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Results tagged with lo.logic
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user 2926
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
20
votes
What is neutral constructive mathematics
You'll probably have better luck with the phrase "intuitionistic higher-order logic" (IHOL). A good place to start is the book by Lambek and Scott, Introduction to Higher Order Categorical Logic. But …
- 53.3k
8
votes
Categories of recursive functions
Regarding question 1.: it's hard to find the statement spelled out in black and white in the literature. Apparently Gavin Wraith proved it in unpublished notes titled "Notes on arithmetic universes an …
- 53.3k
3
votes
Does the exact pair phenomenon for partial orders occur in your area of mathematics?
In trying to understand better a certain weak choice axiom, I once concocted an example in sheaf theory which involves an exact pair in the sense of this question, without knowing it was a "thing" els …
- 53.3k
15
votes
Accepted
Logical complexity of algebraically closed fields
From Dirk van Dalen's Logic and Structure: the theory of algebraically closed fields is not finitely axiomatizable (see page 109 and preceding).
- 53.3k
2
votes
In which sense "closure" is a closure?
Actually, there is a topological meaning to closure of logical formulas, if one represents formulas by string diagrams. A description of such a string diagram calculus, interpreting Peirce's existenti …
- 53.3k
2
votes
Injecting premises into two implicational premises connected by a tensor (multiplicative con...
Well, this is very easy, but because linear logic might be considered a little too specialized for Mathematics StackExchange, I'll answer.
Since the natural semantics of MLL (multiplicative linear l …
- 53.3k
18
votes
Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?
To my aesthetic sensibilities, the Calkin-Wilf tree response is pretty close to optimal, but I'll add some additional glosses. (I only noticed later that Vladimir Dotsenko wrote something similar befo …
- 53.3k
10
votes
Accepted
Primitive recursive arithmetic via universal algebra
According to unpublished notes by Gavin Wraith ("Notes on arithmetic universes and Gödel incompleteness theorems" (1985)), PRA can be described as an equational theory or as a Lawvere theory, and is a …
- 53.3k
4
votes
Brouwer vs. Cantor
Troelstra is a well-known exponent of intuitionism. Here are two online articles that contain philosophical and historical material that may be useful to you:
Remarks on Intuitionism and the Philos …
1
vote
Self-similarity for simple algebraic structures
I haven't read your description (and don't know the area well enough) to make an evaluation, but as far as algebraic treatments go... have you seen Tom Leinster's Theory of Self-Similarity? This is ba …
3
votes
"Introduction to mathematical logic" book from a formalist perspective
I don't know of logic textbooks which adopt an explicitly formalist viewpoint (which is not to say I think none exist), but for what it's worth, I'd say that books on categorical logic tend not to ado …
8
votes
Accepted
Adjoining an arrow to a CCC
They mean this: given a cartesian closed $\mathcal{A}$ and objects $A, B$ of $\mathcal{A}$, the inclusion $i: \mathcal{A} \to \mathcal{A}[x]$ is universal with respect to strict cartesian closed funct …
- 53.3k
40
votes
Accepted
Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?
It seems to me that Schanuel's conjecture (which is a kind of article of faith in transcendental number theory, but of course very far from proven itself) ought to imply that $\pi$ is not definable in …
- 53.3k
7
votes
Accepted
Further relation between monads and theories
As noted by Zhen Lin, monads (on $\mathbf{Set}$) should be thought of as morally equivalent to algebraic theories of fairly general type. By an algebraic theory I mean a single-sorted theory specified …
- 53.3k
17
votes
Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ?
If you accept that toposes are models of constructive set theory, then another way to answer the question is to give a (non-Boolean) topos where the CBS theorem fails; that would show that this theore …
- 53.3k