# Comprehension Question

#### [email protected]

I thought about it and just figured out a way to count $A$. Each element of $A$ is a finite set constructed of left brackets, $\{$, and right brackets, $\}$, so in replacing each left bracket with a $0$ and each right bracket with a $1$, we can convert each element of $A$ to a unique finite binary string:

Let $f(\{) = 0$ and let $f(\}) = 1$.

Arbitrarily, let $a = \{ \{ \{ \} \}, \{ \{ \}, \{ \{ \} \} \} \}$.

Then $f[a] = f(\{) f(\{) f(\{) f(\}) f(\}), f(\{) f(\{) f(\}), f(\{) f(\{) f(\}) f(\}) f(\}) f(\}) = 00011001001111$.

Define $A = \{ x : \text{ there exists some model of ZF set theory such that } x \text{ is an element of the model and } f[x] = \text{ a finite binary string} \}$.

Now that my definition is clear, I am curious, what is the difference between $L_w$ and $A$?
I don't see how any of this ressembles $L_\omega$ in the slightest.

#### AplanisTophet

I'm experiencing restricted comprehension.
Yup. I don't adhere to restricted comprehension (such as the axiom schema of specification, etc.) in defining $A$, but since $A$ is enumerable I don't see the point of doing so.

It matters not that I chose to apply a function to elements of the Alphabet as opposed to sets within the model. My definition is precise. Give me any set and we are able to tell precisely whether or not it is an element of $A$.

The difference between $L_{\omega}$ and $A$ as I understand it would be any infinite elements of $L_{\omega}$. That's it.

#### Maschke

Yup. I don't adhere to restricted comprehension ...
I think you're the one suffering from restricted comprehension here.

#### AplanisTophet

I think you're the one suffering from restricted comprehension here.
Not if you still don't get what $A$ is. At this point, that would be you. I'm not sure you know what $L_{\omega}$ is either at this point, though I'm asking because I myself would like clarification. I've noted that $L_{\omega} = V_{\omega}$ and I want to make sure I understand why.

#### Maschke

Not if you still don't get what $A$ is. At this point, that would be you. I'm not sure you know what $L_{\omega}$ is either at this point, though I'm asking because I myself would like clarification. I've noted that $L_{\omega} = V_{\omega}$ and I want to make sure I understand why.
You haven't responded to any of my direct questions so there's not much left for me here.

#### AplanisTophet

You haven't responded to any of my direct questions so there's not much left for me here.
I gave you the domain of $f$. What else did you ask that I havenâ€™t given you?

#### AplanisTophet

A listing of the finite binary strings, $F$, could be used to construct $A$.

$f^{-1}[0] = \{$. This is not a set, so not in $A$.

$f^{-1}[1] = \}$. This is not a set, so not in $A$.

$f^{-1}[01] = \{\}$. This is a set, so in $A$.

.
.
.

$A = \{ x \in F : f^{-1}[x] \text{ is a set} \}$.

So what sets are in $L_{\omega}$ that are not in $A$?

#### Maschke

I gave you the domain of $f$. What else did you ask that I havenâ€™t given you?
It would be immensely helpful to me if you would go over each of my posts in this thread, and each time you see a sentence ending in '?', please give a clear, straightforward response in simple, declarative sentences. Don't add anything and don't assume anything. Just answer each question.

#### AplanisTophet

It would be immensely helpful to me if you would go over each of my posts in this thread, and each time you see a sentence ending in '?', please give a clear, straightforward response in simple, declarative sentences. Don't add anything and don't assume anything. Just answer each question.
Sounds fun. Will do at some point.

#### Maschke

Sounds fun. Will do at some point.
Meanwhile, did you find the link on the hereditarily finite sets of interest?

Is your idea different in some way?

Also can you please say in English what #5 means?

What is the domain of f?

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