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https://en.wikipedia.org/wiki/Natural_density

Hadn't heard of this concept before but it sums it up well

> Pick any large bound you want from the set of integers, whether it's from 1 million to 10 million, or negative a trillion to positive a trillion trillion trillion. It's always going to converge to integers being twice the size of even integers.

The fallacy in this argument is basically the same as the fallacy used in the "troll math" post that claims pi = 4 <https://qntm.org/trollpi> <https://math.stackexchange.com/q/12906/355349>.

But how can there be more integers than the are even integers when for each and every even integer there exists a corresponding integer? And how can there be more rationals, when we can find an integer for every rational number?

You are thinking in terms of very large sets, but literally infinite sets are simply not the same as very large finite sets. And they are not even the same as the limits of cardinality N sets as N grows to infinity.

For example, if we take the set of all integers less than N and the set of all even integers less than N, we can show that no mapping can exist between the two, even as N grows to infinity - so, one is indeed larger than the other. And yet, a mapping appears if N is literally infinite.

The problem with these discussions is some of the same problem we have in public discourse. One or both parties are using 2 definitions of the same word and trying to make them equivalent.

Are there integers that do not exist in the set of all even integers? Yes. The set of all even integers is incomplete, with respect to the set of all integers.

They unfortunately also have the same number of elements. So there are not 'more' elements in one set than the other, and yet there are elements missing from one of the sets. When we are assembling something and find that it's missing bits, we tend to compare it unfavorably to other things. It's 'less' than they are. But if you substitute other things to make up the difference, it's not less or more it's just different.

> Pick any large bound

Yes, if you tautologically select a finite subset of an infinite set, it tautologically has a comparable size which doesn’t equal the same constraints to select a finite subset of another infinite set. It would be a little bit more accurate to describe these comparisons as “faster” vs “slower” (ie you’ll reach the end of the conceptual subset of integers “before” you reach the end of the conceptual subset of even integers), but that belies the tautology.

My instinct is to send the person to count their infinite set and when they get back I'll give them a billion dollars.

If I can find an interest bearing bank account with a $5 minimum I should have plenty of money to pay off that bet.

> Pick any large bound you want from the set of integers, whether it's from 1 million to 10 million, or negative a trillion to positive a trillion trillion trillion. It's always going to converge to integers being twice the size of even integers.

I think you have to understand this: Even the largest of numbers you can fathom (ie: 10e5000, as in gazillions of trillions) is still closer to 0 than to infinity.

We know the set is smaller (or half) for a determined infinity. What we don't know, is how big the size is going to be when you "go" to infinity.

I've also been convinced that any comparative might have issues. Let's take the set of all primes.

On the one hand, I can trivially demonstrate they are less numerous than the set of integers

On the other hand, Cantor's exercises for showing larger cardinality can be exercised equally on the set of prime numbers

At least as far as I've been able to see. This shows a contradiction that can be easily accommodated for with multi-dimensional or an otherwise richer descriptor set. Any refutation that starts like "assume there exist a function f that generates all the primes as a sequence of integers" I reject. The whole point is that I'm not assuming that - maybe a proof that no such function can exist for the set of primes is an undiscovered property.

If you decide to be extremely disagreeable and stubbornly throw out any speculative, at least I'm left with only with the contradiction. They seem to behave way closer to the irrationals then anything else.

I think a more useful exercise is whether the peano successor function can be defined. For the irrationals, I think the answer is "not a chance". For the primes the intuition is "maybe" but when you start stating why, nearly every argument you give (statistically you can do it - with enough information you can do it, it is a discrete number and so on) you can defend for the irrationals with the same reasons.

Here's a simple function from positive integers to primes.

Let f : Z+ -> Z+

Let f(0) = 2

Let f(n) = p, where p is not divisible by f(n') for all positive integers n' such that n' < n, and p <= p' for all positive integers p' such that p' is not divisible by f(n'') for all positive integers n'' such that n'' < n.

Sure. You just used jargon and formality to define it. That's not an actual algebraic definition in the same way you can express a peano successor function for say, all natural numbers.

My claim is that's not an equivalent class of functions. You can define the irrationals in an almost identical way to that which you proposed

You could claim that one can come to an agreement on the Nth prime but we cannot come to an agreement on the Nth irrational. But I say sure you can. You can number both of them off however you please.

