Explaining Infinite Sets, Measures, and Mappings for Quantum Physics

Alan Forrester wrote in a discussion of multiple universe quantum physics and David Deutsch's books:

The stuff [in the multiverse] doesn’t come from anywhere. There is a continuous infinity of fungible frogs, like the real number line. That continuous infinity of fungible frogs differentiates over time. The number of instances of frogs doesn’t increase or decrease as a result of the differentiation. You’re just taking the set that already exists and dividing it up. The same amount of stuff exists before and after a division.

And Justin Mallone asked:

I've found stuff involving infinity really difficult to wrap my brain around in a way I felt like I could understand and explain and apply. This post reminded me of that fact.

And I don't have trouble understanding complex concepts generally, so I assume I have some systematic mistake or set of mistakes I'm making in my thinking about infinity in particular, but I don't know what it is.

I can easily conceive of some actual set number of universes differentiating many times. Like you have 1024 universes, and they differentiate in some way where something happens in half the universes and something doesn't happen in half the universes (lets call the thing Event A or something). And then you look at the 512 Event A universes and something happens in 3/4 of them (Event B) and so you have 384 Event A + B Universes, 128 Event A Only Universes, and 512 No Event Universes. And on and on.

But the endless differentiation of infinities of universes seems weird to me. I guess cuz one of the things about the sort of differentiation I was just describing is that you can easily assess the ratios of stuff happening in them. You can think of stuff in terms of probability of happening across the set of universes you're considering -- like above you could say it's equally likely that neither Event occurs and that at least 1 Event occurs.

But doing that with infinity seems harder. If you differentiate an infinite set of universes in some way, you still have an infinity after the differentiation, although both are smaller (?!) than the infinity that you had before the differentiation. What??? 樂

To compare sizes of sets you need ways to measure the sets.

For finite sets a standard measure is the integer number of elements.

It's not the only measure people care about. Another thing you can do is weight the elements of the set and then measure the total weights of the sets and compare that. This kind of comparison can be more useful. E.g. you could look at the total weight of the set if you were looking at carrying sets of tools in a backpack. In that case, volume would be an issue too. And you could come up with measures of the utility of each tool and add up the total utility of each set.

Real scenarios are often more complicated than easy-to-calculate measures. Just because your tools have X total volume, and your backpack has X+1 total volume doesn't mean they will fit. One could be too long, or there could be no way to pack them together without empty space in gaps between some tools. And the utility of a set of tools isn't just the sum of the utilities of the individual tools. If a screwdriver is 4 utility, that doesn't mean 2 of that screwdriver is 8 utility. And nails have more utility if you also have a hammer.

The utility of a set of tools is too complex an issue for people to define a measurement function that's very accurate, so they do critical thinking instead (which may involve some utility measures as loose approximations). The weight and volume issues are both simple enough someone could do a good job defining it and inputting their measurement function into a computer which would accurately calculate it. This isn't trivial but it's doable today and people could get good, useful results with it. Companies like FedEx have computer programs that worry about packing objects into spaces and considering weight and volume. FedEx cares especially about putting boxes into cargo holds of trucks and planes. And there are mathematicians who like to calculate how to tightly pack spheres and other shapes into spaces with the least wasted volume, and they know some stuff about how to do that.

---

So some facts:

• You can define dozens (or trillions) of different measures on one set. You can measure the same set multiple ways.

• Some measures are more useful to human concerns than others. Some are harder to measure than others.

• There is no canonical single measure for sets provided by the universe's instruction manual. There isn't like the one objective measure. There are often a variety of worthwhile measures for a set, and also many, many more arbitrary and dumb measures.

• Some measures help address many different human problems, as well as more important problems. Their reach is a sign of their objective value.

• Inches and meters are measures created by humans to help solve human problems. They are relatively useful and objectively valuable compared to many other measures.

• Quantity is a more complex measure than people take for granted, and it's not really a singular measure across all types of sets. In order to measure quantity of iPhones you need a definition of what sets of molecules constitute one iPhone. Then if you want to measure quantity of hammers, you need a totally different definition of what is one hammer. Quantity is only measurable with a supplemental function defining what is "one" of something, and that supplemental function varies for different objects. This complexity is something humans take for granted as common sense

---

OK now let's talk about numbers.

Different infinite sets can be measured and get different results.

You can look at infinite sets and measure them as "Is it finite or infinite?" and come to the conclusion "infinite" for all of them, and think they're the same.

