Slider's Shortform

Slider

This is a special post for quick takes by Slider. Only they can create top-level comments. Comments here also appear on the Quick Takes page and All Posts page.

Let me muse about infinite amounts so that I am not waiting to spill them on a random infinite-adjacent thread.

Someone offers you a choice of 10 fish now or 1 fish a day. Which option lets you eat more fish?

Questions of these kind seem to have properties of:

Even a ludicrous amount of one-off fish will lose to a steady income of fish
Bigger fish streams are bigger. Having 3 fish per day is better than 1 fish per day

Now one could make the case that there is a scheme where each day you grab the new fish and give them a number. For any finite fish per day this can clearly be done. Somebody could come to say that "Hey I can put any fishstream in 1-to-1 correspondence to the natural numbers therefore there is an equal amount of fish in any stream". Yet it seems if somebody says that their 3 fish per day stream is not an advantage over my 1 fish per day stream, I would not be convinced and think they still have got more.

This makes me think that 1-to-1 correspondence doesn't really fit with what my brain points to be size. Or some of the intuitive notions come appart when we enter infinity land.

There is also the construction if somebody promises on each day start a new stream of fish. Then for any stream of fixed finite amount of fish per day promised there is a day where the "growing promise" gets you that and then some more afterwards.

I am guessing/feeling that 2 streams of 1 fish vs 1 stream of 2 fish is connected/represented by ω+ω=2ω and the "growing promise" might be connected to ω*ω

Is there/could there be some notion of size that would let me say that ω+ω is smaller than ω*ω and could I deal with people saying that "mappable to integers" is better notion of size?

2Slider5y

Why 12ω2+12ω and not any other? What kind of stream would correspond to ω2 ?

1Taleuntum5y

You can just pretend that ω is finite and plug it into the formula for the partial sum.n∑i=1i=12n2+12n, so ω∑i=1i=12ω2+12ω. If they were to give the ith odd number amount of fish on the ith day (1,3,5,7,9...), then you would have ω2 amount of fish, because n∑i=12i−1=n2. The two links I posted about the handling of infinite divergent series go into greater detail (eg. the question of the starting index).

The links are very on point for my interest thanks for those. Some of it is in rather dense math but alas that is the case when the topic is math.

At one point there is a constuction where in addition to having series of real numbers to define a hyperreal (r1,r2,r3...)=h1 we define a series of hyperreals (h1,h2,h3...)=d1, in order to get a "second tier hyperreal". So I do wonder whether the "fish gotten per day" is adeqate to distinguish between the scenarios. That is there might be a difference between "each day I get promised an infinite amout of fish" an... (read more)

Slider's Shortform

4

Slider's Shortform

4