For every prime, we take a square root and get an irrational. There's a really simple one. Of course there's lots of holes but the point is the behaviors are far closer then they are to the rationals.

I bet the farm that when Riemann Zeta is solved it will show just this connection.

I can only see it in a blurry fuzzy way. But I don't think it's an illusion

> That's not an actual algebraic definition in the same way you can express a peano successor function for say, all natural numbers.

I guess it's not clear to me how you define an "actual algebraic definition" of a function.

Can you provide a formal definition for this? Like some rules that I could use to determine whether a function is algebraically defined.

I think it'll clear up whatever misunderstanding I seem to have.

Refresh my memory. Cantor's argument is that you can number the primes in order, right? So as long as primes are infinite (we believe they are but they just get more sparse), you can map every single whole number to a distinct prime number.

Sure. You can do that with the irrationals as well. Whenever you have a new irrational, I'll increment my counter and assign it thusly.

You show that there's irrationals between two assigned ones that were missed and I will continue to increment the counter accordingly and assign it to the missed irrationals.

Then we will have our mapping to the whole numbers like you claimed.

The point is that rule is insufficient to demonstrate the distinction we wish for. There's other ones such as the diagonalization or the column proof, you cannot do these directly with the primes, only with this middle step of assigning them a cardinality

If you can only do them if you abstract your representation and assume you can assign them like that which I've showed, then you can do for any other set and it makes it no longer valid.

It's only about what proofs can be demonstrated and what proofs cannot directly.

Cantor's work is, in this way, more about the nature of the "generators" of the sequences then the sequences themselves and categorizing them like algorithmic complexity. The act of mapping is contingent on the sequences peano successor being algebraic.

When you assign them whole number cardinality, you apply another "generator" that normalizes the thing you're inspecting and removes it from inspection.

There's a distinction here that's not appreciated. But I've honestly lost hope

>> if we can deal with ratios in calculus due to infinitesimal sizes using limits

Cauchy proved this in the 19th century using analysis. You did not cite a proof that we can somehow use analysis to do something vaguely similar for infinities.’

>> I know there are supposed theoretical problems with random sampling

Yes, no uniform distribution exists on the integers.

>> But I don’t care

Clearly. It drives me bananas when people knowingly post comments in mathematical discussion that have no rigor and instead are appeals to emotion.

> What drives me bananas is when anybody starts using the words "size" or "larger" or "smaller". I will insist to my dying day that while the cardinality of even integers is the same as that of integers, the set of even integers is still smaller than the set of integers. That's it's still half the size.

Note that size of a set has nothing to do with individual set members and subset relationship. This is true even for finite sets:

Set {'A', 'C', 'G'} has size 3, while set {1, 2, 5, 6 } has size 4, It is clear that the second set is larger than the first set, despite i have no way how to compare individual elements between the sets.

But what does even mean that set {'A', 'C', 'G'} has size 3? We get such result by counting the elements, which is just process of defining one-to-one mapping to {0, 1, 2} (or traditionally {1, 2, 3}, it does not matter), canonical representation of size 3.

From this point of view, the concept of infinite cardinality is just a natural extension of this.

So your only problem is with the word "size" (and relatedly "smaller" and "bigger")?

That seems like a fairly "trivial" issue. I don't think there is any particular benefit to using those terms except that they are evocative of the kind of thing we are talking about when we compare sets. Saying "size" is a useful way to get an idea into someone's head, the objective reality behind it is the definition of cardinality, nothing else.

Since the set of all even integers is a set of all integers, there's at least one kind of 'lesser' built into the definition.

Both have infinite size. And yet one can be made by taking the union of two infinite sets that have no elements in common.

> if we can deal with ratios in calculus down to infinitesimal sizes using limits, we can sure as heck go the other way,

You can absolutely say lim_(x->infinity)x/2x = 1/2

I.e. in I, the density of evens is 50%.

But that's not what people (mathematicians) mean when they compare sets.

You happen to be comparing two sets whose have the same members.

But what about when they don't?

How does the set of real numbers compare to the set of curves?

What "ratio" can be constructed from that? Your proposed methodology falls over.

> there are supposed theoretical problems with random sampling from an infinite set but honestly I don't care

Ah