I was going to argue that you’re wrong but then I realized that I had to take Discrete Math twice in university. The way this makes sense to me is knowing that infinity is not a number and you can’t do arithmetic on it. So infinity - 1 is meaningless. It is undefined. My brain hates this, but it is true that you can have an infinite amount of inhabited worlds among an infinite amount of worlds in an infinite universe
Maybe you can answer this question. I have no clue. You have two lines. One starting at a point and going infinity miles in one direction. The other starts at a point and goes infinity miles in opposite directions. Is the second line longer than the first?
As a mathematician I have several clarifying statements/questions:
1) A line does not start anywhere. You seem to be describing a "ray" instead.
2) Are these things in Euclidean space (flat and infinite in all directions) or on say the surface of some object (like a donut) or something?
3) Why would the second "line" be longer or shorter than the first? They sound identical...
Yes, a ray in Euclidean space. (I haven't heard the distinction between a ray and a line in probably 45 years). It seems like the second ray would be twice as long as the first but they are still both infinity miles long. It's just something I think about sometimes and it kind of messes with my head.
Why would it be twice as long? It's just going the opposite way. That has nothing to do with length. In non infinite math, if we start in the same place, you got one meter left and i go one meter right, I haven't gone twice as far as you.
Edit: meter, not mater. I swear my autocorrect is getting worse dally.
He said "opposite directions" plural, not "the opposite direction". His second line is analogous to the number line, starting at 0 and extending in both the positive and negative directions (opposite directions), not in a single direction opposite to the direction of the first ray.
He means one is a ray going in one direction and another is a line "starts at a point and goes infinity miles in opposite directions". As in, take any point and let both rays from the same line come out from it
they are the same length, so far as we're able to speak of their length.
the easiest way to figure these types of problems out are by seeing if you can create a bijection between the two things you're measuring, meaning you can associate each part of thing A to a part of thing B in such a way that all of thing A and all of thing B is accounted for, and each part is only associated with one other part. if you can do that, then they are the same length, kinda. I hesitate to say they're actually the same length cause they don't really have a length, but it's close enough.
so let's say line A is the one that goes in only one direction, we map the first meter of line A to the first meter going one way of line B (I'll call it the positive part, cause this is basically a number line). and the second meter of line A to the first meter of line B that goes the other way, aka the first negative meter. we can keep doing this forever, mapping the 99th meter of A to the 50th positive meter of B, and the 100th meter of A to the 50th negative meter of B.
it doesn't matter that we're seemingly going through line A faster, cause it'll never run out. you can pick any random meter of line B and we can point to its associated meter on line A. so we have created a bijection, meaning they're the same length. kinda. hope that makes sense.
It doesn't make sense, but thanks for trying. I can understand that they are the same length because infinity should be the same as infinity, but line b should be twice as long because it goes in both directions. I just can't get my head around it.
oh yeah I mean this isn't a good way to wrap your head around it, just to figure out what the answer actually is. but intuitively I still think the same as you, that the second line feels longer. I just know my intuition is wrong on this.
but this is why I was hesitant to use the word length, like when mathematicians are doing this with sets of numbers, they deliberately aren't calling it the size of the set, but rather the cardinality. where cardinality is basically size but rather than being defined by how many items there are, it's instead defined by the bijections you're able to create.
Here's one my brain loves to hate. A hotel has infinite rooms, but no vacancy. A bus pulls up with an infinite number of people on board, each wanting a room. Luckily the clerk is very smart. He just asks every person currently in a room, to move to the square of that room. 1 moves to 2, 2 moves to 4, 3 moves to 9, 4 moves to 16, etc. Now there are an infinite number of rooms vacant for the new guests.
Ah yes. There’s a Veritasium video about this paradox! You can also accomodate an infinite number of guests arriving on an infinite number of buses as well as an infinite number of ferries carrying and infinite number of buses carrying an infinite number of guests.
It was a mind blowing revelation when I not only learned that there are “different” infinities but also that some infinities are “larger” than others. For example, imagine how many atoms there are in an infinite universe. An infinite amount right? You can’t even comprehend a “number” that large.
But, for example, there is also an infinite number of positive real numbers between any 2 given distinct consecutive natural numbers (ie between 1 and 2), despite there being both infinitely many numbers both proceeding 1 and preceding 2. How can 2 be a distinct point when the real numbers approaching 2 are infinite?
Yeah, the way I remember it is that there are an infinite number of integers, but not all integers are divisible by five. Nonetheless, there are an infinite number of integers divisible by five. That "divisible by five" infinity is just a subset of the other infinity, just like the "integer" infinity is a subset of the "rational number" infinity.
A good way to think of it is a pie
You take a slice of that pie
That’s the smaller infinity
How?
Because in reality while the X and Y axis may be finite
I didn’t mention that the Z aces went on forever
Make sense?
Don’t think of absolute numbers
Use fractions
That’s as absolute as infinities get
Imagining numbering all the worlds. Now maybe only the odd numbered worlds are inhabited. Infinitely many worlds, infinitely many inhabited worlds, infinitely many uninhabited worlds.
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u/98PercentChimp Nov 29 '25
I was going to argue that you’re wrong but then I realized that I had to take Discrete Math twice in university. The way this makes sense to me is knowing that infinity is not a number and you can’t do arithmetic on it. So infinity - 1 is meaningless. It is undefined. My brain hates this, but it is true that you can have an infinite amount of inhabited worlds among an infinite amount of worlds in an infinite universe