Thanks! I’ve been debating infinity for years and finally decided to consolidate information into one place to avoid rehashing the same old arguments eternally.
From what I can see so far, you keep saying you’re talking about mathematical infinity, but as arguments against it you immediately cite physical analogies. But just because something is physically impossible doesn’t mean we can’t work with it as an abstract entity. You constantly fall back on physical arguments even when you claim to be arguing against abstract mathematical infinity.
Well, I’ve defined infinity as boundless and asserted that what is without boundaries is not a thing neither in reality nor imagination. Just like i (sqrt -1), infinity may have productive uses, but likely it’s indicative of some underlying mechanism that we don’t yet understand.
You are absolutely correct that the nub of the matter is the Axiom of Infinity. It is indeed arbitrary, in the sense that both it and its negation are perfectly consistent with the other axioms of math. There is no absolute truth of the matter; and no logical reason to prefer one to the other.
However, there is a pragmatic, practical reason. When you assume the axiom of infinity, you can construct the real numbers and do all of modern math and physics. When you deny the axiom of infinity, you get a far more paltry universe that can’t be made to serve the needs of mathematical foundations.
I’m not sure this is true because we routinely use PI, but never all infinite digits of it.
The axiom of infinity is no more “true” or “false” than whether the knight in chess “really moves that way.” What a ridiculous question! It’s a formal game. It’s not true and it’s not false. The rules are what they are, and if they are consistent and interesting, we accept them. Chess is fun to play, so we play. Math is fun to play, and the physicists and engineers find it useful. That’s as far as the ontology goes.
Right, the concept of “knight” is only relative to the game of chess and the concept of truth is only relative to the duality of the universe.
I mean, forget about an infinite set. Does the empty set exist? Ponder that.
No, the empty set does not exist unless it contains potential, but then it wouldn’t be empty. I’m saying neither zero nor infinity exists and they are equally absurd.
I hope you’ll take some of my points to heart. You’re tilting at windmills. You think somebody thinks the axiom of infinity is true. On the contrary. People who think about the question at all, understand that the reason we accept infinity in math is because it’s useful.
How is infinity useful? I googled it and here is what I came up with:
[i]We Don’t Need the Infinite
Let’s face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 10^80 objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about seventeen decimal places. Yet real numbers, with their infinitely many decimals, have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics. We describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.
Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.
Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.[/i] blogs.discovermagazine.com/crux/ … 8soKiXwaHs
ps – You may be interested to read a pair of papers by Penelope Maddy, Believing the Axioms parts I and II. You can find pdfs if you Google around. She walks through each axiom of ZFC and discusses the history and philosophy of how and why it came to be adopted.
cs.umd.edu/~gasarch/BLOGPAP … xioms1.pdf
We must have axioms in order to have a foundation for any construct.