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
- Understanding a proof in propositional logic May 25, 2024I would like help understanding two related claims being made in the following proof, found in the fourth edition of Elliot Mendelson’s introduction to mathematical logic, which I am reading on my own time to help my studies in philosophy of math and philosophical logic. Here is the proof: What I do not understand about […]Joa
- Is there a Gödel sentence for every model? May 25, 2024Gödel’s first incompleteness theorem states that in a formal system of arithmetic there is at least one formula that can be neither proven or disproven within that system. The Gödel sentence is an example of such a unprovable formula. It is (implicitly) about natural numbers where it is (indirectly) true (although unprovable). As the system […]Barney
- How to justify why succession and addition cannot be circularly defined like this? May 24, 2024I am reading Tao's Analysis I, in which he states: One may be tempted to [define the successor of $n$ as] $n + 1$. . . but this would introduce a circularity in our foundations, since the notion of addition will be defined in terms of the successor operation. Say we did define $S(n) = […]Princess Mia
- another werewolf logic problem where werewolves lie and humans tell the truth [closed] May 23, 2024you meet three londoners named Wess, Xandra, and Yuri, each of whom makes a statement. Each of the londoners is either a human or a werewolf. Humans always tell the truth and werewolves always lie. Wess says "There is exactly one werewolf and it is not me" Xandra says "Yeah, Wess is a human" Yuri […]Lynn Wooldridge
- Functions which commute with arbitrary unions May 23, 2024Let $A,B$ be sets and $f: \mathcal{P}(A) \to \mathcal{P}(B)$ be a function. What does it mean to say that such a function "commutes with arbitrary unions"? I read this expression in Exercise 1.1, p.11 of the book "Abstract Algebraic Logic", by Font. I presume it means for that for any $X \in \mathcal{P}(A)$ we have […]user65526
- Necessity of universal quantifier to represent a theorem with logic symbols May 23, 2024I have a preference to reduce the proof steps of a theorem, and the theorem itself, into logic symbols as much as possible. Not just because it is aesthetically appealing, but because it makes makes things simpler and faster when studying, or when one wants to write down any proposition without excess. Since i started […]Davi1399
- Is a single unprovable formula enough for consistency? May 23, 2024In axiomatic logic, the principle of explosion says that any formula of a system can be proven to be true if the system contains a contradiction. So, in an inconsistent system, both a statement and its negation are derivable from the axioms. I am wondering whether we can say that a single unprovable formula of […]Barney
- Linear logic with contraction May 23, 2024What is the proof system obtained when adding a contraction rule to linear logic or removing the weakening rule from intuitionistic logic? In other words, what goes in the missing cell of the following table? No contraction Contraction No weakening Linear logic ? Weakening Affine logic Intuitionistic logic Intuitively, this will mean that one has […]Georg
- "Irreducible" axiomatic systems? May 23, 2024So I was playing with axiomatic / formal systems and wanted to look into this property about formal systems which could be called "irreducibility". It is similar to independence in that it restricts the axioms from having redundant information. Here is how I define the property. Suppose we have a theorem $T$ in our system, […]qucchia
- How can something be both a primitive notion and be axiomatically defined like it is here? May 23, 2024This resource states that In mathematics, the notion of a set is a primitive notion. That is, we admit, as a starting point, the existence of certain objects (which we call sets), which we won’t define, but which we assume satisfy some basic properties, which we express as axioms. I am confused why we can […]Princess Mia
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