Comment by Derek Elkins left SE on Why does Archive of Formal Proofs use...
... foundations in 1922 and have learned nothing of value in the last century and the advent of digital computers in no way changed things. Acceptance is not particularly relevant. The meta-theoretical...
View ArticleComment by Derek Elkins left SE on Is this proof of existence and uniqueness...
"consider the $a$ in" should have been "consider the smallest $a$ in" two comments back.
View ArticleComment by Derek Elkins left SE on Is there any standardized "practical...
Except for the use of higher-order abstract syntax (HOAS), you can take the Twelf description as the constructors of the relevant types, i.e. set, prop, and pf : prop -> Type. The HOAS can be...
View ArticleComment by Derek Elkins left SE on Is just one variable count as minimum...
Plagiarism is when you present someone else's work as your own. If you give proper credit, then you won't be plagiarizing. There may be some other reason not to, but plagiarism is not it.
View ArticleComment by Derek Elkins left SE on Relation between classical implication and...
There's no avoiding this. They're defining their logic via the given semantics. Even if this logic could be understood via a translation into intuitionistic logic (which seems very plausible), you...
View ArticleComment by Derek Elkins left SE on What is the difference between Formal...
@WorldSEnder github.com/sorear/metamath-turing-machines gives an indication of the Kolmogorov complexity of a ZF proof checker is surprisingly low. More practically, twelf.org/wiki/Zermelo_Frankel is...
View ArticleComment by Derek Elkins left SE on What is the category of compact Hausdorff...
The category of abelian groups is a full subcategory of the category of groups, but that doesn't make it "trivial" or uninteresting. The category of abelian groups also has very different properties...
View ArticleComment by Derek Elkins left SE on Is equality in PA just an equivalence...
I realize now that I totally misinterpreted the question somehow, though I do implicitly answer it. The question this answer is addressing is something more like "Does it make a difference if we...
View ArticleComment by Derek Elkins left SE on Are all sets 'regions'?
You should formalize the predicate "$X$ contains points from $Y$". When you have a candidate formalization, you should verify that it behaves correctly in cases where you know the answer. Then you can...
View ArticleComment by Derek Elkins left SE on Two-sorted first-order set theory
I always find these disjointness constraints unnatural. The only reason I can see for why they'd be necessary is that we don't also split equality into multiple per sort predicates. If we do, then...
View ArticleComment by Derek Elkins left SE on Could quantum computing help resolve some...
To be clear, as spaceisdarkgreen states, even if quantum computers could solve NP-complete problems (precisely, if BQP contained NP), this would not resolve P = NP. That said, there would probably be a...
View ArticleComment by Derek Elkins left SE on Composition of a monad with itself.
@AsadSaeeduddin $G(A)=A^M$ can be made into a comonad whenever $M$ is a monoid. Since $G(G(A))=(A^M)^M\cong A^{M\times M}$ and $M\times M$ is a monoid when $M$ is, the composition can always be made...
View ArticleAnswer by Derek Elkins left SE for Sheaves In the Category...
This is not an answer but a collection of likely relevant facts, references, and concerns.Categories of sheaves are locally presentable. This means that every object of such a category is a...
View ArticleAnswer by Derek Elkins left SE for Clarification on Natural Transformations...
It seems that you do more or less understand what's going on. A natural transformation $\alpha: F\to G$ where $F,G:\mathcal C\to\mathcal D$ is a family of arrows of $\mathcal D$ indexed by objects of...
View ArticleAnswer by Derek Elkins left SE for What is the appropriate title of a false...
The simplest thing may be to just say what you mean. That is, if your theorem is that some statement is false, then just make that the theorem you're claiming. The only reason to display the false...
View ArticleAnswer by Derek Elkins left SE for How to determine if given function is...
The section you link to tells you. All the properties that identify the five clones of Post's lattice are mechanically checkable. You can simply provide all possible inputs to your operator (i.e. build...
View ArticleAnswer by Derek Elkins left SE for Being vacuously true is a definition?
This is arguably a definition of what "vacuously true" means, namely that it is an implication that holds because its antecedent doesn't. Of course, that $p\to q$ actually does hold when $p$ doesn't is...
View ArticleAnswer by Derek Elkins left SE for If $F$ is representable, then $G$ is...
Write $K:\mathcal D\to\mathcal C$, for the "inverse" of $H$, i.e. $HK\cong Id_\mathcal D$ and $KH\cong Id_\mathcal C$.Now we can note, unsurprisingly, that your questions are symmetric. If one holds,...
