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Results tagged with lo.logic
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user 3154
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.
13
votes
What was Gödel's real achievement?
Hilbert, in his 1922 "New Grounding of Mathematics" and subsequent papers, developed an approach to axiomatisation of proof that Goedel's result can be seen to have continued, whilst at the same time …
- 2,283
12
votes
3
answers
7k
views
Is functional programming a branch of mathematics?
In Theory mainly concerned with lambda-calculus?, F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics:
That would never stick unless there's another good reason …
10
votes
Accepted
What does the disjunction elimination rule say?
The first rule is not the regular disjunction elimination rule, but is known as disjunctive syllogism, and is essentially the modus tollendo ponens rule of term logic. The two rules are mutually admi …
- 2,283
10
votes
2
answers
2k
views
Scott on the consistency of the lambda calculus
I have twice heard it attributed to Dana Scott that he said something to the effect that the consistency of the lambda-calculus was an accident.
Does anyone have a reasonable-sounding source for this …
- 2,283
9
votes
Accepted
Is there a formal notion of what we do when we 'Let X be ...'?
Kieffer, Avigad, & Frideman, 2008 A language for mathematical knowledge management, which I mentioned in the Proof formalization thread, discusses DZFC, an extension of ZFC with definitions of terms a …
- 2,283
9
votes
Most general formulation of Gödel's incompleteness theorems
Theories can be be represented recursion-theoretically by an encoding of the language as natural numbers (most simply, a bijective encoding, which I assume), and a Turing machine that accepts all and …
- 2,283
9
votes
Alternative axiom to induction
The literature on this question is large!
First recommendation: take a look at George Boolos (1984/1998)'s "The Justification of Mathematical Induction", in Logic, Logic, And Logic, pp. 370—375, Har …
- 2,283
8
votes
Do you know any good introductory resource on sequent calculus?
Gentzen, 1934, 'Investigations into Logical Deduction' — This is very readable, and introduces so many ideas that later synthetic works invariably miss some. If you're serious, this, and some other p …
- 2,283
8
votes
Accepted
Reducing ACA₀ proof to First Order PA
Chapter nine of Simpson (1999) Subsystems of Second-Order Arithmetic proves (a) by showing how to construct a second-order model for ACA0 from a first-order model of PA.
(b) The "second-order" we are …
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7
votes
How much of the current logic is about syntax?
We don't know how to abstract away from syntax in proof theory. If we say there are three main branches in proof theory:
Axiomatics seem to be necessarily syntactic: formulae are what it is about;
…
- 2,283
7
votes
Accepted
Weakest subsystems of second order arithmetic for mathematical logic
In fact, the incompleteness and completeness theorems can be proven in subsystems of second-order arithmetic weaker than RCA-0: incompleteness can be proven in EFA (first-order elementary arithmetic), …
- 2,283
7
votes
Accepted
Is there a relationship between model theory and category theory?
Between model theory and category theory broadly conceived: not anything really compelling, because a category, on its own, does not stand as an interpretation for anything.
Between model theory and …
- 2,283
6
votes
1
answer
691
views
What notions of universe does predicative type theory admit?
Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in R …
- 2,283
6
votes
Theory mainly concerned with $\lambda$-calculus?
I don't know of one that seems sufficiently general. The theory's at an intersection:
It (in its untyped guise) is one of the four most important Turing-complete computation systems;
It is algebra …
- 2,283
5
votes
What does it mean to 'discharge assumptions or premises'?
In the spirit of Kenny's observation, note also that we can formulate classical logic using a Peircian inference rule (equivalent to the usual theory in the presence of ex falso quodlibet) which clear …
- 2,283