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This tag is for questions about Gödel's constructible universe $L$, and related constructions such as $L[X]$ and $L(X)$.

2 votes

Can this semi-constructible structure satisfy existence of a measurable cardinal?

This answer is not correct, because it is a subtler matter to ensure that the levels are closed under relative constructibility while also maintaining $L_{\alpha+1}\cap\mathfrak{L}_\alpha=L_\alpha$. …
Joel David Hamkins's user avatar
12 votes
Accepted

Does this ZFC+V=L like theory, have a limit on large cardinal properties?

The answer is no, you cannot have measurable cardinals consistently with your theory. Your theory includes the axiom "V=L or V=L[c] for an $L$-generic Cohen real $c$". This statement is provable from …
Joel David Hamkins's user avatar
11 votes
Accepted

Complexity of definable global choice functions

There is no such phenomenom for $n\geq 2$. The reason is that if a model of ZFC has a definable choice function, of any complexity, then it actually has one of complexity $\Delta_2$. This is because t …
Joel David Hamkins's user avatar
7 votes
Accepted

Impact of coining $L$ in $\mathcal L_{\omega_1, \omega}$ on which large cardinals it can sat...

It seems to me that if you build the constructible universe using $\mathcal{L}_{\omega_1,\omega}$ logic, you will get the inner model $L(\mathbb{R})$. The reason is that every $\mathcal{L}_{\omega_1,\ …
Joel David Hamkins's user avatar
20 votes
Accepted

Are some interesting mathematical statements minimal?

The minimal model, when it exists, also known as the Shepherdson-Cohen model, is the smallest transitive model of ZFC. This model will have the form $L_\alpha$ for some countable ordinal $\alpha$, and …
Joel David Hamkins's user avatar
2 votes

Is the power set axiom essential for constructing L?

Let me address the question in the body of the post, rather than the title question. Namely, you asked whether your theory interprets ZFC. Negative answer with axiom as stated. The natural reading of …
Joel David Hamkins's user avatar
15 votes
Accepted

Can $L$ be defined without parameters?

Yes, the parameter-free version of $L$ gives rise to the same constructible universe $L$. You will still get all of $L$ this way, but it will come more slowly. The reason is that at stage $\alpha+1$, …
Joel David Hamkins's user avatar
7 votes
Accepted

Are all constructible from below sets parameter free definable?

There are two issues with your question. First, your statement "in other words" is not correct, since there are theories whose models have the property that whenever a statement holds of every paramet …
Joel David Hamkins's user avatar
6 votes
Accepted

Terminology for ordinals whose constructible level is the least one satisfying some formula

Let me say first that your concept is similar in spirit to the notion of sententially categorical cardinal appearing in my joint paper J. D. Hamkins and R. Solberg, Categorical large cardinals and th …
Joel David Hamkins's user avatar
8 votes

Elementary countable submodels in Gödel's universe

No, there are many instances of $L_\alpha\prec L_\beta$ without $L_\alpha\prec L_{\omega_1}$. Here is one easy way to construct one. Consider the smallest $\alpha$ that has an elementary extension $L_ …
Joel David Hamkins's user avatar
8 votes

When is $A$ "$L$-ish" whenever $B$ is "$L$-ish"?

This is a fascinating question! I really like your relation. Here is some small progress. (I am hopeful that more definitive answers will appear later). First, you didn't mention it, but for definiten …
Noah Schweber's user avatar
7 votes
Accepted

If we have a class like $L$ but allowing a set number of unbounded quantifiers, is it strict...

Every $L^{\Sigma_n}$, for $n\geq 2$ will be the same as HOD, the class of hereditarily ordinal-definable sets. This is a consequence of the Myhill-Scott theorem, which asserts that if you form the con …
Mohammad Golshani's user avatar
6 votes
Accepted

How similar are large cardinals, over $L$?

Updated answer. I claim that none of the familiar large cardinal notions consistent with $V=L$ are provably $\gamma$-decisive for any $\gamma$. This includes the cases of wordly cardinals, inaccessibl …
Joel David Hamkins's user avatar
6 votes
Accepted

Fine structure question: when do levels of $L$ look "a lot" like each other?

Under your assumptions, the set $E_\kappa^+$ is empty. Indeed, we don't even need the predicates that you mention. (In any case, since $\kappa$ is definable in $L_{\kappa^+}$, the predicate for $L_\ka …
Joel David Hamkins's user avatar
4 votes

Is every ordinal potentially definable?

Great question! The answer is yes. Start with $V=L$, and fix any ordinal $\alpha$. Now go to the forcing extension $V[G]$ obtained by collapsing $\aleph_\alpha^L$ to $\omega$, so that $(2^\omega)^{V[ …
Joel David Hamkins's user avatar

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