Besides philosophical issues about mathematics we’ve already discussed, I do not assent to the premise that all existents are finite. I see no reason why that should be, and I see no reason why the law of identity in particular has anything to say about this particular matter since it merely tells us that one thing is itself and not another. If it is a part of a thing’s identity to have an infinity of parts, in any sense, then it still retains identity and the law of identity says nothing substantial about this. So unless you can give a more explicit statement of the law of identity beyond A equals A, namely an explanation of how the law bears on the issue of infinite quantities, this road seems like a dead end. However, I do suggest that this discussion–if we’re to pursue it further–be moved to a more appropriate topic. I’m sure there’s one in the archives.
Let’s see: Objectivism Online Forum.
Philosophy.
Metaphysics and Epistemology.
What does .9999999999… mean?
(subquestiona: What does infinity mean?)
What possible observations give rise to the concept of infinity in this case? (Hopefully this is taken as rhetorically as it was meant.)
What observations give rise to the necessity of either of these inter-related issues?
These items either reduce down to, that is create a logical heirarchy of the first level abstractions that give rise to them, or they would be invalid.
It is within the context of mathematics that give rise to both methods inquired about here. What gives them validity outside of the realm of mathematics? You wish to apply infinite to quantity. What gives rise to the concept of quantity? It is the relationship of a group of entities relative to one of its units. (Corvini, 2,3,4 and All That.) We can observe and grasp 2. We can observe and grasp 3, 4, and some of us 5, maybe 6. After that, we need a method to expand and deal with larger quantities.
We look out at the our world and see trees, and forests? We are able to extrapolate the square area of growable area on its surface. Would you suggest that an infinite amount of trees could be within a forest of that specific area?
We look at our galaxy. We observe stars. Via instruments and calculations we can assess its size. Would you suggest that our galaxy has an infinite amount of stars within it?
We look through microscopes and other instruments to conclude that our bodies are made up of cells. Would you conclude that our bodies have an infinite amount of cells within it?
Every observation that you are able to quantify and relate back to the existent that made it up - of them, have you discovered any entity without specific, finite, contextual limits?
Every system, and subsystem you are able to either observe directly, or relate the data back to the observable, have you discoverd any without specific, finite, contextual limits?
How many observations do you need to integrate before you permit yourself to induce the Law of Identity applies not only to the Universe as an entirety, but to each existent within it, within the framework of the context which gives rise to it, including the concept of infinity, and .9999999999…?
By chopping off the logical hierarchial chain part way down, and then trying to apply it within a context which it was not validated for, could account for some of the confusion that appears to be present here.
, that is for any 
there exists an integer N , so that 
for any .
is not.
