Written work: The following problems are SUGGESTED for practice (they could be on the final!), but will NOT be collected.
Sec 12.6 p216: 1, 2
Sec 13.1 p222: 1, 4, 5
WeBWorK – none
OpenLab – none
Project Reflection – Due before the final exam, Tuesday 12/19.
Handy Links
Logic on Math StackExchange
- 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? (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
- Can an Infinite Set of Finite Elements Exist? [closed] June 15, 2024Where have I gone wrong? I aim to demonstrate a contradiction in the assertion that an infinite set of finite elements can exist, specifically focusing on the claim that the set of natural numbers comprises an infinite set of finite numbers. The natural numbers used in the following arguments do not include zero. ${\aleph_0}$ is […]Igor Zimbler
- 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
- How to formalize unary intersection operator? June 14, 2024How can we formalize the $\cap$ (unary intersection) operator? Following is my attempt. I define/construct the function $~\cap : P(P(U)) \to P(U)$ such that: $\forall a\in P(P(U)): \forall b\in U: [b\in \cap a \iff \forall c\in a:b\in c]$ where $U$ is the underlying set being considered, and $P$ is the powerset operator. $P(P(U))$ can be […]Dan Christensen
- the difference of explicit and implicit definition of functions logically June 13, 2024The fundamental analysis book I'm reading has a section on the difference between the implicit and explicit definitions of functions. The implicit definition of a function f specifies what property $ P(x,y) $ links the input x with the output $f(x)$. How is the implicit definition different from an explicit one, showing how one generates […]roro
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