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What does it mean to be a "standard" theory? By any account, the theory ZFC + V=HOD already is one of the "standard" theories. The axiom V=HOD is intensely studied by set theorists; it appears as a hy …

The answer is Yes. The simple fact is that it is much easier to interpret ZFC from low-complexity assertions than one might expect. For example, even PA+Con(ZFC) can already interpret ZFC, since one c …

These axioms thus have independent interest, and yet fulfill the predicted tower of consistency strength. …

If I understand you correctly, the answer is yes, providing that you don't just add those axioms directly to $T$, but instead add the assertion $\phi^D$ for each axiom of $T$. …

Your transfer principle contradicts the axiom of foundation. To see this, observe that under foundation, every nonempty finite set $x$ has an $\in$-maximal element, a set $z\in x$ with $z\notin u$ fo …

No. Consider the model $\langle V_{\omega_1},\in\rangle$. This is a model of Zermelo's theory, but all ordinals in it are countable. Thus, it satisfies the ordinal-replacement axiom, since the image o …

set-theoretic analogue of mereology to be the inclusion relation $\subseteq$ rather than $\in$, since after all, $\subseteq$ is reflexive and transitive and admits relative complements, and these are all axioms … of Classes outlines an approach to mereology by which the parthood relation is understood as $\subseteq$ in a system of class theory similar to Gödel-Bernays, and Lewis's system satisfies many of the axioms …

In my blog post Transfinite recursion as a fundamental principle in set theory, I prove that the principle of transfinite recursion is equivalent to the replacement axiom.

Really, what you want to have is a set-theoretic structure on top of the universe, with classes and meta-classes and hyper-classes and so on, in a set-theoretic realm continuing to build into ranks ab …

Now, by the usual Zermelo axioms, we can project onto the second coordinate (this does not need replacement), and get the desired instance of replacement. Update. …

The assertion that there is (or is not) a truth predicate is expressible in the second-order language of set theory, but assuming consistency, not by any first-order assertion. Second-order. In the …

Thus, one writes down the axioms of the situation. … And most applications of nonstandard analysis that I have seen can be undertaken using only the usual nonstandard axioms. Nonstandard Axiomatic approach. …

We require only that $T$ has a computably enumerable list of axioms, that $T$ is strong enough to express the decision problem $A$, and that $T$ is sufficiently sound. … This argument works even as you strengthen your axioms, and so it doesn't matter whether you use PA or ZFC or ZFC plus large cardinals or what have you. …

In weak set theories, using classical logic and interpreting "small subclass" as "set", this principle amounts to an alternative formulation of the collection axiom. For example, in Zermelo set theory …

Assuming that ZFC is consistent, then your system will not settle the question of whether 1cof is an element of 2cof or vice versa. One can see this by observing that from any model of ZFC we can make …