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Results tagged with lo.logic
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user 5442
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
Difference between constructive Dedekind and Cauchy reals in computation
Once we look at computational content, rather than constructive content, things are easier to answer. When we work on the level of individual reals, all the representations are equivalent - a real is …
- 4,713
11
votes
On Applications of Forcing in Domain Theory
It may not be well known outside domain theory that there are several different groups who work in domain theory for different reasons. People interested in domains as modeling partial information in …
4
votes
Cauchy real numbers with and without modulus
Let $(a_n)$ be a Specker sequence - a computable, bounded, increasing sequence of rationals so that the limit is not a computable real.
First, assume for simplicity that we work in any system of s …
- 9,683
4
votes
Is any Cauchy sequence for completion of rational semicomputable?
I wonder if there is constructive theories of the real which is based on Cauchy sequence, specifically, how to overcom the non-computable problem of some (in fact,uncountably many) Cauchy sequence …
- 9,683
10
votes
Accepted
Matiyasevich's theorem and Gödel's first incompleteness theorem
If you are writing a research paper on this, the usual practice in mathematics is that you do not need a reference for such an obvious result which is known to everyone in the field already. For thin …
- 9,683
2
votes
Interpreting peano arithmetic without parameters
Here is one possible relationship. One common reason to interpret a theory $T_1$ into a second theory $T_2$ is to establish a relative consistency result: if $T_2$ interprets $T_1$ in a syntactic way, …
- 9,683
4
votes
Henkin semantics for second-order logic
There are two ways for a Henkin model of second-order arithmetic to be nonstandard. 1: it could have a standard first-order part of $\omega$, but less than the full powerset of $\omega$ as its second …
- 9,683
3
votes
Accepted
A question on the provability predicate of Q
Robinson arithmetic is $\Sigma^0_1$ complete - every true $\Sigma^0_1$ sentence is provable in Q. If you are asking whether $Q \vdash A$ implies $Q \vdash \text{Pr}(A)$, that completeness results says …
- 9,683
13
votes
Accepted
Looking for a source for Intended Interpretation
Here are quotes from three well-known sources.
Shoenfield, Mathematical Logic (1967), page 23:
We construct a model of $N$ by taking the universe to be the set of natural numbers and assigning t …
- 9,683
3
votes
Compactness for countable models?
The statement for the completeness theorem is due to Harvey Friedman, 1976, "Systems of second order arithmetic with restricted induction II", p. 558 of: Meeting of the Association for Symbolic Logic, …
- 9,683
12
votes
What are some important but still unsolved problems in mathematical logic?
A recent list of open problems in Reverse Mathematics is:
Antonio Montalbán, Open questions in reverse mathematics, Bulletin of Symbolic Logic 17:3 (2011), 431-454. Preprint
A slightly older lis …
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10
votes
Accepted
Did Bishop, Heyting or Brouwer take partial functions seriously?
I don't believe that Bishop explicitly assumed all functions are continuous. I think that "no discontinuous function can be proved to be total in Bishop's constructive mathematics" is actually a very …
- 9,683
6
votes
What can be achieved by liberalizing induction for $RCA_0$?
There is at least one way in which adding induction for arithmetical formulas does "add sets" in the context of Reverse Mathematics: induction is sometimes required to show that certain "finite" sets …
- 9,683
11
votes
Accepted
Does Peano's existence theorem admits a constructive proof?
I think that the heart of the question is "Can one give a rigorous meaning to 'there is no constructive proof to the Peano's theorem'?" The answer to this is yes, but the answer is not as simple as …
- 9,683
12
votes
Forcing is intuitionistic
It seems to me that the "more philosophical" reason why forcing is intuitionistic is that forcing and intuitionistic logic have similar interpretations of the logical connectives and quantifiers. This …
- 9,683