I agree with the fact that one cannot see or touch infinity. It was Aristotle that said actual infinities couldn"t exist, so the idea has been around for quite some time. Infinity is a theoretical abstraction as are philosophical constructs relating to infinities and I think the real philosophical problem of the existence of actual infinities is that they are logically inconceivable. If they did exist we could not confirm it through observation because there would be no way to measure them. If an actual infinity were to exist there is no way for us to know it.
~;> indeed
thats why some people made some claimef about this so called infinite
that happeneds to be weird
:read: (as it is written)
What are some weird facts about infinity?
Every infinite set can be put in one-to-one correspondence with an infinitesimally small (but still infinite) subset of itself!
For example the Natural numbers, N={0,1,2,3,…}N={0,1,2,3,…}, and the powers of ten, {1,10,100,1000,…}{1,10,100,1000,…}, have a rather obvious 1-1 correspondence via the bijection n↔10nn↔10n.
In a rigorous sense, almost all natural numbers are not a power of ten, and yet there are just as many powers of ten as natural numbers. From there ∞∞ just gets infinitely weirder. Hence Bustany's Rule of Infinity:
Infinity and Intuition do not mix
Edit: It seems my use of "infinitesimally small" subset is too intuitive for some, so here is a possible definition. Let the original set be AA and the subset be BB then it is "infinitesimally small" if given N∈NN∈N there exists a function f:A→Bf:A→B such that the pre-image of every element is large, that is
∀b∈B:∣∣f−1(b)∣∣≥N∀b∈B:|f−1(b)|≥N
I leave it as an excercise for the reader to find such a function for the Cantor Set which is an infinitesimally small subset of [0,1]⊂R[0,1]⊂R created by repeatedly removing the open "middle third" of each remaining segment beginning with (13,23)(13,23). This subset also happens to be both uncountable and of measure zero.
If you try to handle infinity with computer arithmetic, you quickly run into some head-scratching situations. I have some favorites, but first: The term "NaN" means "Not a Number" and is the standard response a computer gives if asked "What is zero divided by zero?" and other questions where neither a real number nor ±∞ is the answer. "What is 1/0?" also returns NaN, though some would like to say the answer to that one is "±∞". Hmm… does that mean 0 times ±∞ is exactly 1? Why 1, and not some other finite number? If 2/0 is also ±∞, does that mean 1 = 2? Down the rabbit hole we go.
So here are some of my favorite "weird facts" involving infinity.
• Infinity minus infinity is NaN. So is infinity minus half of infinity. Imagine putting numbers 1, 2, 3,… into a hat, forever, and every time you put an even number in, someone else takes the smallest number in the hat out. How many numbers are in the hat? Your intuition would be an infinite number, but one could also say there are zero numbers in the hat. Pick any number, n. It was removed from the hat when 2n was put in! By this argument, the hat with an infinite number of numbers cannot have any numbers in it. This is why ∞ – ∞ has to be NaN.
• Zero to the power –2 is ∞. It looks like it should be NaN since we divide by zero, but for even negative integer powers of zero, the result is (±∞) squared, which is ∞.
• We know exactly who introduced the ∞ symbol for infinity, and when: The English mathematician John Wallis invented it in 1655, in his Treatise on the Conic Sections.
• Zero times infinity is NaN. Try as you might, there is no way to make sense of the question "What is 0 multiplied by ∞?" and the only reasonable answer is "Not a Number."
:ty:
godbless
unto all always
