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Please don't get put off by the length, all the questions are quite simple, but given the quasi-mathematical context I tried to be precise with the formulation. The more mathematically interesting title question (and the one that's most important for my purposes) is the last one, so if anything please take a look at the end.

Here goes. In his paper Models and Reality (1980) pg. 468 (http://www.jstor.org/stable/2273415) Hilary Putnam states and proves the following theorem:

Theorem. $ZF$ plus $V=L$ has an $\omega$-model which contains any given countable set of real numbers.

I suspect his terminology might be idiosyncratic, so I'll point out that by an $\omega$-model he means "a model of set theory in which the natural numbers are ordered as they are 'supposed to be'; that is, the sequence of 'natural numbers' of the model is an $\omega$-sequence."

My first (not-so-interesting) question is this:

Is this a proper model-theoretic theorem?

By "proper" here I mean a theorem that is about the properties of a countable set of (first-order) sentences as they are reflected in the properties of their models. That is to say, in the case of Putnam's theorem (which I think is improper), it seems to me that it doesn't really say anything about $ZF$ - instead it describes the metatheory in which it is interpreted using model-theoretic language (and succeeds in doing so by assuming an outer $\omega$ and $\mathbb{P}(\omega)$ which exist independently of the metatheory or the particular set theory being interpreted.)

And the related question:

If I am wrong and it is a proper theorem, then how would one put it in more contemporary model-theoretic terminology?

Now to the title question. In his "proof" (quotation marks because I'm not yet sure whether it is a proof or a plausibility argument) of the Theorem, after reducing the statement to something equivalent, he writes:

Now, consider [the sentence '$\mathcal{M}$ is an $\omega$-model for $ZF$ plus $V=L$ and $s$ is represented in $\mathcal{M}$'] in the inner model $V=L$. For every $s$ in the inner model-that is, for every $s \in L$-there is a model-namely $L$ itself-which satisfies "$V=L$" and contains $s$. By the downward L-S Theorem, there is a countable submodel which is elementary equivalent to $L$ and contains $s$.

So here's my question:

Are we allowed to use the downward LST on a proper class-sized inner model such as $L$ in the above case?

In general it seems to me obviously not - any version of the strong LST uses some notion of cardinlaity cardinality which surely cannot apply to $L$. What am I missing here? Putnam does add that strictly speaking the Skolem hull construction is also needed, but I don't see how that would help. Can it?

I will tag this as a reference request too, in case someone knows whether this theorem has been published elsewhere - by Putnam or otherwise. (His footnote says that he proved it in 1963 but provides no more information.)

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# Is this a proper application of the Lowenheim-Skolem Theorem to a proper class?

Please don't get put off by the length, all the questions are quite simple, but given the quasi-mathematical context I tried to be precise with the formulation. The more mathematically interesting title question (and the one that's most important for my purposes) is the last one, so if anything please take a look at the end.

Here goes. In his paper Models and Reality (1980) pg. 468 (http://www.jstor.org/stable/2273415) Hilary Putnam states and proves the following theorem:

Theorem. $ZF$ plus $V=L$ has an $\omega$-model which contains any given countable set of real numbers.

I suspect his terminology might be idiosyncratic, so I'll point out that by an $\omega$-model he means "a model of set theory in which the natural numbers are ordered as they are 'supposed to be'; that is, the sequence of 'natural numbers' of the model is an $\omega$-sequence."

My first (not-so-interesting) question is this:

Is this a proper model-theoretic theorem?

By "proper" here I mean a theorem that is about the properties of a countable set of (first-order) sentences as they are reflected in the properties of their models. That is to say, in the case of Putnam's theorem (which I think is improper), it seems to me that it doesn't really say anything about $ZF$ - instead it describes the metatheory in which it is interpreted using model-theoretic language (and succeeds in doing so by assuming an outer $\omega$ and $\mathbb{P}(\omega)$ which exist independently of the metatheory or the particular set theory being interpreted.)

And the related question:

If I am wrong and it is a proper theorem, then how would one put it in more contemporary model-theoretic terminology?

Now to the title question. In his "proof" (quotation marks because I'm not yet sure whether it is a proof or a plausibility argument) of the Theorem, after reducing the statement to something equivalent, he writes:

Now, consider [the sentence '$\mathcal{M}$ is an $\omega$-model for $ZF$ plus $V=L$ and $s$ is represented in $\mathcal{M}$'] in the inner model $V=L$. For every $s$ in the inner model-that is, for every $s \in L$-there is a model-namely $L$ itself-which satisfies "$V=L$" and contains $s$. By the downward L-S Theorem, there is a countable submodel which is elementary equivalent to $L$ and contains $s$.

So here's my question:

Are we allowed to use the downward LST on a proper class-sized inner model such as $L$ in the above case?

In general it seems to me obviously not - any version of the strong LST uses some notion of cardinlaity which surely cannot apply to $L$. What am I missing here? Putnam does add that strictly speaking the Skolem hull construction is also needed, but I don't see how that would help. Can it?

I will tag this as a reference request too, in case someone knows whether this theorem has been published elsewhere - by Putnam or otherwise. (His footnote says that he proved it in 1963 but provides no more information.)