It seems rather paradoxical in nature, but I was just curious as to what some of the responses would yield. If it is, explain why... if it isn't, explain why. I am really curious to know.
I think you'll need to clarify a bit. As it stands, I'd say the number 6 is infinitely less than infinity, because \(\infty-6=\infty\). I assume I'm missing something here.
You of infinity in terms of " numbers " mathematics , your thinking But I think in terms of physical things , hence geology
Yes, it's possible. Assume somewhere exists infinitely far away. The mid-point in your journey is infinitely less far than that, yet still infinitely far away.
The question as stated is ill formed. Within mathematics you can only make precise statements if you have a precise definition of something. In this case we're talking about infinity and the notions of addition, multiplication etc do not apply to it any more than "purple + elephant = table" makes sense. Despite the common misconception, infinity is not a number, you cannot do arithmetic with it. Likewise with 1/0, a closely related notion. 1/0 is no more a number than "elephant" is. Why? Because we define "The set of Real numbers" (where 'Real' doesn't mean physically extant, it is just an arbitrary label) using certain rules for combining other numbers, which do not allow the combination of 1 and 0 in that manner. There are many definitions of 'infinity', depending on what area of mathematics you're in. The usual arithmetic-like definition views it as being 'larger than any Real', treating it as if it were huge. In more abstract arithmetic it is used to construct the "one point compactification of the Reals", but this means you can no longer say "X is less than Y" in this construct, as it cannot be ordered fully. The most rigorous and relevant domain to your question is set theory. We can construct sets which have a cardinality (a generalised notion of 'size') that is infinite. But more than that we can construct sets which are of infinite cardinality but we can absolutely say that one set is strictly larger than the other. Perhaps I'll elaborate with an example... Consider the set of positive integers, the Natural numbers, N = {1,2,3,4,...}. It has infinitely many elements so its cardinality is infinite. Consider the set of Evens, E = {2,4,6,8,10,...}. It too has infinitely many elements and obviously E is a subset of N, ie each element of E is found in N but not all elements of N are found in E. But their cardinality is the same because I can construct a function which links each and every element of N to one and only one element of E, namely n in N is linked to e=2n in E. If such a map exists then the two sets (which must be necessarily infinite for this sort of thing to occur) have the same cardinality. For the same reason N has the same cardinality as Z = {0,-1,+1,-2,+2,...}, the Integers. Let's try something harder, the Rationals, Q = p/q for all p,q integers except q cannot be 0. Now you might think "Well now you have TWO copies of Z to make Q so surely Q is strictly bigger than Z!". Wrong! If I can find a map which sends each element of Q to a distinct element of Z then I have shown the cardinality of Q cannot exceed the cardinality of Z. And since the cardinality of Z is obviously not greater than the cardinality of Q (since Z is contained within Q) that would mean they have the same cardinality. Suppose p/q is positive, then the map \(\frac{p}{q} \to 2^{p}3^{q}\) is one which maps a rational to a unique positive Z since by prime factorisation changing p and/or q necessarily changes the output. If p/q <0 then map to \(-2^{p}3^{q}\). Job done. Now I could daisy chain these maps together so I can end up sending each element of Q to a unique element of N. This means I can define a list of all of the elements in Q which does not miss out a single element in Q. Specifically if I can link p/q in Q to n in N then my list, \(a_{1},a_{2},\ldots\) has \(a_{n} = \frac{p}{q}\). Doesn't matter if I don't have a p/q mapping to say m, it just means I can put anything there, I'm still listing everything but now with redundancy. But what about the Reals? Well it transpires that it is impossible to construct a list, the Reals are "uncountable" (ie the list is counting through the elements). The proof is more involved but Googling for Cantor's Diagonal Argument will turn it up. What this means is that the set of Natural Numbers N is strictly smaller than the set of Real numbers R. If you were willing to have a dubiously phrased statement then this could be viewed as "N being infinitely less than R". This is obviously a bit of a different take on things from "What is \(x-\infty\)?", which is more what your original question was asking but at least this approach is logically sound.
