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Define the predicate 🤦‍♀️ as the predicate defining 🍕. To be precise, 🤦‍♀️(🧀) means

"for any set 🍌, if ∅ ∈ 🍌 and (for all 🧁 ∈ 🍌 we have 🧁 ∪ {🧁} ∈ 🍌), then 🧀 ⊆ 🍌".

By the problem statement, we are given that 🍕 is the unique set such that 🤦‍♀️(🍕).

Claim 1: for any set 🧀 such that 🤦‍♀️(🧀), we can conclude that 🍕 = 🧀. This follows from the given that that 🍕 is assumed to be unique such set.

Claim 2: 🤦‍♀️(∅). For any set 🍌 satisfying the two properties, we have ∅ ⊆ 🍌, since ∅ is a subset of any other set

Claim 3: 🤦‍♀️({∅}). For any set 🍌 satisfying the two properties, we have ∅ ∈ 🍌, therefore {∅} ⊆ 🍌.

From Claim 1 & 2 we obtain 🍕 = ∅. From claim 1 & 3 we obtain 🍕 = {∅}. Therfore, ∅ = {∅}.

This is false, so we derive at a contradiction.

From a contradiction we can prove anything, in particular that there is no set 🌭 such that |🍕| < |🌭| < |🍔|.

The problem is that there is a mistake in the characterization of 🍕, which is supposed to represent the natural numbers. It is given that it is a unique set satisfying a particular property, but actually every subset of natural numbers satisfies this property. So the assumption that 🍕 is unique is false, and from a given statement that is false we can prove anything.

@FlorisvanDoorn Yes, I see I forgot to specify that 🍕 also has to satisfy the banana property.

@JosephNoonan Yes, that would have fixed it.

@FlorisvanDoorn Man, math with emojis as variables is greatly underrated.

Can I assume that the problem statement is exactly what is written in the current image (copied below)?

Define ❤️(🍎) = { 😎 | 😎 ⊆ 🍎 }

Let 🍕 be the unique set such that, if {} ∈ 🍌 and 🧁 ∈ 🍌 → 🧁 ∪ {🧁} ∈ 🍌 for all sets 🧁, then 🍕 ⊆ 🍌, and let 🍔 = ❤️(🍕).

Prove that there is no set 🌭 such that |🍕| < |🌭| < |🍔|.

@FlorisvanDoorn Yes

This market is surprisingly bullish on someone finding the solution to the Continuum Hypothesis.

@JosephNoonan Dangit I had a feeling this would be a well-known unsolved provlem 🤣

@AshleyDavies Even worse, an unsolvable problem

@JosephNoonan My several pages of scribbling feel vindicated. It’s tantalisingly solvable-looking to my 4-years-recessed-from-set-theory eyes 😄

@JosephNoonan You find it “surprising” that people believed you when you said “Just a simple problem, shouldn't be too hard”?

@JosephNoonan I think there is a typo, 🍌 ⊆ 🍕 instead of 🍕 ⊆ 🍌

@fejfo You're right, I fixed the image.