What separates infinity from the largest number less than infinity?

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The largest number less than another number only has meaning in certain groups of numbers. Infinity isn’t a member of any of those groups, so your question is meaningless.

Consider what the largest number less than 1 would be. On the integers, it’s 0. But what about rational numbers? 0.5 is closer to 1 than 0. 0.9 is closer still. You can get repeatedly closer to 1 by averaging 1 and whatever number you choose less than 1, and you’ll remain in the rational numbers. So the largest number less than x has no meaning in the rational numbers.

It is possible to treat infinity as a cardinal value, similar to the natural numbers (1, 2, 3…), so you may be tempted to ask what infinity – 1 would be. It’s fairly easy to show that infinity – 1 = infinity in the sense that removing one element from an infinite set doesn’t change the number of elements in that set. In order to get back into real numbers, you have to perform an operation like infinity – infinity, which is what’s called an indeterminate form – it can take on any real value, depending on the context.

That might be the answer you’re looking for: there is infinite distance between infinity and any real number, including any you might declare the “largest number less than infinity”.

If I take some liberty with your question and interpret “largest number less than infinity” as the largest finite number that has ever been used for any calculation, written down, named, or theorized. The big key is finite. Meaning if you had a pen and paper, you could start writing the number, and your task would eventually finish. (Perhaps it might take longer than a human lifetime, paper than exists in the world, and more ink than there is water in our oceans, but if you know how many numbers per second you can write, you can predict how long it would take). If you started a stopwatch, and every second that ticked by, you added 1 to that number. Even if you waited 14 billion years, you would still have a finite number. Even if every second you doubled the number instead of adding one. Even if you squared it, raised it to its own power, or ran it through some rapidly embiggening function… You would still have a finite number.

Infinity is not *really* a number as much as it is a concept or description of something that is unending, but let’s say there was a decimal representation of infinity. It might look like a 1 followed by an infinite number of zeros (an unending number of zeros by definition). So go back to our loosely defined largest finite number… If you could manage to write that many zeros per second, how long do you think it would take you to finish writing infinite zeros? When could you end your task if your task is literally endless? That’s the difference between finite and infinite.

This is actually interesting with a very complex hour but infinty-th is a decent start

(There are differing levels of infinity such as infinity + or * infinity. or -infinity.)

The reason this works is because infinty-th is an infinite amount of zeros then a one at the end. Although you could get smaller by dividing infinity-th by infinity an infinite number of times. But get this you can get even smaller by doing this infinity-th^-infinity and even smaller by doing this infinty-th^-1(infinity^infinity) and raising the inside of that parenthesis o the power of infinity an infinite number of times. It could get smaller then that with some more math but you get the point

The reason you want a infinitely small number is to subtract from from infinity to make the largest number less then infinity.

This concept is the solution of the infinite hotel paradox. There are infinite rooms and every room is occupied. What do you do when a new guest wants a room? You shift everyone to there room number plus one thus freeing one room. There is a lot of interesting mathematics involved with that and Vsauce did a great video on it fulling explaining it if you’re interested

Edit: I didn’t realize there were smaller numbers then infinity-th

To keep this simple, numbers are written out of series of numbers (and maybe decimal point, too), but the infinity have it’s own symbol: ∞. Infinity can be anywhere, including after decimal point at any number. So, numbers have their own position, but infinity haven’t, it can be literally anywhere…

There is no largest number less than infinity. In fact, there is no largest number, including infinity. Infinity is a concept meant to say it’s higher than any conceivable number.

If you had a candidate “largest” number, you could add 1 to it and make a larger number.