And the notion of non-set classes does apply to this odd way you want A = infinity to work. If you can't be clear, you should learn to interpret answers as analogies. You need to consider this, even if it is not directly in the right domain. – jobermark Jan 4 '18 at 22:58