View ArticleAnswer by Derek Elkins left SE for The empty set is a subset of every set
The "domain" is the class of all sets if you are working in the first-order theory of ZFC. Indeed, notation like $\forall x\in \mathbb R.P(x)$ in a set-theoretic context, is usually defined as...
View ArticleAnswer by Derek Elkins left SE for Confusion about definition of a Limit by...
Intuitively, prelimD is the limit, because it satisfy the universal property of limit. But according the existence condition of limit, it is not, because there is a morphism m'' : c'' -> prelimD...
View ArticleAnswer by Derek Elkins left SE for Bijection between $\textbf{Cat}(A \times...
Your proof is fine enough as far as it goes, namely showing bijection but not yet naturality. I don't know if it is intentional or not, but you don't actually show surjectivity, just state what you...
View ArticleAnswer by Derek Elkins left SE for What is the reason we usually don't use...
tl;dr There is little reason to make paper-and-pencil formal proofs other than learning about formal logic. Machine-checked proofs take less effort to produce and are more valuable than...
View ArticleAnswer by Derek Elkins left SE for Does a false statement always have a...
As the comments have indicated, this isn't a very clear question. The problem lies in what "false" means or what "example" in "counterexample" means or, possibly, what "statement" means? Or I should...
View ArticleAnswer by Derek Elkins left SE for Why Induction Axiom is needed if...
If you are working in a framework that allows you to define inductive types/inductively defined sets, like the Calculus of Inductive Constructions, then you wouldn't need to add induction on naturals...
View ArticleAnswer by Derek Elkins left SE for Substructural Logic: Understanding the...
You are using confusing notation. You are using $\longrightarrow$ for two different things and then, arguably, conflating it with $\to$ which is a third thing. A more common notation, using weakening...
View ArticleAnswer by Derek Elkins left SE for A clarification about...
Using the name $U$ is not the best here. Often $\iota$ or something similar will be used for inclusions. This need not be trivial. Nevertheless, for a typical definition of $\mathbf{Ab}$ and...
View ArticleAnswer by Derek Elkins left SE for Formulation of the preservation theorem...
I doubt there's a conflict here, though it's hard to tell without more context. The "formal" definition does state what you say it does, if a term of type $T$ reduces, the reduct has the same type $T$....
View ArticleAnswer by Derek Elkins left SE for Is there a conjecture suggesting if some...
It's not clear what you mean by "numerically testable", but presumably it would contain at least the $\Pi_1$ statements which are formulas (of PA) that are logically equivalent to formulas of the form...
View ArticleAnswer by Derek Elkins left SE for What's the difference between a linear sum...
I have a blog post I've mostly written that I need to finish on this and other "formal" constructions. This is a topic I find many people only have a slippery grasp on and is somewhat hard to explain,...
View ArticleAnswer by Derek Elkins left SE for Given this definition of the vector form...
To start, none of the colored formulas would represent the set $\{\vec x\mid \exists t\in\mathbb R.t\vec d + \vec p = \vec x\}$ in a general context. The closest, if anything, would be the first red...
View ArticleAnswer by Derek Elkins left SE for A category admitting pullbacks and...
The following is more or less the same as the other answers but is packaged in a different manner that I, personally, find more convenient/understandable.Given any category $\mathcal E$, we have the...
View ArticleAnswer by Derek Elkins left SE for Finding inhabitants in Lambda P
There's a rule-by-rule correspondence between rules of natural deduction for first-order logic and the typing rules of $\lambda P$.1 Here's a proof of the latter formula (written as...
View ArticleAnswer by Derek Elkins left SE for Is this the correct usage of the...
Your use of the commutative property of $\lor$ is fine enough, though you are also implicitly using a substitutability property as well. Commutativity says $\varphi\lor\psi\equiv\psi\lor\varphi$ but...
View ArticleAnswer by Derek Elkins left SE for Interpreting Category axioms in Set Theory
There are alternatives such as type theory. You could also simply consider the formal first-order theory of categories itself and its extensions without considering (set-theoretic) models. If you wanna...
View ArticleAnswer by Derek Elkins left SE for Defining Material Conditional
A definition doesn't have to be convincing or appropriate; you can define things however you want. Of course, it is better to have a convincing argument on why a definition is appropriate and...
View ArticleAnswer by Derek Elkins left SE for Why $\{0\}$ is linearly dependent?
Quoting the first line of the Wikipedia page on linear independence:[...A] set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the...
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