Tag: Open Lab 8
Logic on Math StackExchange
- Equivalence of axiomatic and semantic definition of Presburger arithmetic January 21, 2026From Wikipedia: Presburger arithmetic can be viewed as a first-order theory with equality containing precisely all consequences of the above axioms. Alternatively, it can be defined as the set of those sentences that are true in the intended interpretation: the structure of non-negative integers with constants 0, 1, and the addition of non-negative integers. How […]user1868607
- Can mathematics be done, in some form, in complete and decidable foundations? Is foundational incompleteness a consequence of our epistemic standards? [closed] January 21, 2026[An apology: I am deeply sorry for the confusing title, it is rather difficult to formulate my question on such limited character count. Also, I must advise I am far from decently lectured about these topics (I am a fresh undergraduate) and this is a very exploratory question. I also apologize in advance for both […]rutabulum
- To what extent can non‑decidability be viewed as the result of an inadequately specified or insufficiently precise space? [closed] January 21, 2026Consider the following question: Is 3.5001 closer to 3 or to 4 ? Toto is in “Mat Sup” (Maternelle SupĂ©rieure) and only keeps two digits after the decimal point. In his space (“Toto’s space”), 3.5001 is represented as 3.50. The distance to 3 is 0.50, and the distance to 4 is also 0.50. He cannot […]Saint-Martin
- Syntactic $\Sigma_1$ formuals January 21, 2026I have not been able to find a great source for this. In general, I understand that $\Sigma_1$ formulas are defined as formulas of the form $\exists z_1 \dots \exists z_k \theta(z_1, \dots, z_k, \bar{x})$ where $\theta$ is $\Delta_0$ (contains only bounded quantifiers). Speaking of only syntax, would a $\Delta_0$ form be considered $\Sigma_1$? Of […]cspurposesonly
- Does the modal logic $\sf GLS$ have a finite axiomatization? January 20, 2026The modal logic $\sf GLS$ is quasi-normal in that it validates the K axiom $\Box (A \to B) \to \Box A \to \Box B$ but does not validate the Necessitation Rule From $\vdash A$, infer $\vdash \Box A$. In particular, $\sf GLS$ is axiomatized by taking the validities of $\sf GL$$=K+(\Box (\Box A \to A) […]PW_246
- The meaning of "provability" in wikipedia's definition of completeness of a theory January 20, 2026I am looking at this Wikipedia page about complete theories. What does provability condition mean? Wouldn't this definition then be, in parts, dependent on what we consider as a proof system?Clemens Bartholdy
- Understanding Herbrand expansions: How do predicate logic formulas turn to propositional logic ones? January 20, 2026When we do Herbrand Expansion, we say that over the Herbrand universe our class of formulas are equivalent to the Herbrand expansion of them over this universe. Firstly, my question is, How many Herbrand expansions are there for a given set of fomrulas? I think there should be only one, but I haven't found anywhere […]Clemens Bartholdy
- Proper classification of type theories January 19, 2026I read that some authors classify type theories by: simple type thoery, polymorphic type theory and dependent type theory. They differ in the dependency of types. But there is also polymorphic dependent type theory and full higher order type theory. Personally I understand them as following: we have a universe $\mathrm{Type}$ and a higher universe […]Westlifer
- What happens if a proof by contradiction turns out all correct? January 19, 2026Suppose we have a statement X. It is a hypothetical statement after doing an experiment. I don't know whether it is to prove or disprove. After failing to prove it via a direct proof, I attempt to disprove it via a proof by contradiction. However, it turns out that there is no contradiction. Is this […]user516076
- Is it possible that the statement "$\pi$ is normal in base $10$" is independent of ZFC? January 19, 2026What puzzles me is the following intuition. The decimal expansion of $\pi$ is a well-defined sequence of natural numbers: for each $n\in\mathbb N$, the $n$-th digit of $\pi$ is a definite value, and in fact is computable. Normality, however, is a statement about the asymptotic frequencies of finite digit blocks in this fixed sequence. If […]Elvis
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