Hi everyone,
I have a commitment tomorrow morning until 9:30, but I will be in my office (or in the classroom) after that. We’ll be doing exam review in class for the first half hour or so, so bring your questions!
See you tomorrow,
Prof. Reitz
Logic on Math StackExchange
- Could ZFC be proven consistent for good by some other system that can prove its own consistency? September 11, 2024Is it impossible that Con(ZFC) could be once upon a time proven by a formal system S that is able to prove its own Con(S), i.e. proving Con(ZFC) for good? S would need to avoid Gödel 2, i.e. needing to be weaker than our arithmetic with natural numbers. But that would mean Cn(S) $\subseteq$ Cn(ZFC) […]Pippen
- Mathematical Skepticism September 11, 2024Let S be an arbitrary formal system. We wanna prove if S is consistent, Con(S), or not, ~Con(S). There are two possibilities only: Some S‘ proves Con(S) but in that case we‘d need to know if Con(S‘) because if ~Con(S‘) then efq and it follows ~Con(S). But so we need S’‘ to prove Con(S’) and […]Pippen
- Discrete Mathematics logical equivalence laws of logic September 10, 2024I'm stuck with a problem and I'm having trouble understanding the problem and I keep getting stuck. Prove that ∼[ (c∨t)∨(p∨∼p)]≡c. Don’t use a Truth Table. Prove a list of logical equivalencies citing a logic law (such as DeMorgan) at each step (see Thm 2.1.1 of our textbook) So far, I have this written out […]FighterJet1
- apparent nonsense in truth table [duplicate] September 10, 2024The truth table of implication of two statements is known as: p q p --> q T T T T F F F T T F F T My questions: Consider p as "it rains" and q as "it is cloudy". It is not necessarily true that if it does not rain then it is […]Hair80
- Graph theory approach to find the best sequence of implications to prove the equivalence of N statements September 10, 2024Suppose we have N statements that are known to be equivalent and suppose we know how difficult it is to prove every implication. which is the easiest way to prove that they are equivalent? This problem can be stated in terms of graph theory in the following way: We have a complete directed graph G […]stebev
- Is a "logical argument" a "rule of inference", and an "inference rule" the same thing? September 10, 2024Difference sources and diciplines use different terminologies, but they all seem to describe the same logical structure like the following: Premise 1 Premise 2 ... --------- Conclusion So are all those terms describe the same thing? Or do they have some subtle differences?Lesley Lai
- Consistency-proof of ZFC September 9, 2024According to Gödel's Second Incompleteness Theorem, ZFC cannot prove its own consistency. However, I have read that a stronger system S, i.e., a system that can fully represent/simulate/formalize ZFC and even go beyond it, can prove Con(ZFC) (of course, only if such a proof results from what S can do beyond ZFC). But this seems […]Pippen
- The prenex form doesn't seem equivalent to the original sentence September 9, 2024I was reading this stackexchange post and am confused about the following logical equivalence. I have tried to generalize what they've done as follows: ∀x[∀y[f(y)] → g(x)] ; for all x, if for all y, f(y) is true, then g(x) is true ∀x[~∀y[f(y)] V g(x)] ∀x[∃y~[f(y)] V g(x)] ∀x∃y[~[f(y)] V g(x)] ∀x∃y[f(y) → g(x)] ; […]work work
- 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? [closed] 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
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