# Tag: “Math Improve”

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- Why do theories extending $0^\#$ have incomparable minimal transitive models? June 17, 2024This question says that the theory ZFC + $0^\#$ has incomparable minimal transitive models. It proves this as follows my emphasis): [F]or every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^M = α < ω_1$, the intersection of all such $M$ equals $L_α$, and furthermore a subset of $L_α$ is definable (with parameters) in all […]Arvid Samuelsson
- Question on Theorem 1.11 in Baby Rudin June 17, 2024Definition 1.7 in Baby Rudin says: Suppose S is an ordered set, and E $\subset$ S. If there exists a $\beta \in$ S such that $x \leq \beta$ for every $x \in E$, we say E is bounded above and call $\beta$ an upper bound of E. In the proof of Theorem 1.11 in Baby […]Dr. J
- Are Heyting algebras models or structures? June 17, 2024On the nlab page on Heyting algebras https://ncatlab.org/nlab/show/Heyting+algebra, it is written "a Heyting algebra is precisely a model of the intuitionistic propositional calculus". Just to be sure, is this really an abuse of terminology to say that they are structures for LJ?Anserin
- A statement that seems to be neither true nor false June 16, 2024Let us consider a statement A that says "Statement A is false". Now is the statement A true or false? If it's false then statement that says "Statement A is false" is true therefore statement A is true which is a contradiction. If statement A is true then statement that says "Statement A is false" […]Filip Mazurek
- Why can't we define arbitrarily large sets yet after defining these axioms? (Tao's Analysis I) June 16, 2024In Tao's Analysis I I am very confused why he says we do not have the rigor to define arbitrarily large sets after defining the below 2 axioms: Axiom 3.4 If $a$ is an object, then there exists a set $\{ a \}$ whose only element is $a$, i.e., for every object $y$, we have […]Princess Mia
- Gödel theorem as mentionned in Harthorne's Geometry: Euclid and Beyond June 16, 2024I am reading Geometry: Euclid and Beyond by R. Hartshorne and there is a section discussing the possible axiomatizations for planar geometry. In the following paragraph, Hartshorne mentions Gödel's results (p. 71): Finally, one can ask whether the axiom system is complete, which means, can every statement that is true in every model of the […]Weier
- Is the naive axiomatization of a sort of classes in ZFC, equiconsistent with ZFC? June 16, 2024Is the following extremely naive axiomatization of ZFC+Classes equiconsistent with ZFC? If so, can my attempted proof be turned into a correct argument of this fact? If not, I'm really curious why not. I deliberately chose the axioms so that interaction between the $S$ and $C$ sorts would be "unidirectional" and our new classes would […]Greg Nisbet
- Showing equal things are the same object (Tao's Analysis I) June 16, 2024I am reading Tao's Analysis I, and throughout he uses the notion of an object to define even axioms, treating it as an intuitive notion without explaining it. Based on the axioms that he has given for equality, I am trying to reason that if 2 things are equal, they are the same mathematical object, […]Princess Mia
- Why can't three-valued logic (ternary logic) simply have only two truth values? June 15, 2024Consider the statement: P ∧ ¬P ⊢ Q where: P is any proposition, ¬P is the negation of P. Q is another proposition. Wouldn't proving both P and ¬P to be true simply lead to a new proposition Q, rather than introducing a third truth value? Even if we follow the principle of explosion, wouldn't […]Sam
- A generalized algorithm to convert a formula in algebraic normal form to an equivalent formula that minimizes the number of bitwise operations June 15, 2024In this question, “bitwise operation” means any operation from the set {XOR, AND, OR}. The NOT operation is not included because it can be represented as a single XOR with 1. Given an arbitrary formula $f$ (in algebraic normal form), how to find an equivalent formula $g$ that minimizes the number of bitwise operations? For […]lyrically wicked

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