I have this irrational cringe every time that people say infinite when they actually mean unlimited. I hear and read it everywhere, multiple times per day. In English and in German.

"You can do this infinitely often"

"AI bots have infinite APM"

"The number of items you can store is infinite"

Aaaaargh! Stop it! It hurts! 😫

Well, infinite and unlimited are quite the same, at least mathematically xd.

But yeah, none of the statements are valid 😂

@Demolishun

Simple. Read on the distinction between countable and uncountable infinites.

For bonus points, hierarchies of infinites.

If ya can't be bothered, I can sum up.

@CoreFusionX what do different infinities have to do with (finite) state machines?

@Demolishun @Lensflare

With finite state machines, nothing.
I was just commenting it on the semantic side.

As for undecidable state machines, (which are essentially reducible to a Turing machine), there's a slight difference between countable and uncountable.

If there were to be uncountable states, the machine is pointless. The universe itself is it.

If they were countable, it is decidable.

Take for example String.endsWith() from the point of view of a Turing machine (that is, you don't know the string length beforehand, that would be trivial).

You have countable infinite possible states, given enough resources, you could solve it.

(And that is valid, because an ideal Turing machine has infinite tape.)

Again, it's just mathematical nuances, don't mind me, for all practical purposes, what Lensflare said is all true.

@Demolishun

As for the infinite hierarchy, it stems from the very definition of infinite as we rationalize it.

Initially, all infinites are the same (lol, no end, right?). But mathematically, it's not the same to take the limit of a linear function than an exponential function.

Sure, they both (and many others) are defined up to infinity, but you can still study their properties, and one such is that

lim x->inf x < lim x->inf e^x

That holds true as per L' hopital rule.

And the moment you can establish absolute ordering relations between (for all intents and purposes) infinities, some infinities are less infinite than others 😂

@CoreFusionX my favorite part about infinity is that while there is the countable infinity and the larger, uncountable infinity, the question if there is a different infinity between the countable and the uncountable is apparently a very weird one. In mathematics it is debated if it is true, false, undecidable or just a meaningless question that doesn‘t make sense to ask within the realm of our mathematics.😆

Source: Matt Parker, Stand up maths.

@Lensflare

The way I see it, a countable infinity loses to any other because it is in essence, linear, which is the "lowest" hierarchy of infinities.

All uncountable infinities could be what have you, but essentially, (again, math), they can never be less than a linear infinity.

@CoreFusionX yeah but the question is if there is an infinity below the countable. That‘s obviously false. But the question if there is one above the countable and below the uncountable :)

@Demolishun

That's actually an uncountable infinite.

By definition, if you can not prove two real numbers are the same, there are infinite numbers between them.

(Hint, that's the actual discriminator, if you can find an homeomorphism to N on any infinite set, it's countably infinite.)

Math is alluring, stupid, lovable and hateable all at once, that's the beauty.

@Demolishun let me break your intuition and tell you that there are infinitely many fractions between 0 and 1 and also infinitely many real numbers between 0 and 1 but the fractions are countable while the reals are uncountable 😄

@Lensflare

Sorry, you posted while I was writing .

As I said, linear growth is the lower bound of infinities. Literally every infinite that is not countable is uncountable by definition.

@CoreFusionX I see. That explains why the question of one in between makes no sense :)
It’s like asking if there is a temperature between cold and not cold :)

@Lensflare

Ah, but that is a not so clever mathematical trick.

Incredibly enough, we knew about irrational numbers back in ancient Greece.

In fact, for those not Latin-derived speaking, rational comes from ratio, as in, integer division.

Of course, infinite numbers can not be expressed as a simple ratio, but they exist just like the ends with problem. 😁

@Lensflare

Also, fractionals are countable just because they are an operation on two countable numbers, which makes it automatically countable too, because its result is also a natural number.

@CoreFusionX but there are operations on countable numbers which make them uncountable. Like square root of 2.

@Lensflare

Nope. Don't confuse with irrational.

If the operation is reversible (and all roots of natural numbers are), then the resulting set is also countably infinite by virtue of homeomorphism.

@Demolishun

Probably not in any way XD

And as for the second question, 42, of course.