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The lattice of sets of natural numbers is rich (2021)

jdh.hamkins.org

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Re: The lattice of sets of natural numbers is rich (2021)

#4

Post by michael0church »

munchler wrote: post #3

What a beautiful illustration. It makes intuitive the very abstract concepts discussed in the text. It’s fun to zoom in and browse around the structure.

It’s also genuinely surprising. We’re used to thinking of the countable as the small infinity, which it is, and yet a structure we feel like we can visualize contains so much complexity.

There is also, weirdly, a way in which massive finite numbers like TREE(3) “feel” larger than N, and large countable infinities “feel” larger than w_1, even though the opposite is clearly true.

Re: The lattice of sets of natural numbers is rich (2021)

#5

Post by voidmain »

michael0church wrote: post #4
munchler wrote: post #3

What a beautiful illustration. It makes intuitive the very abstract concepts discussed in the text. It’s fun to zoom in and browse around the structure.

It’s also genuinely surprising. We’re used to thinking of the countable as the small infinity, which it is, and yet a structure we feel like we can visualize contains so much complexity. There is also, weirdly, a way in which massive finite numbers like TREE(3) “feel” larger than N, and large countable infinities “feel” larger than w_1, even though the opposite is clearly true.

The visualization is of the power set, which is uncountable.

Re: The lattice of sets of natural numbers is rich (2021)

#8

Post by michael0church »

voidmain wrote: post #5
michael0church wrote: post #4

Earlier quoted context omitted.

It’s also genuinely surprising. We’re used to thinking of the countable as the small infinity, which it is, and yet a structure we feel like we can visualize contains so much complexity. There is also, weirdly, a way in which massive finite numbers like TREE(3) “feel” larger than N, and large countable infinities “feel” larger than w_1, even though the opposite is clearly true.

The visualization is of the power set, which is uncountable.

Right. But because it’s the smallest structure of its type (speaking loosely) it feels like something we should have a grasp on, even though it contains more complexity than we could ever describe or compute with (since both of those are countable.)

Re: The lattice of sets of natural numbers is rich (2021)

#9

Post by zaebal »

michael0church wrote: post #4
munchler wrote: post #3

What a beautiful illustration. It makes intuitive the very abstract concepts discussed in the text. It’s fun to zoom in and browse around the structure.

It’s also genuinely surprising. We’re used to thinking of the countable as the small infinity, which it is, and yet a structure we feel like we can visualize contains so much complexity. There is also, weirdly, a way in which massive finite numbers like TREE(3) “feel” larger than N, and large countable infinities “feel” larger than w_1, even though the opposite is clearly true.

TREE(3) is unimaginably small, compared to ω

Re: The lattice of sets of natural numbers is rich (2021)

#10

Post by zygentoma »

zaebal wrote: post #9
michael0church wrote: post #4

Earlier quoted context omitted.

It’s also genuinely surprising. We’re used to thinking of the countable as the small infinity, which it is, and yet a structure we feel like we can visualize contains so much complexity. There is also, weirdly, a way in which massive finite numbers like TREE(3) “feel” larger than N, and large countable infinities “feel” larger than w_1, even though the opposite is clearly true.

TREE(3) is unimaginably small, compared to ω

Well, any natural number is unimaginably small, compared to ω …
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