Our finite minds have ways of comprehending mathematical infinities. These ways are extensions of what we do for large finite sets. We usually don't try to list every element of them, but instead find some rule which generates all of them and no others. Limited imagination is not much of an argument.
(ETA: some of what I had posted here I've moved to the "Infinite Sets" thread, where it belonged)
I can see extremely finite rules but no infinities.
To see something requires at least some time.
To see infinite elements though can't be done even if you have infinite time to do it in.
Infinite elements is by definition an amount of elements that can never be expressed. The end of them can never be observed.
It's more like
"The first hundred positive integers"
vs.
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100}
The first description is much shorter than the second one, and by untermensche's argument, there is no such thing as a large finite set, since he seems to consider only the second kind of description a valid description of a set. Once one accepts rules for generating set elements, like "The first hundred positive integers", it is a small step to infinite sets: "All positive integers".
In theory there is a large finite set.
And in theory any finite set can be expressed.
Even a set with more members than there are elementary particles in the observable Universe? A number which is approximately 10^{86}.
So I can write "the first 10^100 positive integers" without having to write them all down, because doing so is a physical impossibility. What is the fundamental difference between "the first 100 positive integers" and "the first 10^{100} positive integers"? Or between those two descriptions and "all positive integers"?