Quite the opposite imo, if there was a definite “largest number”, it would be an engine limitation. It is literally an actual issue with numbers representations in computers.
Now, some aspects of quantum physics really feel like IRL engine limitations though
“Countably infinite” means an infinitely-large set of numbers that could be generated by infinitely following an algorithm with a finite number of steps. For example, natural (positive whole) numbers are countably infinite because they could be generated by following this simple algorithm:
Start with the number 1
Add 1 to your number
Repeat step 2
The set of real numbers, on the other hand, is uncountably infinite because you can have an infinite number of digits after the decimal place. You can’t define a finite generation algorithm like the one above simply because any precision you use wouldn’t cover the full range. In other words, if you wanted to modify the above algorithm, and chose 0.1 as your starting number, your algorithm would miss 0.01. If you chose to start at 0.01, you would miss 0.001, and so on
That is the way it is often taught but actually both sets are infinite that is have no ends or in other words are not bounded.
The thing that is confusing to understand is that the question how many there are and how much there is diverges at infinity.
Our intuition (as finite beings) is broken here. Both sets are infinite but in one is more than in the other. That does not make one set more infinite than the other. You cannot be more unending than to literally have no end.
This is incorrect. There is not a one-to-one and onto mapping from the natural numbers to the real numbers ergo the sets have a different size. We have defined words to describe this. We can put uncountably many copies of the natural numbers inside of the real numbers so there are arguably infinitely more reals than naturals.
Granted you have to accept the axiom of choice for any of this.
I know. I’ve studied this extensively. I am specialized in formal logic and by extension set theory. I’ve worked with and help write actual research papers in this field where this is basic knowledge.
I’ve never claimed there to be a bijection between the reals and the natural numbers. Please point out what statement I made that is wrong. I would very much like to know.
Also no you do not have to accept choice for this to be true. ZF is perfectly acceptable to study various infinite sets with differing cardinality.
Edit: This is what I mean when I say that our intuition is broken. One set can be larger than the other but both be non-ending that is infinite.
That does not make one set more infinite than the other. You cannot be more unending than to literally have no end.
Your use of language is incorrect. But, since you’re clearly the only published expert with any experience in this topic on the internet, it’s really not worth pointing out that we fall on two sides of the standard axiom of choice debate since you already knew that. Have fun!
Our mathematical definitions say that it does not end. We’ve defined addition so that any number + 1 is larger than that number (i.e. x+1 > x).
You’re probably confused, because you think infinity is a concrete thing/number. It’s not.
In actual higher-level maths, no one ever does calculations with infinity.
Rather, we say that if we insert an x into a formula, and then insert an x+1 instead, and then insert an x+2 instead, and were to continue that lots of times, how does the result change?
So, very simple example, this is our formula: 2*x
If we insert 1, the result is 2.
If we insert 2, the result is 4.
If we insert 82170394, the result is 164340788.
The concrete numbers don’t matter, but we can say that as we increase x towards infinity, the result will also increase towards infinity.
(The result is not 2*infinity, that doesn’t make sense.)
Knowing such trends for larger numbers is relevant for certain use-cases, especially when the formula isn’t quite as trivial.
The question doesn’t make sense, there are many things which have an infinite quality (like infinite cardinality) or are called infinite/infinity (like infinite cardinals and ordinals). They’re not contradictory. They coexist the same as all finite things do.
I’m not sure anyone has really provided a complete explanation of what is the difference between working with an absolute infinity and the way we do math normally in science and such.
Basically, no one has found the idea of using an absolute infinity to explain the world to be better than the way we deal with infinity in college courses. In college, you run across the idea that some infinite sets are larger than others (countable numbers vs uncountable). Edit - I think you could have the idea of different sized infinities and a final largest absolute infinity. It’s just that this concept isn’t useful. It would be like claiming God is purple. Nobody can prove you wrong and it doesn’t matter.
Of course, an infinite set makes sense in math, and has practical uses in the sciences, but nothing can truly be demonstrated to be unending. Another poster put it nicely - infinity is a direction, not a destination.
I recommend this video How to count past infinity by Vsauce (about 20 minutes long). It is closer to entertainment than a lecture but its pretty good. I’m only an undergrad math major but I haven’t found any real problems with this video (though, he does start talking about ordinal numbers which aren’t terribly useful to anyone that I know of, yet, except for some really complicated number theory stuff cryptographers might use, don’t ask me. cryptographers are basically wizards imho).
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