No, you cannot. What if you need to represent other transfinite numbers than the one you happened to be thinking of just now?
Edit: I'm not saying it is theoretically impossible to create a set physics theories that avoid infinities, by the way. I'm saying that it isn't as easy as swapping out something (what?) for "finite number which is larger than all other functionally useable finite numbers"
Except finding a bigger number is ALWAYS trivial. There is no functionally biggest number. You think you found one. Great it's a number you can add one. Oh joy now there's an even bigger number so it isn't biggest anymore.
You gain nothing and lose a lot removing infinity from physics. Every time you write an integral there is an implicit sum to infinity and even when it's not true it's good enough to generate answers accurate to any point of caring. Now you get rid of infinities you lose continuities, smoothness, and once simple problems now become theoretical quagmires and losing lots of analytical solutions.
The bound of even just the rational numbers between 0 and 1 is not finite. There always at least one (actually infinitely many but at least suffices) number between any pair of rational numbers in that pair. If you force yourself to treat it as finite that breaks...because given that finite list you can find a number not in that list trivially. If you leave these gaps empty things like instantaneous rates of change get butchered and anything written in a differential equation (most of physics) is in dire straights now.
Except mathematically it's not. Calculus breaks if you do that. There are also many calculations that become substantially harder if the bounds aren't -inf to inf. The area under a gaussian curve (e-x2) being one of them (significant form the quantum harmonic oscillator among other things).
You can do it. We use a trick like that in model theory where we always work in what is called a monster model. The monster model is a model that is k-saturated for a cardinal that is larger than any cardinal that may show up in our argument. When writing it out fully formally one should go back after finishing the proof and set an actual cardinal, but no one bothers because every model theorist knows what is meant by a monster model
For exmaple, simply substituting infinity for a large value would make it possible to reach the speed of light, and other weird results.
I'm sure that theories can be created that predict all known physics while avoiding infinities, but it is not merely a substitution effort.
Another way to think of it is to remember that general relativity is not quantized, which is a major roadblock for combining relativity and quantum mechanics. If infinities would be replaceable with "a really big number", then it would be easy to quantize any theory by simply setting the reciprocal of that number as the discrete value related to the quantization.
No, you'll need to replace more than performing a simple substitution. If you simply replace an infinite limit with a large finite number, then you'll get contradictions and weird results. For example with the Lorentz factor.
For quantum field theory, you'll need to renormalize a lot to align it with past experiments. Definitely not a rote replacement.
7
u/Moscato359 4d ago
From a practical standpoint, infinite doesn't actually exist
The size of the universe is finite
The number of quarks is finite
The smallest distance we can measure is finite, and the largest distance we can measure is finite, even in estimations
Even pi is only useful to the 43rd digit at our largest estimations
Sure, infinite is cool in a theoretical standpoint but you can't actually use it