But there are other measures. The set of all real numbers has more stuff in it than the set of all integers, even though they're both infinite. This fact is then actually used for measuring other infinite sets -- is it the same size as all integers, or all reals, or something else?

The way to compare infinite sets is to consider whether there is a mapping that puts the sets in one-to-one correspondence.

I will show you a mapping from the set of all integers to the set of all odd integers. This shows they are the same size! (Same size according to the normal, useful way mathematicians measure infinite sets.)

The mapping function is: n*2-1

You take an integer, n, and you double it and subtract 1. Now you have an odd number, m.

This is a one-to-one correspondence. No m repeats and no m is left out. You can pick any odd number m and find that there is exactly one integer n which corresponds to it via this mapping function.

The function is also reversible and the reverse gets you a one-to-one complete correspondence from odd integers to integers. You just take an odd integer, add 1 and halve it and boom you've got an integer. And you can get any integer by doing this.

BTW you only need to find one mapping function to make your point. There are other mapping functions which work just as well. Such as n*2-27. That also maps integers to odd integers with complete one-to-one correspondence.

This set mapping stuff is counter-intuitive because people initially feel like there are twice as many integers as odd integers, so the measure should say it's twice as big. But this measure says they are the same size! Yet this is the measure that's actually useful, has reach, has lots of objective value, is used in math a bunch. You could define some measures which assign an integer size to infinite sets and assigns twice as large an integer to the integers as the odds, but that's either not useful or sufficiently obscure/niche that I'm not familiar with it.

And people feel like you would run out of odd integers when trying to have one odd integer for every integer. And you would with finite sets. There are twice as many integers from 1-100 as odd integers in that range. And the same goes for 1 to a trillion.

But you don't run out of numbers with infinite sets. That you're mapping 101 to 50 and 10001 to 5000 is not a problem. You can just keep going up and never run out of odd integers to map to all the integers. I think the best way to look at this is just that you can pick any integer and clearly see there is exactly one odd integer it corresponds to with our mapping.

A complete one-to-one correspondence means sets are the same size in some meaningful sense. It's like putting one set on the left, one on the right, and drawing a line between every element on the left to every element on the right. And every single element in each set has exactly one line connecting it to the other set. Just like if I have 4 goats and you have 4 cows and we drew 4 lines and saw our sets of animals were the same size. But if you had 5 cows then we couldn't draw lines in this way, you'd have an extra cow with no line touching it, or else i'd have a goat with 2 lines touching it.

There is no one-to-one correspondence from the integers to the reals, which is why the reals are considered a bigger set. That thing you can do to correspond odds to integers is not possible with integers to reals. That makes reals a bigger set than integers in a way which integers aren't bigger than odds. And this particular way of measuring set size turns out to be valuable, useful, helpful -- it's a good way to think about things for many purposes.

---

You're having trouble because you're using your intuitions about quantity measures of finite sets, including I presume that there's only one way to measure a set.

Physicists have a measure of infinite sets of universes. The results are fractions like 0.2 or 0.01. And it means what you would expect: e.g. you measure the universes where event A happens, get 0.5, and that means it happened in 50% of universes.

You could measure as a fraction of the whole multiverse but you normally don't want to. You can also measure as a fraction of a region of the multiverse, like the ones where you exist and are doing this experiment at this time and everything else is the same too. Then of those initially identical universes, you end up with 3 sets of universes and one has a measure of .5 and one has a measure of .375 and one has .125 (these are the numbers from Justin's example quoted above for No Event, A Event Only and A+B Events.)

How can an infinite set get a finite measure? Well it's just like if there were hypothetically infinite points in an inch, it could still be measured as one inch. And another distance could be two inches, and they both have infinite points but there's still this useful measure in which one is twice as long as the other. Current physics doesn't say an inch of space is infinitely divisible into infinite points, but that is a familiar classical physics scenario which people's intuitions don't struggle with so much.

---

Anyway this stuff is really confusing without the right background knowledge: having some general understanding of sets, measures, mappings, infinite sets, etc. So this should help give you some leads to think about and lead to some followup questions.

BTW I don't even know where this stuff is taught or good books explain it. I learned most of this from David Deutsch personally.

PS I am not an expert on math or physics terminology and may have used a technical term a little bit wrong or omitted a technical term that is normally used. I've tried to focus on clearly and understandably presenting useful, important concepts.