**Week 4 Assignments**

**Written work** – none

**WeBWorK **– Assignment #3 and Assignment #4, due Tuesday, September 20th, at midnight.

**OpenLab **– none

**STUDY** – for your first exam, taking place next Thursday, 9/22, during the first hour of class.

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**Week 4 Assignments**

**Written work** – none

**WeBWorK **– Assignment #3 and Assignment #4, due Tuesday, September 20th, at midnight.

**OpenLab **– none

**STUDY** – for your first exam, taking place next Thursday, 9/22, during the first hour of class.

- Is it believed that statements exist whose independence cannot be proven even if we start using large cardinal axioms? September 8, 2024Here are some observations: Gödel's proof of his theorem invokes an unprovable statement. In a stronger system that accepts the original system was consistent (which may be the result of a large cardinal), it is possible to prove the undecidability of this statement. Secondly, AC and CH are provably independent of ZF and ZFC when […]Pineapple Fish
- How could I formally express: System F cannot express universal quantification over things that are not types? September 8, 2024I'm trying to understand exactly why it is that https://ncatlab.org/nlab/show/computational+trilogy states that quantification requires dependent types, and why this wouldn't be possible to achieve with System F. From what I've managed to gather, System F can express quantification, but only over types, and not over elements or predicates. I'm looking for a way to formalize […]shintuku
- Can we always prove whether a statement is undecidable by adding consistency strength? September 8, 2024So, if a statement is independent of a list of axioms then the system cannot prove it. For example, CH is independent from ZFC. But if we accept ZFC is consistent then we can prove CH must be independent of it. Of course because of Gödel's incompleteness theorem we can cook up more statements that […]Pineapple Fish
- construction of truth table for $(P \implies Q) \vee R)$ September 8, 2024construction of truth table for $(P \implies Q) \vee R)$ So here's my attempt :- \begin{array}{|c|c|c|c|c|} \hline P & Q & P \implies Q & R & (P \implies Q) \vee R \\ \hline T & F & F & {F\\T} & {F\\T} \\ \hline T & T & T & {T\\F} & {T\\T} \\ […]math and physics forever
- Are Compactness for FOL and the Pumping Lemma for RL/CFL two instances of the same phenomenon? September 7, 2024As title states, I'm curious whether my intuition for the Compactness result for FOL and the Pumping Lemma for RL/CFL being two expressions expressions of the same phenomenon (that is: an attempt to capture the infinite through finitary means) might be wellfounded. The idea is precisely that: In Compactness one gets that, due to the […]Sho
- What is a predicate in first-order logic, formally? [duplicate] September 7, 2024Recently, I am trying to gather together a consistent 'foundation of mathematics' from several sources/books/scripts which satisfies me as a 'working mathematician'. When, for example, I want to start with set theory, I have to formulate the axioms of ZF (or ZFC). For this - in particular for the axiom schema of specification - I […]mrpotato
- how to prove well-ordering isomorphic theorem? September 7, 2024I am reading well-ordering isomorphism theorem in Kunen's "set theory : an intro function to independence proofs(1980)" here, "pred" refers to initial segment for an element x in A under relation R. at first glance, if we define f in the above way, it seems that if A's cardinal less than B, then (b) will […]유준상
- Inclusion in a set using formal logic September 7, 2024I am having trouble understanding the definition of a subset in terms of formal logic, I want to prove that these statements are logically equivalent: Let $A,B$ be subsets of $E$ $$ A\cap B = \emptyset \\ \equiv \\ A\subset (B\cap \bar{A}) \\ \equiv \\ B\subset (A\cap \bar{B}) $$ Here is my question: Take $A\subset […]realreal
- Continued fractions inspired propositions: What do these have in common in terms of truth values? September 6, 2024Suppose a single logic binary relation (i.e., any of those in $\{\lor, \land, \to, \leftrightarrow\}$) were to take the place of addition in a continued fraction, with $\neg$ as taking multiplicative inverse and the values in the fraction being propositions. For example, for $\lor$, a "continued fraction" would be $$[A; B, C]_\lor:=A\lor\neg(B\lor\neg C).$$ (Here the […]Shaun
- Help on a proof using implication [duplicate] September 6, 2024I am trying to prove the following statement: Let $a,b\in \mathbb{R} \setminus \{0\}$ $$ [(\exists n \in \mathbb{N} : a=nb) \wedge (\exists n \in \mathbb{N} : b=na)]\Rightarrow a = b $$ Here is what I've come up with: $$ [(\exists n \in \mathbb{N} : a=nb) \wedge (\exists n \in \mathbb{N} : b=na)]\Rightarrow a = b […]realreal

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