Set Theory - Blog Posts

Here's another weird model of ZFC relating to the axiom of regularity that fucked me up when I learned about it. The axiom of regularity implies that there can be no set {x_n | n ∈ ω} such that x_{n+1} ∈ x_n for each n ∈ ω.

Let's construct our model. Add to the language of ZFC countably infinitely many constants c_0, c_1, c_2, ... and let Γ be the set of sentences Γ = {c_{n+1} ∈ c_n | n ∈ ω}. We will use the compactness theorem to show that there is a model of ZFC ∪ Γ.

Let Δ be a finite subset of Γ and Let J be any model of ZFC. Since Δ is finite, there is a maximum k such that the sentence c_k ∈ c_{k-1} is in Γ. Add to J the definitions, for each n ≤ k, c_nJ = k - n, and for each n > k, set c_nJ = 0. Then for all 1 < n ≤ k, c_n = k-n ∈ k-n+1 = c_{n-1}, and so J is a model of ZFC ∪ Δ.

Thus, by the compactness theorem, there exists a model of ZFC ∪ Γ.

This is very surprising, and at first glance seems to contradict the axiom of regularity! But what it really means is that the sets x_n from the first paragraph can exist, but they cannot be gathered together in a set.

downward lowenheim-skolem is so fucked up to me. what do you *mean* there's a countable model of first-order set theory

enough with the squares and rectangles

everyone knows that all squares are rectangles and not the other way around, right? well enough with that example. that's just subsets. you can use literally anything.

all squares are rhombi, but not all rhombi are squares

all legs are limbs, but not all limbs are legs

all natural numbers are integers, but not all integers are natural numbers

all of the bodies are buried in wyoming, but not everything buried in wyoming are the bodies

all phones are devices, but not all devices are phones

this is set theory, bitches! X⊆Y does not necessarily imply Y⊆X.

stop saying squares and rectangles, its tiring, say literally anything else, be original.

all squares are non-triangles, but not all non-triangles are squares.

wanted to see if the ai would be able to answer set theory questions. it was pretty close actually! (maybe my wording was confusing? idk)

im going to edit to add more examples btw

YOOOOOOOOOO

(i didn't have the actual => symbol ready to use so i used → instead, but it worked!)

AND IT UNDERSTANDS WHY Q/⊆P omg

Y'all,

I took a class on conlanging this semester, and today was the last day, so the prof presented snippits from conlangs.

My snippit was that really sad quote I made in an earlier post, and I had to explain how the Sêi (the Time God) works and how she exists at all times kinda deal.

But tone of my classmates made a language with classifiers from set theory and discrete math.

Snippits from his conlang included phrases like "for all boys they are smarter than him" and "there exists a boy such that he is smarter than him."

I failed to ask him about what possessed him to do this, but I may get back here after that.