Intereting Posts

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$\sum a_i \ln(b_i) \leq \sum a_i \ln(a_i)$ with $\sum a_i = \sum b_i = 1$
approximating a maximum function by a differentiable function
Joseph Kitchen's Calculus (reference)

I have a question about a passage of Just/Weese. In the following, let $\mathcal M = \langle M, E \rangle$ be a model of ZFC. Here is the first half of what I’m about to ask a question:

So I asked myself the question of whether the other way around is also true: does every element $a \in M$ correspond to a subset $A \subset M$? Given that the argument for the other direction is purely based on the size of the sets one would expect that yes. But since I constructed a counter example, I think the answer is no (unless my counter example is incorrect (please correct me)):

- The cardinality of the set of all finite subsets of an infinite set
- Why is there no function with a nonempty domain and an empty range?
- Question about sets and classes
- Cardinality of power set of empty set
- How to Understand the Definition of Cardinal Exponentiation
- In a Venn diagram, where are other number sets located?

Let $M = \{ \{\varnothing\}, \{\varnothing, \{\varnothing\}\}, \{\{\varnothing\}\} \}$, $E = \langle \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\rangle, \langle \{\{\varnothing\}\}, \{\varnothing, \{\varnothing\}\}\rangle \}$.

Then $A = \{ \{\varnothing\}, \{\{\varnothing\}\} \}$ corresponds to $a = \{\varnothing, \{\varnothing\}\}$ But the element $a=\{\varnothing\}$ does not correspond to any subset $A \subset M$.

Now look at this:

In particular, “…Let $A$ be the subset of $M$ that corresponds to $a$. …”. But there doesn’t necessarily have to be such an $A$, so this passage is not right.

What am I missing? Thanks for your help.

Corresponds is given as follows:

- How to show if $A \subseteq B$, then $A \cap B = A$?
- Cardinality of $\mathbb{R}$ and $\mathbb{R}^2$
- What does the Axiom of Choice have to do with right inversibility?
- How to represent each natural number?
- Correct formulation of axiom of choice
- How can an ordered pair be expressed as a set?
- What are good books/other readings for elementary set theory?
- Is it true that $A \cup (B - C) = (A \cup B) - (A \cup C)$?
- How do I prove $A \cup\varnothing = A$ and $A \cap\varnothing = \varnothing$
- How many ways to merge N companies into one big company: Bell or Catalan?

Each $a \in M$ will correspond to $\{ b \in M : b \mathrel{E} a \} \subseteq M$. (All that is required here is that the relation $E$ is *extensional*, so that different elements of $M$ will have different sets of elements $E$-below them.)

Note, also, that the model you have constructed does *not* satisfy extensionality. Indeed we have that

$$\begin{gather}

\{ \varnothing \} \mathrel{E} \{ \varnothing ,\{ \varnothing \} \} \\

\{ \{ \varnothing \} \} \mathrel{E} \{ \varnothing , \{ \varnothing \} \}

\end{gather}$$

and there are no other relations, so that $\{ \varnothing \}$ and $\{ \{ \varnothing \} \}$ cannot be differentiated by what is $E$-below them.

First we need to figure out what “corresponds” means.

It means that $A=\{b\in M\mid b\mathrel{E}a\}$. Indeed there are only a few of those. One example is that if $\langle M,E\rangle$ has non-standard $\omega$, then there is an $A$ which corresponds to the “true” $\omega$, but that $A$ is definitely not going to be any element of $M$, as we discussed in a previous thread.

As for the second question, *assume* that there is such $A$. For example, if $a$ is the set $\langle M,E\rangle$ sees as the real numbers, $A$ itself might be countable, but $a$ is uncountable in $M$.

(You may be interested in the questions of pichael and the answers I have provided to some of them about arguing internal vs. external points of view)

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