Written work – None.
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OpenLab – OpenLab #5, due Thursday, October 12th, before class.
Logic on Math StackExchange
- Questions on using an ultrafilter to "collapse" a boolean valued model and show that CH is independent of ZFC July 12, 2026I just finished going through a proof that CH is independent of ZFC using a boolean valued model. Here's the proof: Let $P$ be the poset of functions from finite subsets of $\aleph_2 \times \aleph_0 \rightarrow \{0, 1\}$ ordered by reverse inclusion. Let $B$ be its completion. For $a \in \aleph_2, n \in \mathbb{N}, i […]David Lui
- What exactly do we obtain from a concrete formal proof? July 11, 2026Suppose I find a formal proof of a sentence $\varphi$ in PA. What exactly have I obtained from this? Can I say that the mathematical statement expressed by $\varphi$ has actually been proved? At first sight, it seems that all I have shown is a syntactic fact about a formal system: there is a finite […]mathlogic2025
- Is it possible to define a total order on arbitrary arithmetic expressions of a single real variable that respects eventual sign? July 10, 2026By arithmetic expression I mean one consisting of arbitrary combinations of natural number constants, a single real variable x (which may appear any number of times in the expression), and the five arithmetic operators - addition, subtraction, multiplication, division, and exponentiation. I would like to define a total order on such expressions $f$, $g$ such […]stackshifter
- Is there a construction where the universal quantifier is more natural than the existential quantifier? July 9, 2026Universal and Existential quantifiers are dual, so it is usually said that when defining a first-order language, one can take either (or both, of course) one as a primitive. But when taking the categorical/topos theoretical perspective, only the existential quantifier seems to admit a natural construction. Concretely, $\exists_f$ can be defined as a composition with […]prime235711
- What is Tarski's Thesis Saying about logical operators? July 8, 2026Following Gila Sher's treatment of Tarski's Thesis. She defines a structure as tuples $$\langle A,\alpha_1,\ldots\rangle$$ where $A$ is a set (presumably in ZF of some kind, Sher hints at Principia as foundation or "similar theory") with each $\alpha_i$ a "set-theoretic constructs of elements of $A$". Then two structures are isomorphic, denoted $$\langle A,\alpha_1,\ldots\rangle\cong \langle B,\beta_1,\ldots\rangle$$ […]Algeboy
- Exist a unique homomorphism from term algebra to a model? [closed] July 7, 2026Exist a domain $\mathbb{K}$ $\\\mathbb{Q}⊆\mathbb{K}⫋\mathbb{C} \\ ∀a,b∈\mathbb{K}， a^b∈\mathbb{K} ~(if~a=0, ~Re(b)>0)$ Note that $a^b$ is main-branch-choosing exponential function and isn't defined when $a=0 ~Re(b)≤0$. But it doesn't influence saying $a^b$ is closed in $\mathbb{K}$ Because consider {$\mathbb{C},+,\times ,\mathbb{0},\mathbb{1}$}the theory of algebraically closed fields, which is a countable language and has countable elementary submodels. Therefore, we can […]chinesemathnoob
- Reducing adjacent quantifiers of the same type in Peano Arithmetic July 6, 2026I'm currently reading through Reverse Mathematics by John Stillwell. In section 2.8, while defining what it means for a formula in Peano Arithmetic to be $\Pi_m^0$ or $\Sigma_m^0$, he writes that we can "reduce adjacent quantifiers of the same type to a single one," since $$\forall x\forall y\,\varphi(x,y)\iff\forall z\,\varphi(P_1(z),P_2(z))$$ and$$\exists x\exists y\,\varphi(x,y)\iff\exists z\,\varphi(P_1(z),P_2(z))$$ where $P_1$ […]homiamorphism
- The proof-theoretic strength of axiomatic hierarchies relative to $\Delta _1^{ZFC}$ formulas July 3, 2026Consider a $\Delta _1^{ZFC}$ formula $\phi$ such that there exists a transitive model $M$, $M⊨ZFC$ with $M⊨ϕ$ , i.e., true sentences that are absolute for transitive models of ZFC. Is there a well-founded set of theories $\lbrace Z F C （T_ 0）, T_ 1 , T_ 2 , … \rbrace$ such that Each being an […]chinesemathnoob
- Expected proof of $\vdash_{L_1}B\lor(C\lor D)\to((C\lor(B\lor D))\lor B)$, Introduction to Mathematical Logic 4th ed by Mendelson, exercise 1.54 (h)? July 1, 2026What is the expected derivation of $$ \vdash_{L_1}B\lor(C\lor D)\to((C\lor(B\lor D))\lor B) $$ in Introduction to Mathematical Logic 4th ed by Mendelson, exercise 1.54 (h)? In the theory of $L_1$, we define $$A\to B:=\lnot A\lor B.$$ In my approach, I investigated my previous theorems to see if I could construct them in such a way as […]Jasper
- If a transitive proper class is a model of ZFC, is it also a model when constructed "inside" another model? June 18, 2026Let $C = \{x : \phi(x)\}$ be a transitive class model of ZFC, and $M$ be a (set) transitive model of ZFC, where the $\in$ relation in both $C$ and $M$ are the "real" $\in$ relation. Is $C$ constructed in $M$ also a transitive model of ZFC? That is : $\{m \in M : (M, […]David Lui
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