Written work: NOTE: The following suggested problems are for practice only, and will NOT be collected.
Section 11.4 p194: 2, 3, 5, 6, 7
Handout: Theorem NT 6.2, 6.3
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Project Reflection – Due before the final exam, Thursday 12/20.
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Logic on Math StackExchange
- Inaccessible cardinals and consistency of ZFC May 18, 2024I don't see how to state and prove rigorously the following result. Which of the following statement makes sense and is true ? For all $\phi$ in $ZFC$, we have : $ZFC+\exists \kappa'(\kappa' inaccessible)\vdash \forall\kappa(\kappa inaccessible\rightarrow \phi^{V_{\kappa}})$ For all $\phi$ in ZFC and for all inaccessible cardinal $\kappa$, we have : $ZFC+\exists \kappa'(\kappa' inaccessible)\vdash \phi^{V_{\kappa}}$ […]Mamoun Aich
- Can we build multiple models of PA within one the same model of ZFC? May 18, 2024Just to check my understanding I have the following (possibly dumb) question: can we build multiple models of PA (one standard and multiple non-standard) within one same model of ZFC?user341
- Clarification on model satisfaction definition and satisfaction over classes May 18, 2024By the completeness theorem of logic, we have "A theorem $T$ is consistent if and only if it has a model $M$" - whereby $M$ is a model of $T$ if and only if all the non-logical axioms of $T$ are valid in $M$. Now, suppose that $M$ is a model of $T$, and $\phi$ […]Link L
- Graphs of recursive functions May 18, 2024Recently I've been studying the relationship between recursiveness of a function and recursiveness of its graph (i.e. recursiveness of the characteristic function of the graph). First, we have Theorem 1: If $f:\mathbb{N}\to\mathbb{N}$ is primitive recursive, then its graph is primitive recursive. The converse is false, as witnessed by the Ackermann function for example. In an […]blargoner
- Why can't we derive the Left Contraction Rule in predicate logic? May 17, 2024Suppose we only have all the standard left and right logical inference rules ($∧L_{1}$, $∧L_{2}$, $∨L$, $→L$, $¬L$, $∨R_{1}$, $∨R_{2}$, $∧R$, $→R$, $¬R$) and on top of that 4 quantifer rules (see the Wikipedia page: https://en.wikipedia.org/wiki/Sequent_calculus) My professor said that in predicate logic we cannot prove the Left Contraction Rule unlike in Propositional logic (without […]Alessandra12342
- Models of ZFC and inaccessible cardinals May 17, 2024For a homework I have to show that working in $ZFC$, we have $(ZC)^{V_{\lambda}}$ for all limit ordinal $\lambda>\omega$. Now in my lecture notes the relativization is defined for a class $C$ characterized by a formula $\psi_C(x)$ (i.e. $x\in C$ if and only if $\psi_C(x)$ holds) as follows : $(\forall x\phi)^C:=\forall x(\psi_C(x)\rightarrow \phi^{C})$ and $(\exists […]Mamoun Aich
- Languages with quantification over *formulas*? May 17, 2024The 4th page of this paper mentions: A first-order language that contains “substitutional quantifiers” (with formulas as substituends) in addition to ordinary quantifiers turns out to be precisely the sort of “infinitary” language that we need So, it seems to be describing a sort of extension of first-order logic where you can quantify over formulas […]NikS
- Clarification on "absolute" property May 17, 2024I read that if some property $P(x)$ defined by a formula $\phi$ is absolute for some class $M$, then $\phi(P(x)) \leftrightarrow \phi^M(P(x))$, where $\phi(P(x))$ is interpreted in $V$, and $\phi^M(P(x))$ is relativized to $M$, which is a subclass of $V$. In most texts in set theory (i.e. proof that $R$ (axiom of regularity) is consistent […]Link L
- Godel's Second Incompleteness Theorem on $L$ May 17, 2024Am an absolute beginner here. But I've read that if a sentence $A$ is a theorem of a Theory $T$, then $T[\neg A]$ is inconsistent (where $T[\neg A]$ means that $\neg A$ is added as a logical axiom to $T$. Now, if say some axiom $A$ were shown to be not provable in $ZF$ then […]Link L
- Confusion on Section 1.2 of Rosen's Discrete Math Textbook May 16, 2024So I was able to deduce based on the rule that p implies q is the same as q unless (not p) that this is same as: (not s) -> (r -> (not q)) I could use the logical equivalence (A -> B) = (A or (not B)) to get Rosen's result, but up to […]Bob Marley
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